# Finite volume approximations for some equations arising in mathematical physics by

Finite volume approximations for some equations arising in mathematical physics by Aderogba Adebayo Abiodun Submitted in partial fulfillment of the requirements for the degree Philosophiae Doctor In the Faculty of Natural and Agricultural Sciences University of Pretoria Pretoria May, 2014 © University of Pretoria i Declaration I, Adebayo Abiodun, Aderogba declare that the thesis, which I hereby submit for the degree Philosophiae Doctor at the University of Pretoria, is my own work and has not previously been submitted by me for a degree at this or any other tertiary institution. SIGNATURE: DATE: © University of Pretoria ii To τ &Ω © University of Pretoria Acknowledgement You know how it feels when you pick up a book, flip to the acknowledgement page, and discover, once again, the author missed out your name. This time it is different, therefore, I appreciate everybody who contributed in one way or the other to the success of this work. I will not forget to mention the following: The Almighty God for life and all provisions. Drs. Chapwanya M and Djoko JK for guidance. University of Pretoria for financial support and conducive research environment. The Dean, Faculty of Natural and Agricultural Science. Department of Mathematics and Applied Mathematics, University of Pretoria staff, friends and colleagues. Obafemi Awolowo University for support and grant of my study leave. Department of Mathematics, Obafemi Awolowo University, staff, friends and colleagues. All my colleagues for advise and counsels. DCLM, Pretoria. CGMC, Ibadan, Nigeria. My Parents and the entire Aderogba’s and Adebayo’s family. Praise, Goodness, Halleluyah, for understanding. My sweetheart for the love. © University of Pretoria Contents Declaration i Acknowledgement iii 1 Introduction 1.1 Higher order equations . . . . 1.2 Singularly perturbed equation 1.3 Numerical approach . . . . . . 1.4 Thesis outline . . . . . . . . . . . . . . . . . 1 2 5 7 12 2 Fractional splitting method for 2.1 Introduction . . . . . . . . . . 2.2 Numerical approach . . . . . . 2.3 Numerical experiments . . . . 2.4 Conclusion . . . . . . . . . . . higher order differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 14 15 23 41 . . . . 43 43 45 55 68 . . . . 71 71 72 75 82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Nonclassical methods for 3.1 Introduction . . . . . . 3.2 Numerical approach . . 3.3 Numerical experiments 3.4 Conclusion . . . . . . . singularly perturbed equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Nonclassical methods for 4.1 Introduction . . . . . . 4.2 Numerical approach . . 4.3 Numerical experiments 4.4 Conclusion . . . . . . . higher order equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion and future perspective 84 Bibliography 87 iv © University of Pretoria List of Tables 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 4.1 4.2 The values of cri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence rate of the schemes for the nonlinear equation (2.48) . . . . . . Error due to each scheme for the numerical approximation of the linear equation. Convergence rate of the fractional step for the K-S equation with initial data that corresponds to the exact solution. . . . . . . . . . . . . . . . . . . . . . Convergence rate of the fractional step scheme for the K-S equation with Gaussian initial data. The exact solution is taken as the solution of the most refined grid (m = 640) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence rate of the fractional step for the Cahn-Hilliard equation . . . . Convergence rate of the fractional step for the cCH equation with the NSTG scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence rate of the fractional step for the cCH equation with the WENO scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence rate of the fractional step for the cCH equation with the FD scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error values of the nonstandard scheme and the classical scheme for different values of ǫ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error values of the nonstandard scheme for different values of ǫ . . . . . . . . Error between the solution at different grid points as compared with a reference solution at 256 grid points . . . . . . . . . . . . . . . . . . . . . . . . . Error computations for Example 4.2 . . . . . . . . . . . . . . . . . . . . . . . Error computations for Example 4.41 . . . . . . . . . . . . . . . . . . . . . . v © University of Pretoria 20 25 26 28 29 36 37 38 38 59 59 69 77 78 List of Figures 1.1 Finite volume domain discretization . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 2.2 Mean energy density profiles for problem (2.30). . . . . . . . . . . . . . . . . Mean energy density profile for the diffusion term for T = 20, γ = 1 and (a) x ∈ [0, 2π] and (b) x ∈ [−2π, 2π]. . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the exact solution with the numerical solutions. . . . . . . . . Traveling wave solution as standard compared to other schemes at T = 10. . The chaotic solution of the K-S equation with Gaussian initial conditions (Φ(x) = exp(−x2 )) up to T = 40. . . . . . . . . . . . . . . . . . . . . . . . . Mean energy profiles for the full K-S equation. . . . . . . . . . . . . . . . . . Profile of the K-S equation showing the periodicity and zero average of the solution at T = 20 and L = 2π. . . . . . . . . . . . . . . . . . . . . . . . . . ku k2 (a) loglog plot of the kuk2 . (b) loglog plot of Lx 2 . . . . . . . . . . . . . . . Solution of the Cahn-Hilliard (4.24) at time T = 10. Profiles in (a) are given at equally spaced time intervals. . . . . . . . . . . . . . . . . . . . . . . . . . Conservation of mass and dissipation of energy. . . . . . . . . . . . . . . . . Numerical solution of the convective Cahn-Hilliard equation. . . . . . . . . . Numerical solution of the convective Cahn-Hilliard equation. . . . . . . . . . Coarsening process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coarsening process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservation of mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rough behavior of the convective Cahn-Hilliard equation. . . . . . . . . . . . 24 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 3.1 3.2 3.3 Comparison of the solution of the equation (3.50) as computed by the classical scheme and the new scheme on 10 grids and compared with the exact solution on 200 grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the solution of the equation (3.50) as computed by the classical scheme and the new scheme on 40 grids and compared with the exact solution on 200 grids when ǫ = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the solution of equation (3.50) by the schemes (3.6) and (3.47) vi © University of Pretoria 26 29 31 32 33 33 34 36 37 38 39 39 40 40 41 41 58 59 60 vii 3.4 3.5 3.6 3.7 3.8 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Comparison of the solution of equation (3.50) by the schemes (3.6) and (3.47) The slope of the lines say r = ∆(log(error)) represents the convergence rate ∆(log(h)) due to each scheme. The approximation on 32, 64, 128, 256 elements grid are compared with a reference solution on 512 element grid. . . . . . . . . . . . . Comparison of the solution of the equation (3.64) between the classical scheme and the new scheme. Solution was computed with ǫ = 2 × 10−2 on 32 element grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence of the simulations with, ǫ = 1 × 10−2 and several grid choices. . Comparison of the solution of the Schrödinger equation (3.64) on 40 grid nodes (dots) and 512 grid nodes (solid line) with ǫ = 2 × 10−2 . . . . . . . . . . . . 60 Solution profile of equation (4.17) . . . . . . . . . . . . . . . . . . . . . . . . Solution of the Kuramoto-Sivashinsky equation by the NSFV scheme. . . . . Bounds verified for the linear and the Kuramoto-Sivasinsky (K-S) equation by the NSFV scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of the Cahn-Hilliard equation verified by the nonstandard schemes with γ 2 = 0.03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solutions of the Cahn-Hilliard and the convective Cahn-Hilliard equation by the nonstandard schemes. (a)γ 2 = 0.02, (b) γ = 1. . . . . . . . . . . . . . . . Conservation of Mass of the cCH equation . . . . . . . . . . . . . . . . . . . Coarsening properties of the convective Cahn-Hilliard equation verified by the nonstandard schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coarsening process T = 640 . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 79 © University of Pretoria 66 66 69 69 79 80 81 81 82 82 Title Name Supervisor Co-Supervisor Department Degree Finite volume approximations for some equations arising in mathematical physics Aderogba Adebayo Abiodun Dr. Jules Djoko Kamdem Dr. Chapwanya Michael Mathematics & Applied Mathematics Philosophiae Doctor Abstract In this thesis we design and implement finite volume schemes to approximate the solution of 1-dimensional (partial) differential equations. Most of these partial differential equations (PDEs) are made up of not only mathematically interesting but also physically relevant terms such as the hyperbolic convective and parabolic diffusive operators. The coupling of higher order, linear and nonlinear operators and the presence of a small parameter multiplying the highest derivative imposes some stiffness into the equations thereby making both their numerical and mathematical analysis interesting but very challenging. For example, singularly perturbed second order ordinary differential equations (ODEs) possess boundary layers and/or oscillatory solutions which make their numerical approximation by difference-type schemes expensive. We design two uniformly convergent finite volume schemes for a singularly perturbed ODE: the Schrödinger equation. The first scheme is based on the nonstandard finite difference (NSFD) method which is known to preserve the qualitative properties of the physical model and the second is based on boundary layer analysis. We employ fractional splitting method for the analysis of higher order equations in order to isolate the linear and nonlinear terms thereby resolving the stiffness in the equation. The nonlinear hyperbolic term is solved by shock capturing schemes while the fourth order linear parabolic term is handled by A-stable schemes. We also utilize the idea of the NSFD method to design a scheme for the hyperbolic, nonlinear parabolic and the linear fourth order PDEs. Each of the terms is solved sequentially within every time step and their solutions are pieced together in such a way as to preserve the properties of the original equations. We observe uniform convergence with better approximation at relatively low computation cost when the Schrödinger equation was solved by the proposed schemes. We also examined the computational strength of our schemes on two fourth order equations: the KuramotoSivashinsky equation and Cahn-Hilliard equation. We studied the effect of combining different schemes for each of the split sub-problems on the convergence of the fractional splitting scheme. We are able to reproduce all the expected properties of the selected equations. We observed a better convergence when the nonstandard finite volume method was applied to these PDEs. Throughout this work, numerical simulations are provided to validate the computational power of the proposed schemes. © University of Pretoria Chapter 1 Introduction In this thesis, numerical investigations of 1-dimensional singularly perturbed (partial) differential equations are presented. The finite volume method is employed throughout. Each chapter represents scientific contribution in form of published, accepted, submitted or work in progress. The real world is filled with different challenges ranging from social, physical and engineering sciences. Reliable mathematical models offer the cheapest way of investigating these challenges. These mathematical models range from algebraic, differential, integral, algebraicdifferential, algebraic-integral or differential-integro equations. Designing well-possed mathematical models is essential so as to provide realistic solutions under reasonable and dynamically consistent assumptions. Surprisingly, most of these equations, because of their complexity, as we will see later, are not amenable to analytical solutions. For this reason, there is need to develop approximate numerical solutions for these models. The size of system of equations, the number of unknowns, the nonlinearity in the equations and some other complexities in such equations coupled with limited space, time and human capacity constrained individuals to make assumptions which at the end of the day reduces the strength and quality of the results, and consequently their applicability to real life models. Thanks to the advent of computer system, now we only need to develop numerical algorithms which are able to handle such equations with more realistic assumptions. Computational/numerical mathematics has gained much ground over a century now. It is a strong tool for scientific computing and very useful in solving problems from physical, economical, engineering and biological models. There are many different numerical methods for solving mathematical models. We have among many, the finite element method, finite difference method and finite volume method. Some of the common features of these methods are as follows. • Grid generation: This is the process by which the continuous space is represented by discrete points/grids in a numerical domain. 1 © University of Pretoria Chapter 1 Introduction 2 • Time discretization: The whole time interval is sub-divided into finite intervals called time steps. • Lastly, the continuous equation is transformed into algebraic system of equations. In the present work, we will employ the finite volume method to approximate the solutions of three important models 1. The Kuramoto-Sivashinsky equation, 2. The Cahn Hilliard equation, 3. The Schrödinger equation. This will help us to treat each of the differential equation models in their integral forms. The process of discretization will be discussed in the next sections. Appeal is made to the fractional splitting method for its stability and the nonstandard finite difference method for their qualitative stability, monotonicity and positivity preserving properties. 1.1 Higher order equations Pattern formation resulting from phase transition has been observed in alloys, glasses, polymer solutions, binary liquid mixtures, ecological contexts among many. The fourth order Cahn-Hilliard equation, ut + γuxxxx = φ(u)xx + δuux (a < x < L, 0 < t), (1.1) φ(u) = αu3 + βu2 + κu, (1.2) where has been a successful model for such transitions (see for example [18, 19, 62] and the references therein). The research into the Cahn-Hilliard equation (1.1) has been documented in the literature. The authors in [19] established the global existence or blow up in a finite time of problem (1.1)-(1.2) when δ = 0 under some certain conditions. They observed that if α > 0, there exists a unique solution for any initial data u0 ∈ H 2 and satisfying ux (0, t) = 0 = ux (L, t), but the solution must blow up in finite time for large initial data if α < 0. Along side this, they noted that if γ > L2 /π 2 and the initial data is small, there will be a unique global solution which decays to the constant M as t → ∞ no matter the sign of α. This was later verified in [18] via the Galerkin finite element method provided that Z γ (F (u0) − Fm )dx + |u0 |21 , 2 I is sufficiently small, F (u0 ) > Fm ∀x, where Fm is of the local minimal of F (s) on R, M is sufficiently close to um , where Fm = F (um) and the seminorm kD s vk is denoted by |v|s . Next, © University of Pretoria Chapter 1 Introduction 3 a semi-discrete finite element scheme for the solution of the Cahn-Hilliard equation (1.1) (1.2) was derived in [20]. They also derived optimal order error bounds in various norms. The Cahn-Hilliard equation (1.1) with boundary conditions u(0, t) = u(1, t) = uxx (0, t) = uxx (1, t) = 0 was considered in [16]. They showed the existence and uniqueness of solution based Faedo-Galerkin semi-discrete scheme and also presented a new finite element based fully discrete scheme which has a Lyapunov functional. The Lyapunov functional helped them to obtain point-wise estimate of the solution. These theoretical results were verified numerically by several authors (see [12, 13, 14, 22, 48] among others). Eyre [22] studied the solution of the Cahn-Hilliard equation (1.1) employing a splitting method in which the contractive and expansive terms of the equation are separated across the time step. They noted that the flops reported in their work do not directly apply to solving the Cahn-Hilliard equation in more than one spatial dimension. Later, their type of splitting was employed in [13] to solve the Cahn-Hilliard equation subject to free boundary condition and showed that unconditionally gradient stability is achieved in one, two and three dimensions with large time steps. Recently, Cueto-Felgueroso and Peraire [12] studied numerically the two dimensional phase separation problem governed by the CahnHilliard equation using a time adaptive procedure. They were able to provide a quantitative characterization of the different time scales present in phase separation processes. The work in [85] on the Cahn-Hilliard equation with degenerate mobility enjoyed the use of tanh, homotopy perturbation and Adomian decomposition methods. The tanh method was used to find the traveling wave solution while the main equation was solved by the homotopy perturbation and the Adomian decomposition separately. The work of Dehghan and Mirzaei [14] reproduce most of the properties of the Cahn-Hilliard equation. The authors employed a numerical method based on the boundary integral equation and dual reciprocity methods. Most recently, Lee et al. [48] introduced a gradient stable scheme to solve the N-component Cahn-Hilliard system. The method helped them to reduce the N-component system to a system of N − 1 binary Cahn-Hilliard equations which they solved with the aid of a nonlinear multi grid method. √ If δ > − 2, γ = 1, β = 0, the asymptotic solution of equation (1.1) takes the form √ of a valley and is called a kink while it takes the form of a hill (anti-kink) if δ < 2 [26]. The occurrence of a kink and anti-kink together is referred to as kink-anti-kink pair. In 1996, Emmott and Bray [21] studied the Cahn-Hilliard equation (1.1) in the presence of a driving force (i.e. δ 6= 0). They examined the effects of the driving force on the solution of the equation. They observed that it has an asymmetric effect on the solution for a kink and the direction of the field determines whether the analytic solutions derived earlier by Leung [49] are unique. They also studied the dynamics of a kink-anti-kink pair. Later, this problem was investigated by Golovin et al. [26] where they observed that the kink-anti-kink pair actually comes as the final result of coarsening on a periodic domain. √ They noted that such a solution does not exist for δ > 2 and that the far field solution becomes unstable when δ > 0.94. This was also confirmed by the work of Watson [86] and © University of Pretoria Chapter 1 Introduction 4 Watson et al. [87]. Watson [86] derived the convective Cahn-Hilliard equation (1.1) when examining the coarsening dynamics of crystal growth. He performed the direct simulation of the equation with δ = 0.1 while Watson et al. [87] performed a matched asymptotic analysis on the equation (1.1) (δ = 1, γ = 0.01) on a periodic domain. Both teams, Watson [86] and Watson et al. [87], agree on the impossibility of binary coalescence of phase boundary and the occurrence of ternary coalescence only through the kink-ternary interaction (i.e., two kinks meet an anti-kink resulting in a kink). They also concluded that when δ ≫ 1 the solution of the convective Cahn-Hilliard equation behaves like that of the KuramotoSivashinsky equation. This was shown for a two dimensional case in [26]. Recently, the dynamics of domain walls (kinks) governed by the equation (1.1) with special interest in the dynamics of kink pairs and triplets that play crucial role in the coarsening process was investigated by Podolny et al. [69]. They were able to derive an analytical formula that describes the motion of the kink pairs and the triplets. Scaling the equation (1.1) by δ and letting ū = uδ gives rise to another interesting fourth order unsteady equation [26], the Kuramoto-Sivashinsky equation, ( ut + uux + αuxx + γuxxxx = 0, ∀ (x, t) ∈ R × (0, ∞), (1.3) u(x, 0) = ψ(x), as δ → ∞ after dropping the bar. Here, α, γ > 0 are constant coefficients accounting for the long wave instability (gain) and short wave dissipation, respectively. Equation (1.3) is a well known model of one dimensional turbulence which was derived in various physical contexts including chemical-reaction waves, propagation of combustion fronts in gas, surface waves in a film of a viscous liquid flowing along an inclined plane, patterns in thermal convection, rapid solidification, and many others (see for example [41, 44, 80, 74, 75, 76]). Various approaches have been presented in the literature to find the properties of the solutions of the Kuramoto-Sivashinsky (K-S) equation, with special attention on the energy bound derived theoretically in the form lim sup ku(x, t)k2 = lim sup(LE(t))1/2 ≤ CLp , t→∞ where (1.4) t→∞ 1 E(t) = L Z L u2 dx. (1.5) 0 For example, Nicolaenko et al. [61] determined p = 5/2 with the assumption that initial data is L-periodic, antisymmetric about the origin and of zero mean. In Collet et al. [10] removed the antisymmetry requirement and observed that p = 8/5. A 1-dimension version of the equation was considered by Goodman [27] where they removed the requirement of odd solutions and arrived at the same value of p following a generalization of the Nicolaenko et al. ’s Lyapunov function argument. Giacomelli and Otto [24] obtained two integral identities in L4 − norm of u and employed these to determine a bound for equation (1.4). Through © University of Pretoria Chapter 1 Introduction 5 their analysis, they arrived at p = 3/2. A weaker bound which was proved to be necessary in the presence of a linear destabilizing term was later introduced in [9]. Recently, a Lyapunov argument was followed in [63] to obtain bounds that are independent of the system size. The numerical solutions of the K-S equation have been widely investigated in the literature. In particular, we highlight the Galerkin method [89], the Chebyshev spectral methods [43], the B-splines [46], the meshless method of lines [28], etc. The aim of these investigations have been on the accuracy [89], and/or how these solutions compare with the well documented benchmark solutions [64]. In [89], an explicit Runge Kutta method was used to avoid the restrictive stability limit of the fourth order derivative. Further advantage of the method is that the approach can easily be tweaked to obtain any required order of accuracy. Other approaches are based on simplifying the partial differential equation in order to reduce difficulties in its numerical approximation. For example, the B-spline approach by Lakestani and Dehghan [46] reduced the problem to a set of algebraic equations, while in [43] the equation was reduced to a system of ODEs that were solved by implicit-explicit BDF method. In this theses, we employ the fractional splitting method to simplify the analysis of equations (1.1) and (1.3). 1.2 Singularly perturbed equation The steady state of the Cahn-Hilliard equation (1.1) shows that it is probably a singularly perturbed equation [70]. Equations of such type are known to be either strongly oscillatory or possess boundary, interior or inversion layers. As γ → 0 (see equation (1.1)) they become very stiff and computationally expensive to be handled by classical methods. Here, we motivate the numerical study of the steady state CH equation via a well known second order differential equation. A general expression of a second order singularly perturbed boundary value problem is −ǫ2 u′′ (x) + a(x)u′ (x) + b(x)u(x) = f (x), x ∈ (a, b), u(a) = α, u(b) = β, (1.6) where 0 < ǫ ≪ 1. Analytical approximation of equation (1.6) has been the subject of many research activities (see for example [71, 88] and the references therein). Wollkind [88] compared the effectiveness of the method of matched asymptotic expansion and the multiple scale method in solving equation of type (1.6) when the coefficient of the diffusion term is positive and a(x) is either only increasing or decreasing in the entire interval. They observed that the two methods yield valid asymptotic approximation. Many algebraic calculations are involved in the multiple scale method while the method of matched asymptotic expansion needs more terms for greater accuracy. He also observed that the method of matched asymptotic expansion is very useful when studying flows past a body at high Reynold’s number © University of Pretoria Chapter 1 Introduction 6 than the method of multiple scales. In fact, later, an asymptotic estimate was derived in [71] for the Reynold’s equation. They also established existence and a uniformly valid asymptotic expansion following the matched asymptotic expansion method. Here, our focus will be on the Schrödinger equation 2 ′′ ǫ u (x) + q(x)u(x) = 0, √ (1.7) ǫu′ (a) + ıpu(a) = 2ıp, ı = −1, p > 0, ǫu′ (b) = ıpu(b), where q(x) > 0, ǫ ≪ 1. It has its relevance ranging from industrial purposes to domestic usage such as design of semiconductor appliances, resonant tunneling diodes, microwaves, in quantum and plasma physics, etc. Therefore, a lot of numerical analysis and simulations have been done on the system (1.7) based on finite difference (both classical and adaptive mesh), finite volume, finite element and Wentzel-Kramers-Brillouin (WKB) methods (see [8] and references therein). Two different iteration techniques for the numerical solution of the self-consistent Poisson-Schrödinger equation subject to Dirichlet boundary condition (with zero values at both ends) was suggested in [79]: the extrapolated-convergence-factor and the perturbation-iteration method. The advantage of the perturbation-iteration method over the extrapolated-convergence-factor method is its fast convergence even though it takes longer computing time per round. This model was employed by Inoue et al. [35] to determine the electronic states in a selectively doped double-heterojunction system. The system of equation was solved by the finite element method. Their observed electron concentration variation with gate voltage agreed with experiment. Later, a two-dimensional SchrödingerPoisson equation was solved by Laux and Stern [47] by the finite difference method on a non uniform mesh using a conjugate gradient method. They employed a Lanczos iteration in order to tridiagonalize the Schrödinger equation and self-consistency is obtained by a damped Newton iteration using an approximate Jacobian matrix. Their aim was to determine the electron states under narrow gate in metal-oxide-silicon. They observed that the states for motion parallel to the silicon-oxide-silicon interface are more closely spaced than the state for motion perpendicular to the interface. The same equation was solved by the finite difference method on a uniform grid in [77] where Snider et al. determined the conduction band, electron states and electron concentration in quantum well wires. Their simulation agreed well with experiment for shallow mesa but not for deep mesa. Next, a finite difference scheme on a non-uniform grid was proposed for the one dimensional Schrödinger-Poisson equation in [83]. Their matrix transformation preserves symmetry, hence reduces computation time. They validated their result by comparing it with exactly calculated eigen-states of GaAs/AlGaAs rectangular wells. Another application of the Schrödinger equation is found in the resonant tunneling diodes. Equation (1.7) was solved subject to open boundary condition coupled with the Poisson equation (subject to Dirichlet boundary condition) in [68]. The authors achieved self-consistence and fast convergence by a Gumel approach. Later, Abdallah and Pinaud [1] improved on this result and were able to approximate the solution © University of Pretoria Chapter 1 Introduction 7 of the Schrödinger-Poisson equation on a coarser grid. They accomplished this by discretizing the equation on a finite element space with WKB oscillating interpolation basis. This scheme was later analyzed in [59] for consistency and stability. Recently, another numerical scheme for the Schrödinger equation (1.7) subject to the open boundary condition was proposed and analysed in [8]. Their scheme based on a WKB-type transformation was able to filter out the dominant oscillations, and the reduced equation was then solved on a coarser grid with less computational cost. Here, we will employ nonclassical finite volume schemes to approximate the Schrödinger equation. 1.3 Numerical approach This section introduces the several numerical methods used in this work. Here we highlight the fractional split method, finite volume method and the NSFD method. All the schemes in this thesis were implemented via the MATLAB programming language. 1.3.1 Fractional Splitting The fractional splitting method originates from Alternating Direction Implicit method (ADI) and the Local One Dimensional (LOD) method used by Peaceman and Rachford [66] and Douglas and Rachford [15]. Yanenko [90] later built on these methods and designed the fractional step method to solve problems in mathematical physics. This method was analyzed by Crandall and Majda [11] when applied to conservation laws in 2-D. Since then, it has been a very effective tool in solving incompressible Navier-Stokes equation [23, 67], nonlinear convection diffusion equations [31, 39, 40], the Kortewg de Vries (KdV) equation [30, 32], the thin film equation [91] and the Fisher equation [45]. The method simplifies the numerical analysis and computation of complicated differential equations. The complicated unsteady partial differential equation is split into simpler equations which may either be steady or unsteady partial or ordinary differential equations. These are solved sequentially within every time step until the final time of integration is reached. The method is always exact when dealing with linear equations while the order of accuracy may be reduced when employed to solve non linear equation. It is essential to note that the stability of each of the solver of the sub problems guarantees the stability of the entire scheme. The idea of the method is best illustrated using examples. Example 1.1 We want to show the accuracy of the fractional splitting method for the solution of the equation ut + ux − 2uxxxx = 0, u(0, x) = g(x), x ∈ RR, t > 0. © University of Pretoria (1.8) Chapter 1 Introduction 8 An exact solution of (1.8) is g(x − t) exp(2t). In order to apply the method of fractional splitting to this equation, we first split it into two subproblems that can be solved independently: ut + ux = 0, u(0, x) = g(x), (1.9) wt − 2wxxxx = 0, w(0, x) = w0 (x). (1.10) The solution of (1.9) is g(x − t), while that of the equation (1.10) is w0 exp(2t). Note that the fractional step method requires the initial data of the second subproblem to be the final solution of the first subproblem within that same time step. Hence, w0 = g(x − t) and therefore, the solution of the entire equation gives g(x − t) exp(2t) which just reproduces the exact solution of the equation (1.8). We are interested in illustrating the general approach of fractional splitting method to linear equations. We pick linear equation just for the simplicity of presentation. We will consider the general equation ut = (A + B)u, (1.11) ∂ and where A and B are differential operators, for example − ∂x A and B are time independent then ∂4 ∂x4 as in (1.8). Given that utt = (A + B)ut = (A + B)2 u. For this case it is generally acceptable that [51] ujt = (A + B)j u. If the operators are time dependent then utt = (At + Bt )u + (A + B)ut , which leads to some other complications. If they are nonlinear operators, we can still appeal to Taylor’s expansion to handle the higher derivatives as long as the solution is smooth. For equation (1.11) 1 u(x, ∆t) = u(x, 0) + ∆tut (x, 0) + ∆t2 utt (x, 0) + · · · 2 1 = u(x, 0) + ∆t(A + B)u(x, 0) + ∆t2 (A + B)2 u(x, 0) + · · · 2 1 = I + ∆t(A + B)(I + ∆t(A + B)) + · · · u(x, 0) 2 ∞ X ∆tj (A + B)j u(x, 0). = j! j=0 By the aid of Taylor’s series this can be written as u(x, ∆t) = e∆t(A+B) u(x, 0). © University of Pretoria (1.12) Chapter 1 Introduction 9 The operator e∆t(A+B) is called the solution operator for the unsplit equation. For the fractional split method, following the same procedure we compute u∗ (x, ∆t) = e∆tA u(x, 0), for problem (1.9) and u∗∗ (x, ∆t) = e∆tB u∗ (x, ∆t), for (1.10). This means that the solution through the fractional split method will be u∗∗ (x, ∆t) = e∆tB e∆tA u(x, 0). Therefore, the splitting error is given by u(x, ∆t) − u∗∗ (x, ∆t) = (e∆t((A+B)) − e∆tB e∆tA )u(x, 0). (1.13) The second term in (1.13) is 1 1 e∆tB e∆tA u(x, 0) = (I + ∆tB + ∆t2 B2 + · · · )(I + ∆tA + ∆t2 A2 + · · · )u(x, 0) 2 2 ∆t2 2 (A + B2 + 2BA) · · · )u(x, 0). = (I + ∆t(A + B) + 2 (1.14) We can easily see that the term I + ∆t(A + B) agrees with the expansion in (1.12) while the term 1 2 (A + B2 + 2BA), 2 will only be the same as in (1.12) only if the operators A and B commute. When dealing with nonlinear operators or whenever the operators do not commute, the above scheme tends to reduce the order of accuracy in time by O(∆T ). We will quickly talk about some of the types of splitting we have. The two major types of splitting are 1. Godunov Splitting: This was introduced in the previous section and we have already ascertain that it is first order accurate unless the operators commute. We will discuss another type which is an improvement on this. 2. Strang Splitting: Instead of taking the solution operator of problem (1.9) over the entire time step, it is taken over half a time step and used as initial data for problem (1.10) which will be integrated over the entire time step and finally use as initial to solve the problem (1.9) again to be integrated over half a time step. We have the Taylor’s expansion of the solution operator as A A ∆t2 2 ∆t2 2 ∆t A+ A + · · · )(I + ∆tB + B + ···) 2 8 2 ∆t ∆t2 2 (I + A+ A + ···) 2 8 ∆t2 2 = I + ∆t(A + B) + (A + B2 + AB + BA) + O(∆t3 ). 2 e∆t 2 e∆tB e∆t 2 = (I + © University of Pretoria (1.15) Chapter 1 Introduction 10 From equation (1.15), one can see that the O(∆t2 ) is captured correctly as in (1.12). Hence, the Strang splitting scheme is second order accurate. Remark 1.1 When dealing with much complicated equations, it has been advised that the Godunov splitting is often sufficient [51]. Also, in the presence of rough solutions it was argued in [11] that the order of accuracy is always reduced to one no matter the type of splitting employed. This may be due to the fact that at the region of discontinuity, the order of accuracy always reduces to one [51]. 1.3.2 Finite volume method The finite volume method, like the finite difference and the finite element methods, is a discretization method for differential equations. Unlike the finite difference method in which the equation is approximated at the grid points and the finite element method in which it is approximated element-wise, the finite volume approximation is evaluated at the intercell. The domain of integration is first subdivided into cells (finite volumes), then the approximation are evaluated at the intercell average. Therefore, the approximation is kept away from shock regions, since discontinuities that leads to shock occurs at grid points. Also, the flux entering each cell (finite volume) is identical to the flux leaving, then the flux in each cell is conserved. Hence, its application to conservation equations. Here, we will introduce this by applying it to hyperbolic conservation laws. Example 1.2 Consider the finite volume discretization of the hyperbolic equation ut + f (u)x = 0, (1.16) by the finite volume method. The idea of finite volume follows the weak formulation. The domain of integration is subdivided into grid cells called finite volumes, say, Ij = [xj− 1 , xj+ 1 ], j = 1, 2, 3, ..., m, 2 2 (1.17) and xj+ 1 = xj + ∆x , see Figure 1.1. m is the number of with interfaces xj− 1 = xj − ∆x 2 2 2 2 volumes and ∆x = xj+1 − xj = xj+ 1 − xj− 1 is the cell size on uniform grids. Here, x 1 2 2 2 and xm+ 1 are the boundaries. We will always employ boundary conditions to handle ghost 2 points where applicable. The integral formulation is approximated in each of the cells at the cell interfaces for one dimensional problems. We will integrate the equation (1.16) over Ij with respect to x. Z ∂ u(x, t)dx + f (u(xj+ 1 ), t) − f (u(xj− 1 ), t) = 0, 2 2 ∂t Cj © University of Pretoria Chapter 1 Introduction 11 t vjn+1 tn+1 vjn tn x1 xj− 1 xj−1 2 xj+ 1 xj 2 xj+1 2 x Figure 1.1: Finite volume domain discretization integrating the above equation over time from tn to tn+1 yields Z n+1 u(x, t Cj )dx − Z n u(x, t )dx = Cj Z tn+1 f (u(xj− 1 ), t) − 2 tn Z tn+1 f (u(xj+ 1 ), t). 2 tn Dividing the above equation by ∆x and letting the average Z 1 n u(x, tn )dx, vj = ∆x Cj we have vjn+1 1 = vjn − ∆x Z tn+1 tn f (u(xj+ 1 ), t) − 2 Z tn+1 tn ! f (u(xj− 1 ), t) . 2 (1.18) However, we can write this as vjn+1 = vjn − where Fj+ 1 2 1 = ∆t ∆t n n , Fj+ 1 − Fj− 1 2 2 ∆x Z (1.19) tn+1 tn f (u(xj+ 1 ), t), 2 is the numerical flux. The definition of this flux gives rise to many finite volume methods for approximating the solution of (1.16). We will discuss more on this in later chapters. © University of Pretoria Chapter 1 Introduction 1.3.3 12 Nonstandard finite difference method NSFD originates from the work of Mickens [57]. It has been employed to solve several differential models including singularly perturbed equations [53, 54]. We will adapt the NSFD procedure to design nonstandard finite volume schemes when solving some of the equations in order to preserve some physical and dynamical properties of the equation. To the best of our knowledge, this has not been employed to equations with complex solutions like the Schrödinger equation and higher order equations. Example 1.3 We will design a nonstandard finite difference scheme for the harmonic oscillator equation [57] d2 y = −λ2 y, λ ∈ R. (1.20) dt2 The derivation is based on the two linearly independent solutions to (1.20), namely y1 = exp(ıλt), y2 = exp(−ıλt). The linear independence of y1 and y2 implies y ıλ∆tj −ıλ∆tj e e j (1.21) yj+1 eıλ∆t(j+1) e−ıλ∆t(j+1) = 0. ıλ∆t(j+2) −ıλ∆t(j+2) yj+2 e e The determinant in equation (1.21) leads to the difference equation yj+2 − 2yj+1 cos(λ∆t) + yj = 0. (1.22) Employing the trigonometric identity cos(θ) = 1−2 sin2 ( θ2 ) and shifting the index j simplifies (1.22) to λ∆t 2 + yj−1 = 0. yj+1 − 2yj + 4yj sin 2 Finally, we have the exact finite difference scheme yj+1 − 2yj + yj−1 + λ2 yj = 0. 4 2 λ∆t sin λ2 2 (1.23) Equation (1.23) is an exact nonstandard finite difference scheme for the harmonic oscillator equation (1.20). This method shall be adapted to the finite volume space in this thesis. 1.4 Thesis outline This thesis is divided into self-contained papers that represent specific scientific contributions. • In Chapter 2, we design finite volume schemes for fourth order equations: the KuramotoSivashinsky and the (convective) Cahn-Hilliard equations. The fractional splitting © University of Pretoria Chapter 1 Introduction 13 method is employed to split the linear higher order terms from the nonlinear lower order terms. We employ shock capturing schemes for the nonlinear hyperbolic subequations; design A-stable schemes for the fourth order equation and a θ− scheme for the nonlinear diffusion equation. This way, we are able to resolve the restrictive stability condition, ∆t ∼ O(∆x4 ), common to most explicit schemes and the stiffness imposed by the coupling of the linear and the nonlinear terms. The physical behaviour of these models are all verified numerically. This is the first time these two equations will be investigated by the fractional splitting method. The results in this chapter are the subject of the papers [2, 3]. The presence of the small parameter in the Cahn-Hilliard equation led us to the study of singularly perturbed equations. This is the subject of the succeeding chapters. • Chapter 3 is devoted to the study of singularly perturbed differential equations with a special interest on the Schrödinger equation. The presence of the reduced Planck’s constant, the oscillatory behaviour of its solution and the complex boundary conditions makes its simulation challenging. Here, two different uniformly convergent finite volume schemes were designed for this equation. These schemes, semi-analytical scheme based on perturbation analysis and the nonstandard finite volume scheme proved efficient in resolving the challenges stated earlier. Their efficiency were shown through a series of computation. This is the first time nonclassical schemes of this type are being applied to the Schrödinger equation and we are able to design exact schemes for the complex dependent boundary conditions. Some of our observations are subject of [4] while our major results are being prepared for submission. • In Chapter 4, we design finite volume scheme for the Kuramoto-Sivashinsky equation and the Cahn-Hilliard equation. The schemes enjoy several properties which include preserving the properties of the original equation. Each of the sub-equations is approximated by nonstandard finite volume schemes. The significance of this work includes the reliability of the scheme in handling the small parameter. We also verified all the properties tested for in Chapter 2. The importance of this includes the construction of a nonstandard finite volume scheme for the fourth order partial differential equation. Our observations are being prepared for submission. • The last chapter is dedicated to discussion, conclusions on and possible further research work. © University of Pretoria Chapter 2 Fractional splitting method for higher order differential equations In this chapter, we implement finite volume schemes to solve the 1-dimensional KuramotoSivashinsky equation and the Cahn-Hilliard equation. We design operator splitting algorithm to resolve the restrictive stability condition (∆t ∼ O(∆x4 )) imposed by the coupling of the linear and the nonlinear terms. Our results on the K-S equation have been published in Aderogba et al. [2] while our results on the CH equation [3] will soon appear in Engineering Computations. 2.1 Introduction We design operator splitting based finite volume schemes to solve some fourth order unsteady partial differential equations. The operator splitting method is used in order to simplify the computation of the solution of these nonlinear equations. We apply the schemes to approximate the Kuramoto-Sivashinsky (K-S) equation and the Cahn-Hilliard (CH) equation. The first equation under investigation in this chapter is the K-S equation (2.1), ( ut + uux + αuxx + γuxxxx = 0, ∀ (x, t) ∈ R × (0, ∞), (2.1) u(x, 0) = u0 (x). Unless otherwise stated, the initial condition u0 (x) is considered to be a L-periodic function with zero average, that is Z L u0 (x + L) = u0 (x) and u0 (x)dx = 0. (2.2) 0 Our focus is to implement finite volume scheme to solve equation (2.1). This is the object of discussion in Section 2.3.4. This method allows us to treat the nonlinear hyperbolic term 14 © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 15 with optimal shock capturing schemes explicitly while the linear terms are approximated by the backward differentiation formula. Consequently, our results validate all the physical properties of this model: the existence of regular and oscillatory shock achieved through the traveling wave solution, the chaotic behaviour of the solution and the bound on the energy. We also examined the convective Cahn-Hilliard equation (cCH) ut + γ 2 uxxxx = φ(u)xx + δuux , where Ω ∈ (a, L) and x ∈ Ω, 0 < t, (2.3) φ(u) = αu3 − u, α > 0, which has been a successful model of phase transition observed in alloys, glasses, polymer solution and binary liquid mixtures (see for example [18, 19, 62], and the references therein). Our numerical approach for these equations through the fractional time splitting method is discussed in Section 2.2 where we introduce the shock preserving schemes for the nonlinear hyperbolic term, the θ-method for the nonlinear diffusion term and the schemes for the linear fourth order term. Section 2.3.4 is dedicated to the analysis of the K-S equation while we study the CH equation in Section 2.3.5 and 2.3.6 and discuss our observations in Section 2.4. 2.2 Numerical approach We will employ the fractional time-splitting method to solve our desired equations. From an abstract point of view, suppose we are dealing with a differential equation ut + A(u) = 0, and the operator A(u) is decomposable to either H(u) + N (u) + L(u) or H(u) + L(u), the space discretized problem can be written as: ( ut + H(u) + N (u) + L(u) = 0, (2.4) u(0) = u0 , or ( ut + H(u) + L(u) = 0, u(0) = u0 , (2.5) where H is the discretization of the nonlinear (convection) operator, N is the discretization of the nonlinear diffusion operator and L is the discretization of the linear operator. Apart from the fact that the fractional step method substantially reduces computational time, it has the advantage of simplifying a complex process efficiently. This method permits the treatment of each segment of the original equation separately, thus ( ut + H(u) = 0, (2.6) u(0) = u0 , © University of Pretoria Chapter 2 Fractional splitting method for higher order equations ( and ( 16 wt + N (w) = 0, w(0) = w0 , (2.7) yt + L(y) = 0, y(0) = y0 . (2.8) Let Ht u0 , Nt u0 and Lt u0 describe the approximate numerical solution operators for each of the split models (2.6), (2.7) and (2.8) respectively. We will use the first order scheme based on Marchuk splitting which can be written as v 0 = u0 for n = 0; and for n > 0, we obtain v n+1 from v n via the solution of vk (x, nk) = [Lk ◦ Nk ◦ Hk ]n v0 (x), (2.9) vk (x, nk) = [Lk ◦ Hk ]n v0 (x), (2.10) or for some time step ∆t = k, where the notation ◦ denotes the composition of two operators [30] and n denotes the time discretization step. Our splitting scheme will follow this algorithm throughout this work. This algorithm can be explained as below 1. We initialise the solution via v 0 = u0 (xj ) for j = 0, 1, 2, 3, · · · , m i.e., m + 1 grid points. 2. For n > 0, we compute the intermediate solution Hk , from the advection equation. 3. Within the same time step, we use the solution from Step 2 to compute the solution of the nonlinear equation to obtain Nk . 4. Within the same time step, we use the solution from Step 3 to compute the solution of the linear equation to obtain Lk . 5. We use the solution from Step 4 in Step 1 to initialise the next time step. 6. We repeat until the final computational time. Remark 2.1 The boundary conditions of each of the subproblems were corrected to first order accuracy in time following a similar procedure in [42], and the references therein. Remark 2.2 In the case where the original equation has only two discretized operator, there will only be two split equations and hence only two solution operators. Consequently, one of the steps 3 and 4 will not be essential in the described algorithm above. In the succeeding sections, we will describe the numerical solution operators that will be employed in our numerical experiments. © University of Pretoria Chapter 2 2.2.1 Fractional splitting method for higher order equations 17 Schemes for the hyperbolic equations (2.6) The design of finite volume numerical schemes for equations of type ut + f (u)x = 0, (2.11) has been introduced earlier on. It is noteworthy that the robustness of existing schemes on equations of this type is embedded in the way the numerical fluxes are handled and the temporal order of convergence. In this work, we employed 1. Godunov scheme 2. The non-staggered scheme (NSTG), [5, 81] 3. Semi-discrete scheme (SemiD) 4. Fully discrete (FD), [82] 5. Weighted essentially non-oscillatory scheme (WENO), [72, 73] 6. Implicit schemes It is worth mentioning here that apart from the implicit schemes, all the schemes listed above are total variation diminishing (TVD), monotonicity preserving and conservative. Remark 2.3 A TVD scheme is one in which T V (v n+1 ) ≤ T V (v n ) where TV(u) is the total variation of the function, u. This type of scheme preserves monotonicity and guarantees the absence of spurious oscillation in our numerical simulations. We will briefly outline these schemes. 2.2.1.1 Godunov scheme Godunov methods, founded by Godunov (1959), rely on the solution of the Riemann problem. The basic version of the scheme is only first order accurate but by updating the reconstruction process with the introduction of slope limiters, there is a possibility of extending this to higher order. Since the flux is convex, the application is direct. The intercell average at the interface xj+1/2 is given as Z x 1 j+ 2 1 n u(x, tn )dx, (2.12) vj = ∆x x 1 j− 2 and the numerical flux is f (unj+1/2 ) ≃ f (vjn ) h ′ n ′ n ∆t σjn , + f (vj ) 1 − f (vj ) 2 h © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 18 where σjn is a slope limiter and h = ∆x = xj+ 1 − xj− 1 is the space width. Several limiters 2 2 are documented in the literature, but throughout this thesis we will employ the monotonized centered (MC) limiter, which is given as vj+1 − vj vj+1 − vj−1 vj − vj−1 . , ,2 σ(j) = minmod 2 h 2h h The minmod function compares the absolute value of the arguments and returns the one with minimum absolute value as the slope. If σjn = 0, the scheme will switch to the basic first order Godunov scheme. For more information on limiters and the Godunov scheme, the reader can consult [51, 84, 91] among others. 2.2.1.2 Non staggered central difference scheme The non staggered central difference scheme (NSTG) is a second order extension of the non staggered version of the central difference scheme by Lax-Friedrich (see for example [81, 82]). It follows a process of reconstruction, evolution and projection step and takes an easily implemented predictor-corrector form, λfj′ , (2.13) 2 λ 1 ′n 1 n ′n n+1/2 n+1/2 n vj−1 − vj+1 f (vj+1 ) − f (vj−1 ) , vj+1 + vj−1 + vjn+1 = − (2.14) 2 4 2 where λ = hk is the CFL number. Furthermore, the non oscillatory behavior of this second order scheme is guaranteed by the choice vj′ = minmod ∆vj+1/2 , ∆vj−1/2 , n+1/2 vj = vjn − and fj′ = a(vj )vj′ , where a(vj ) is the derivative of the flux function with respect to the argument vj which should be interpreted as the Jacobian when dealing with systems of conservation laws. 2.2.1.3 Semi discrete central-upwind scheme The second order semi-discrete central (SemiD) difference scheme (see [5, 6, 82]) in the conservative form is given by [vj ]t = − Fj+1/2 − Fj−1/2 , h (2.15) with the numerical flux Fj+1/2 = + − f (vj+1/2 ) + f (vj+1/2 ) 2 − aj+1/2 + − (vj+1/2 − vj+1/2 ), 2 © University of Pretoria (2.16) Chapter 2 Fractional splitting method for higher order equations 19 ± where the intermediate values vj+1/2 are given by h + vj+1/2 = vj+1 − (ux )j+1 , 2 h − vj+1/2 = vj + (ux )j , 2 and the aj+1/2 term is the maximal local speed given in the generic case as ∂f − ∂f + (v (v ) ,ρ ) . aj+1/2 = max ρ ∂u j+1/2 ∂u j+1/2 In the scalar case with convex flux the maximal local speed simplifies to − + aj+1/2 = max(|f ′(vj+1/2 )|, |f ′(vj+1/2 )|). Also, the slope (ux )j is given by the minmod function vj+1 − vj vj − vj−1 , , (ux )j = minmod h h and the time derivative in (2.15) is evaluated by the Euler’s scheme. 2.2.1.4 Weighted Essentially Non-Oscillatory Scheme The scheme presented here follows the work of [72]. The hyperbolic conservation law (2.11) can be cast into the semi-discrete form as dvj = A(vj ), dt (2.17) where fj+1/2 − fj−1/2 . (2.18) h The left hand side of the equation (2.17) will be handled by the third order Total Variation Diminishing Runge-Kutta scheme as developed in [73]: A(vj ) = − u1 = v n + ∆tA(v n ), 1 1 3 u2 = v n + u1 + ∆tA(u1 ), 4 4 4 1 n 2 2 2 n+1 v = v + u + ∆tA(u2 ). 3 3 3 (2.19) The discretized flux terms fj±1/2 are given as − + fj±1/2 = g(vj±1/2 , vj±1/2 ), where g(a, b) = 1 (f (a) + f (b) − α(b − a)) , 2 © University of Pretoria (2.20) Chapter 2 Fractional splitting method for higher order equations 20 is the Lax-Friedrich monotone flux and α = maxu |f ′ (u)| is a constant. The maximum is − + taken over the relevant range of u. The values of vj+1/2 and vj−1/2 are computed through the intercell average of the conserved variable u given in equation (2.12). − vj+1/2 = l−1 X r ωr vj+1/2 , r vj+1/2 = r=0 and + vj−1/2 = l−1 X cri vj−r+i , i=0 r ω̃r vj−1/2 , r vj−1/2 = r=0 l−1 X c̃ri vj−r+i , i=0 where r = 0, ..., l − 1. The linear weights αr ωr = Pl−1 s=0 αs and l−1 X α̃r ω̃r = Pl−1 s=0 α̃s , αr = dr , (ǫ + βr )2 , α̃r = d˜r , (ǫ + βr )2 where ǫ = 10−6, βr are smooth indicators and d˜r = dl−1−r are constants. The order of accuracy is 2l − 1. We use l = 2 in our computations, consequently, we only need d0 = 2/3 and d1 = 1/3. The values of constants c̃ri = cr−1,i are given in Table 2.1 as extracted from [72]. Also, the smooth indicators are given as β0 = (vj+1 − vj )2 , β1 = (vj − vj−1 )2 . Table 2.1: The values of cri r i=0 i=1 -1 1.5 -0.5 0 0.5 0.5 1 -0.5 1.5 2.2.1.5 Fully discrete scheme The fully discrete scheme for the convective equation (2.11) is given by [5] Z 1 xj+1/2 n+1 n+1 w̃(ξ, tn+1)dξ = λanj−1/2 wj−1/2 + 1 − λ(anj−1/2 + anj+1/2 ) wjn+1 (u)j = h xj−1/2 n+1 + λanj+1/2 wj+1/2 + h n n 2 n+1 (λaj−1/2 )2 (ux )n+1 − (λa ) (u ) x j+1/2 , j+1/2 j−1/2 2 © University of Pretoria (2.21) Chapter 2 Fractional splitting method for higher order equations and (ux )n+1 j+1/2 2 = minmod h n+1 n+1 wj+1 − wj+1/2 n+1 wj+1/2 − wjn+1 , 1 + λ(anj+1/2 − anj+3/2 ) 1 + λ(anj+1/2 − anj−1/2 ) 21 ! , unj + unj+1 h − anj+1/2 ∆t 1 n+1/2 n+1/2 + ((ux )nj − (ux )nj+1) − n = f (uj+1/2,r ) − f (uj+1/2,l ) , 2 4 2aj+1/2 λ ∆t n n+1/2 n+1/2 (aj−1/2 − anj+1/2 )(ux )nj − f (u ) − f (u ) , = unj + j+1/2,l j−1/2,r 2 1 − λ(anj−1/2 + anj+1/2 ) n+1 wj+1/2 wjn+1 with n+1/2 uj+1/2,l = 1 n − λaj+1/2 , 2 1 n n n − λaj+1/2 . = uj+1 − h(ux )j+1 2 ∆t f (unj+1/2,l )x , unj+1/2,l = unj + h(ux )nj − 2 unj+1/2,l ∆t f (unj+1/2,r )x , unj+1/2,r 2 The local speed of propagation is ∂f − ∂f + n (u (u aj+1/2 = max ρ ) ,ρ ) , ∂u j+1/2 ∂u j+1/2 n+1/2 uj+1/2,r = unj+1/2,r − where ρ is the spectral radius, h h n n n − n u+ j+1/2 = uj+1 − (ux )j+1 , uj+1/2 = uj + (ux )j , 2 2 and (ux )nj unj − unj−1 unj+1 − unj , ), = minmod( h h with minmod(a, b) = sgn(a) + sgn(b) min(|a|, |b|). 2 Remark 2.4 It is worth mentioning at this point that the convergence of the above schemes for the convection equation has been documented in the literature as quadratic (see [5, 6, 51, 81, 91] for example). Throughout, we choose the CFL number to be less than 1, which is within the stability requirement of all the schemes considered. 2.2.1.6 Implicit schemes When the intercell time average is 1 n n+1 f (unj+1/2) ≃ ((1 − θ)(f (vjn ) + f (vj+1 )) + θ(f (vjn+1 ) + f (vj+1 ))), 2 then the corresponding numerical scheme vjn+1 = vjn − ∆t n n n+1 n+1 ((1 − θ)(f (vj+1 ) − f (vj−1 )) + θ(f (vj+1 ) − f (vj−1 )), 2h is either fully implicit (θ = 1) or Crank-Nicolson (θ = 0.5). © University of Pretoria (2.22) Chapter 2 2.2.2 Fractional splitting method for higher order equations 22 Schemes for the nonlinear diffusion equation In this section, we implement finite volume schemes for a nonlinear diffusion equation. We solve the nonlinear diffusion equation explicitly and implicitly by considering a conservative scheme in which case the equation is written as ut = α(u3 )xx . (2.23) We discretized (2.23) using the θ-method to give n+1 n+1 n n n vjn+1 = vjn + r θ[φn+1 − 2φ + φ ] + (1 − θ)[φ − 2φ + φ ] j+1 j j−1 , j+1 j j−1 (2.24) where φnj = (vjn )3 , r = hk2 and vjn is the intercell averages. For the purpose of comparison, we also solve (2.23) with second order non conservative schemes, in which case the equation is re-written as ut = α(ϕux )x , (2.25) where ϕ = 3u2 . Similarly, this is discretized using the θ method to give n+1 n+1 n+1 n+1 n+1 vjn+1 = vjn + rθ ϕn+1 [v − v ] − ϕ [v − v ] + j j−1 j+1/2 j+1 j−1/2 j n n r(1 − θ) ϕnj+1/2 [vj+1 − vjn ] − ϕnj−1/2 [vjn − vj−1 ] , (2.26) where ϕj±1 + ϕj , 2 and θ is a weighting factor. The schemes (2.24) and (2.26) will be fully implicit if θ = 1, Crank-Nicolson if θ = 0.5 and fully explicit when θ = 0. The explicit schemes result in a system of linear equations while the implicit schemes give a system of nonlinear equations. While the linear equations can be solved directly, the nonlinear equations are solved by the iterative Newton-Raphson method. Coupled with the corresponding second order boundary conditions, the expected rate of convergence is 2 for all these schemes. ϕj±1/2 = 2.2.3 Schemes for the linear (diffusion) equation In this section, we consider three different schemes for the numerical solution of the linear subproblem ut + uxx + γ 2 uxxxx = 0. (2.27) Via the θ-scheme, we consider the implicit schemes in the following form vjn+1 + θFjn+1 = vjn − (1 − θ)Fjn , j = 1, 2, · · · , m − 1, where n n n n n n Fjn = r(vj−1 − 2vjn + vj+1 ) + µ(vj−2 − 4vj−1 + 6vjn − 4vj+1 + vj+2 ), © University of Pretoria (2.28) Chapter 2 Fractional splitting method for higher order equations 23 µ = γ 2 hk4 and r = hk2 . The scheme is fully implicit if θ = 1 and Crank-Nicolson (C-N) if θ = 0.5. We have chosen not to include the explicit scheme (θ = 0) in the analysis because of its restrictive stability condition which requires a time step of O(h4 ). Scheme (2.28) obviously include ghost nodes which are eliminated via the boundary conditions. This will be explained later in this chapter. We will also consider the backward differentiation formula (BDF2) in the form 3vjn + 2Fjn = 4vjn−1 − vjn−2 , n = 2, 3, 4, 5, · · · , (2.29) where the backward Euler scheme vjn = vjn−1 + Fjn , is used for the n = 1 time step and Fjn is as given above. Lastly, we employ the Diagonally Implicit Runge-Kutta (DIRK) scheme which is given as 1 vj∗ = vjn−1 − Fj∗ , 2 n−1 n vj = vj − Fj∗ . For periodic boundary conditons, we will also solve the linear equation using the Matlab built-in fft-solver to serve as a benchmark for the other schemes. 2.3 Numerical experiments In this section we apply each of the numerical schemes discussed in Section 2.2 to specific examples in order to validate their effectiveness. 2.3.1 Hyperbolic equation In this Section, we apply and compare the performance of the schemes listed in Section 2.2.1 in approximating the inviscid Burgers equations. Example 2.1 Here we solve the following convection problem, ( ut + uux = 0, ∀ (x, t) ∈ (0, L) × (0, T ], u(0, x) = 0.5 + sin(x) , (2.30) with periodic initial and boundary conditions. Solution to the convection term preserves the average energy density until the formation of the shock [24]. The profiles of E(t) in Figure 2.1 show the conservation of E(t) before the development of shock. This is a generic behaviour of the solution of hyperbolic equations. © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 24 0.8 Mean energy 0.75 0.7 SemiD NSTG FImpl CNS Godunov 0.65 0.6 0.55 0 0.5 1 t 1.5 2 Figure 2.1: Mean energy density profiles for problem (2.30). Giacomelli and Otto [24] also claimed that the convection term preserves the mean energy density before the onset of shock. This is confirmed by all the schemes through Figure 2.1. These profiles also show that the shock develops at about time T = 1 except for the Godunov scheme. The semi-discrete central and the implicit schemes agreed with the approximation of time of shock development while the non-staggered central scheme depict shock as setting in a little earlier before T = 1. These all agree with the assertion in [5]. The down-hill sawtooth behavior of the mean energy after the shock in the implicit schemes (which is more pronounced in the Crank-Nicolson scheme) may be due to oscillation about the shock region as observed in the earlier work of Yong-Jung et al. [91]. Moreover, it is note worthy that out of all the schemes, the mean energy climbs uphill after the shock only in the Godunov scheme. Thus from here forthwith, we will drop the simulations based on the Godunov scheme because of its poor performance. 2.3.2 Nonlinear diffusion equation Here we consider the following example. Example 2.2 Solve 3 ut − (u )xx = f (x), x ∈ [0, π], u(0, t) = u(π, t) = 0, u(x, 0) = sin(x), (2.31) where f (x) = − 34 [3 sin 3x − sin x], to test the effectiveness of schemes (2.24) and (2.26). We compare our results with the exact steady state solution u(x, t) = sin(x) at time T = 0.5. The ∞ − norm error, e(mj ), for different number of grid points, mj , were computed and the convergent rates, p, were calculated based on the relation p= log(e(m2 )/e(m1 )) . log(m2 /m1 ) © University of Pretoria (2.32) Chapter 2 Fractional splitting method for higher order equations 25 The errors and the convergent rates due to each of the scheme are given in Table 2.2 for different grid points. It is obvious that all the schemes are equally good. Table 2.2: Convergence rate of the schemes for the nonlinear equation (2.48) Grid points 20 40 80 160 L∞ error×10−3 (p) at T = 0.5 Conservative Non Conservative Schemes CN Explicit CN Explicit 18.1 18.2 23.0 20.0 6.34 (1.46) 6.39 (1.46) 7.86 (1.49) 7.30 (1.50) 2.03 (1.62) 2.02 (1.63) 2.50 (1.63) 2.39 (1.64) 0.58 (1.78) 0.58 (1.78) 0.72 (1.78) 0.70 (1.78) Remark 2.1 We highlight here that it is not necessary to determine the convergence through other norms. This guaranteed by the equivalence of all norms in Rd [52]. 2.3.3 Linear diffusion equation For finite domains, the boundary conditions are either periodic or non periodic. For the non-periodic case we have nonhomogeneous Dirichlet and Neumann boundary conditions as follows u(0, t) = g(0, t), ux (0, t) = gx (0, t), u(L, t) = g(L, t), ux (L, t) = gx (L, t), where g(x, t) is a known function related to the exact solution. In particular, we introduce n n ghost nodes, v−1 and vm+1 , which are eliminated using the following discretization, v0n = g0n , and n n vm = gm , n −3v−1 + 4v0n − v1n + O(h2 ) = gx (0, t), 2h (2.33) n n n vm−1 − 4vm + 3vm+1 + O(h2 ) = gx (L, t), 2h (2.34) respectively. Example 2.3 We solve ( ut + uxx + γ 2 uxxxx = 0, ∀ t ∈ (0, T ], u(x, 0) = Φ(x), subject to periodic boundary condition. © University of Pretoria (2.35) Chapter 2 Fractional splitting method for higher order equations L∞ error×104 at T = CNS Spectral BDF2 74.4087 1516.61 74.4253 19.1106 773.780 19.1108 5.02760 390.024 5.02761 1.28539 195.709 1.28539 Grid points 20 40 80 160 26 1 DIRK 81.1000 19.4500 4.8850 1.6900 Table 2.3: Error due to each scheme for the numerical approximation of the linear equation. We compare the numerical approximations by each of the scheme discussed in Section 2.2.3 x to the exact solution u(x, t) = sin γ and compute the k · k∞ error. These are shown in Table 2.3. This computation is done with Φ(x) = sin γx . It is claimed, see for example [24], that the energy density for this diffusion equation grows exponentially. We validate this claim by considering Example 2.3 with different initial conditions. 70 0.52 Implicit Spectral BDF2 0.51 50 0.505 40 E E 0.515 Implicit Spectral BDF2 60 0.5 30 0.495 20 0.49 10 0.485 0.48 0 5 10 t 15 20 0 0 5 10 t (b) Φ(x) = e−x (a) Φ(x) = sin(x) 15 20 2 Figure 2.2: Mean energy density profile for the diffusion term for T = 20, γ = 1 and (a) x ∈ [0, 2π] and (b) x ∈ [−2π, 2π]. The growth of the mean energy of the diffusion equation as shown in Figure 2.2(b) agrees with the earlier observations (see [24] and the references therein). No growth is expected when the initial condition is a periodic exact steady state solution of the problem as was shown in Figure 2.2(a). From here forthwith, the convection term is solved by any of the above mentioned schemes while the diffusion term is solved by the BDF2 scheme (for K-S equation) or DIRK scheme (for CH equation). The C-N scheme was dropped basically because of the stiffness of the problem under consideration. Hence, we will refer to the method of solution of the K-S equation by the scheme used to handle the convection term. In all the schemes we will © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 27 choose 320 grid points. 2.3.4 Kuramoto-Sivashinsky equation In this Section, we will consider equation (2.1). The split scheme used here is motivated by observing that the K-S equation consists of two different spatial operators: the linear and the nonlinear operators. Interestingly, the nonlinear operator is hyperbolic, i.e., it is known to introduce discontinuity in finite time while the linear fourth order term has a stabilizing effect. Therefore, it is advantageous to split the equation into the nonlinear and the linear equations for easy treatment. Hence, we split equation (2.1) into the following two partial differential models ut + uux = 0, (2.36) and ut + α uxx + γ 2 uxxxx = 0. (2.37) It is noteworthy at this point that while (2.37) accounts for the long wave instability (gain) the equation (2.36) accounts for short wave dissipation. Consequently, the equation is split into two physical processes evident in the equation: the convection (inviscid Burgers) equation and the linear fourth order equation. Each of the equations is solved numerically using finite volume method [50] in such a way as to preserve some qualitative properties of the solutions related to each continuous problem. Our aim in this work is to present reliable and efficient solution approach and, in the process, numerically validate some of the theoretical results of the K-S equation documented in the literature. We verify the results stated above on the bound of the solution, preservation of the periodicity and zero average as observed in [24]. The schemes that we will employ to treat the convective equation have been discussed in the preceding section. We will only test the efficiency of four of these when combined with the solver that will be used for the linear term. The four schemes are the Implicit, Godunov, NSTG and SemiD schemes as outlined in Section 2.2.1 while we will make use of the BDF2 scheme outlined in Section 2.2.3 for the linear equation (2.37). Example 2.4 We consider the computational domain to be [−L, L] with a focus on computing the solution of the entire K-S equation using the fractional time step method described in Section 2.2. We test the convergence of the following problem as given by [89], solving ( ut + uux + uxx + uxxxx = 0, ∀ (x, t) ∈ (−L, L) × (0, T ], (2.38) u(x, 0) = Φ(x). First, we solve equation (2.38) with Φ(x) = g(x, 0) and subject to boundary conditions u(−L, t) = g(−L, t), u(L, t) = g(L, t), ux (−L, t) = gx (−L, t), © University of Pretoria ux (L, t) = gx (L, t), (2.39) Chapter 2 Fractional splitting method for higher order equations where g(x, t) is the exact solution given by r 15 11 (−9 tanh[l(x − ct − x0 )] + 11 tanh3 [l(x − ct − x0 )]). g(x, t) = c + 19 19 28 (2.40) The ghost points are removed via equations (2.33) and (2.34). Here, c, l and x0 are constants p and in the computations we take L = 30, x0 = −12, c = 5 and l = 11/19/2 as documented in [7, 89]. For comparison we also solve equation (2.38) for Φ(x) = exp(−x2 ) subject to periodic boundary conditions. We will follow the fractional step algorithm as given in Section 2.2. The convergence results due to different numerical schemes for the convection equation are all presented in Tables 2.4 and 2.5. The numbers inside the brackets are the convergent rates, q. These are computed from the errors em and e2m corresponding to number of grid points m and 2m respectively, that is q= log(e2m /em ) . log 2 From Table, 2.5 the Godunov scheme admits the largest error followed by the nonstaggered central scheme and then by the implicit schemes, while the semi-discrete central scheme appears to produce the best result. Interestingly, from Table 2.4, when the initial data that corresponds to the exact solution of the K-S equation was employed, the explicit schemes behave far better than the implicit ones. The semi-discrete is consistent in producing the least possible error out of the explicit schemes. It is also evident that the convergent rate non-staggered central scheme is better than any of the other schemes. Table 2.4: Convergence rate of the fractional step for the K-S equation with initial data that corresponds to the exact solution. Grid points L∞ error (convergence rate, p) at T = 1 Godunov SemiD NSTG CNS FUIM 40 1.040 1.373 1.883 1.588 1.608 1.512 (-0.54) 0.535 (1.36) 1.258 (0.58) 0.709 (1.16) 0.688 (1.22) 80 0.950 (0.67) 0.137 (1.97) 0.6011 (1.07) 0.206 (1.78) 0.222 (1.63) 160 0.268 (1.82) 0.178 (2.94) 0.2197 (1.45) 0.056 (1.88) 0.068 (1.71) 320 0.099 (1.43) 0.007 (1.41) 0.0698 (1.65) 0.021 (1.45) 0.028 (1.26) 640 The profiles of the solution generated by the different schemes in comparison with the exact solution including the ’close-up near the peaks’ are shown in Figure 2.3. The deviation of all the numerical schemes from the exact solution is shown in Figure 2.3(a) while Figure 2.3(b) reveals the deviation of each of the schemes at the highest peak. Of particular interest, the NSTG scheme gives the largest deviation from the exact solution at the peak. © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 29 Table 2.5: Convergence rate of the fractional step scheme for the K-S equation with Gaussian initial data. The exact solution is taken as the solution of the most refined grid (m = 640) Grid points L∞ error ×103 (convergence rate, p) at T = 1 Godunov SemiD NSTG CNS FUIM 40 281 241 391 263 263 268 (0.07) 107 (1.18) 178 (1.14) 113 (1.22) 113 (1.22) 80 160 83.6 (1.68) 46.6 (1.20) 60.4 (1.56) 48.1 (1.24) 47.9 (1.23) 34.8 (1.26) 15.6 (1.58) 17.8 (1.76) 15.9 (1.59) 15.9 (1.59) 320 7 6.8 6.5 6.6 6 SemiD Exact NSTG Godunov FUIM CNS 5.5 u 5 4.5 6.4 u 6.2 SemiD Exact NSTG Godunov FUIM CNS 6 4 5.8 3.5 3 −30 5.6 −20 −10 0 x 10 20 30 −5 (a) All schemes with the exact solution. −4.5 −4 −3.5 −3 −2.5 x (b) Close peaks. Figure 2.3: Comparison of the exact solution with the numerical solutions. 2.3.4.1 Comparison with the traveling wave solution The traveling wave solution of every time-dependent partial differential equation gives the solution at all times. Therefore to test the accuracy of the numerical schemes, it makes sense to initialize the solution with the traveling wave solution and check the deviation of the schemes from the traveling wave solution as time advances, [91]. This section is dedicated to traveling wave solution of the K-S equation. This is advantageous over any other solution since the chaotic behavior of (2.38) is restricted to it being integrated over a finite x−domain with periodic boundary conditions. Therefore, following the work of Hooper and Grimshaw [34] and Yong-Jung et al. [91] (and references therein) we use the transformation u(x, t) = u(z) where z = x − st, s is the wave speed, so that the traveling wave solution is defined over the entire z−domain, −∞ < z < +∞. The boundary conditions are such that u → ul as z → −∞ and u → ur as z → +∞. The substitution above reduces equation (2.38) to an ordinary differential equation which can be integrated once to give 1 u′′′ = c + su − u2 − u′, 2 © University of Pretoria (2.41) Chapter 2 Fractional splitting method for higher order equations 30 where the prime denotes the derivative with respect to z. The wave speed s and the constant of integration c are determined by the far field solutions as ul ur ul + ur , c=− . s= 2 2 The wave speed is found via the Rankine-Hugoniot condition to be s= f (ur ) − f (ul ) . ur − ul The spatiotemporal behavior of the solution of (2.42) had been recorded by many authors (see [24, 34, 55, 60] among many). Michelson [55] gave the steady solution of (2.38) and studied the solution as a function of the square of a parameter c. With this, he classified the behavior of the solution as conical (for large value of c2 ), periodic or quasi-periodic (for small values of c2 ). Later, Hooper and Grimshaw [34] classified the solution based on the shock development as either regular shocks, solitary waves or oscillatory shocks. This they did by observing the far field behavior of the solution. They also noted that experiments may show chaotic behavior with respect to traveling waves. Recently, Nickel [60] employed the conditions for solitary and periodic waves to derive an exact solution to the traveling wave. Here we implement the oscillatory shock behavior as given in [34]. Thus, we solve the non homogeneous ordinary differential equation ′′′ ′ u + u − (s − 0.5u)u = c, (2.42) u(−∞) = ul , u(+∞) = u . r The nonlinear boundary value problem (2.42) was discretized and the system of equations derived were solved by the Newton’s method. In the numerical computations, we choose a sufficiently large domain such that u′ vanishes at the truncated boundaries. We highlight here that our numerical approach was able to reproduce most of the different families of solutions predicted in [34]. For the oscillatory shock considered here, we impose the far field boundary values, ul = 1 = −ur , consistent with the work of Hooper and Grimshaw [34]. The results in Figure 2.4 were all generated as outlined in Section 2.2. We highlight that all the schemes produced the same quantitative behavior. Nevertheless, NSTG and the fully implicit scheme solution are the closest to the traveling wave solution with the NSTG giving the least deviation from the traveling wave solution. The Godunov and the semi-discrete scheme solution gave the largest deviation of all the schemes. 2.3.4.2 Chaotic property of the Kuramoto-Sivashinsky equation In this section we verify the chaotic nature of the solution of the K-S equation by solving (using the time-splitting scheme) equation (2.38) subject to periodic boundary conditions. We can see that the numerical simulations in Figure 2.5 are consistent with the work of [89]. We highlight the convergence of the presented scheme in Figure 2.5(a) and Figure 2.5(c). © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 2 1.5 1 0.5 u 1.6 1.4 u1.2 0 −0.5 1 −1 0.8 −1.5 0.6 −2 −30 −20 −10 0 x 10 20 SemiD Travel wave NSTG Godunov FUIM CNS 1.8 SemiD Travel wave NSTG Godunov FUIM CNS 31 −10 30 −8 −6 −4 −2 0 2 x (a) All schemes with the traveling wave. (b) Close peaks of all the schemes. Figure 2.4: Traveling wave solution as standard compared to other schemes at T = 10. 2.3.4.3 Solution properties of the Kuramoto-Sivashinsky equation We begin this section by validating the bound for the mean energy density of the full K-S equation. In particular, Giacomelli and Otto [24] claimed that the effect of (2.30) will balance the exponential growth of (2.35) resulting in a bound for the mean energy of the entire K-S equation. We consider equation (2.38) with Φ(x) = sin(x) subject to periodic boundary conditions. The validation is given in Figure 2.6. We note that all the schemes determine approximately the same bound. However the NSTG scheme gives a largest deviation from the other schemes. In addition to the boundedness of the mean energy density of the full equation, we also performed numerical experiments to validate the periodicity preserving property. In Figure 2.7(a), we show the conservation of the conserved variable. It is shown here by the zero mean in Figure 2.7(b). Next, in order to show the effectiveness of our proposed scheme, we validate the bound as proved in earlier literature. We write the inequality (1.4) as lim sup kuk2 = t→∞ Z 0 L 21 = O(Lp ). u dx 2 (2.43) Hence, we plot log(ku(x, t)k2 ) against log(L) and p is approximated via the slope of the best fit. The results are shown in Figure 2.8(a) for each of the fractional split scheme. The conjectured bound is of O(L) (see [78] and references therein), i.e., p = 1. The value of the slope for each scheme gives a good approximation of this value. For the fully implicit scheme p = 1.0047, for the C-N scheme p = 1.0074, for semidiscrete scheme p = 0.9988. However, we highlight that the largest deviation is recorded for the NSTG scheme (p = 0.9767) which gives a lower bound compared to all the other © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 32 40 35 30 t 25 20 15 10 5 0 −15 (a) Surface plot on 160 grids −10 −5 0 x 5 10 15 (b) Contour plot showing the peaks (160 grids) 40 35 30 t 25 20 15 10 5 0 −15 (c) Surface plot on 320 grids −10 −5 0 x 5 10 15 (d) Contour plot showing the peaks (320 grids) Figure 2.5: The chaotic solution of the K-S equation with Gaussian initial conditions (Φ(x) = exp(−x2 )) up to T = 40. schemes. Figure 2.8(b) shows the system size independence of the quantity kux k22 = Z L 0 u2x dx, (2.44) as proved theoretically in [63]. Hence, the expression in (2.44) should be of O(L0 ). Our computation reveals that the exponent is −0.02025, 0.0001, 0.0063 and 0.0009 for the NSTG, semi discrete, fully implicit and C-N schemes respectively. It is also evident from Figure 2.8(b) that the NSTG scheme deviates significantly from all other schemes. It is obvious from the Figure that the semi-discrete and the implicit schemes behave equally well unlike the non staggered central scheme. © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 0.5 33 NSTG SemiD CNS FUIM Spectral 0.4 E 0.3 0.2 0.1 20 40 60 80 100 120 140 t Figure 2.6: Mean energy profiles for the full K-S equation. 0.05 NSTG SemiD CNS FUIM NSTG SemiD CNS FUIM 0.6 0.4 u 0.2 0 0 −0.2 −0.4 −0.6 0 1 2 3 4 5 6 −0.05 0 5 x (a) Periodicity 10 t 15 20 (b) Zero average Figure 2.7: Profile of the K-S equation showing the periodicity and zero average of the solution at T = 20 and L = 2π. 2.3.5 Cahn-Hilliard equation In this section we apply the fractional splitting method to another relevant equation in physics, the convective Cahn-Hilliard equation (cCH) (2.3). This equation is solved subject to the boundary conditions ux = γuxxx − φ(u)x = 0, (x = a, x = L). (2.45) In particular, we split equation (2.3) using a fractional time step algorithm following the work of, for example, [24, 91]. The stability of the method is guaranteed as long as the properties of each subproblem are preserved, see for example [29]. We split equation (2.3) into three subproblems: the hyperbolic equation ut − δf (u)x = 0, © University of Pretoria (2.46) Chapter 2 Fractional splitting method for higher order equations 34 −0.1 10 −0.2 10 ||u||2/L 0 10 −0.3 10 2 ||u|| 2 NSTG SemiD CNS FUIM NSTG SemiD CNS FUIM −0.4 10 −0.5 10 1 1 10 10 L L (a) Bound of (2.43) (b) Bound on (2.44) Figure 2.8: (a) loglog plot of the kuk2 . (b) loglog plot of kux k22 . L the linear fourth order equation ut + γ 2 uxxxx + uxx = 0, (2.47) and the nonlinear diffusion equation ut − α(u3 )xx = 0, (2.48) where f (u) = u2 /2. For δ = 0, equation (2.3) reduces to the traditional Cahn-Hilliard (CH) equation ut + γ 2 uxxxx = φ(u)xx , x ∈ Ω, 0 < t, (2.49) which is an equation of conservation of mass with flux J = − φ(u) − γ 2 uxx x . (2.50) It has been observed that the critical points of the Ginzburg–Landau free energy form Z L γ2 2 (2.51) E= F (u) + ux dx, 2 0 where F is a Lyapunov functional, F (u) = Z u φ(s)ds, 0 is a steady state solution of the CH equation (2.49) [19, 62] giving the boundary conditions ux = γ 2 uxxx − φ(u)x = 0, © University of Pretoria x ∈ ∂ Ω. (2.52) Chapter 2 Fractional splitting method for higher order equations 35 We believe that this is the first time the equation is handled numerically using a fractional step method of this kind. The hyperbolic subproblem (2.46), and the parabolic subproblems (2.47) and (2.48), being of different natures, can be handled using different efficient numerical schemes. The convective equation (2.46) will be solved by some of the finite volume schemes designed for hyperbolic conservation laws [51], precisely the NSTG, FD and WENO as outlined in Section 2.2.1 while the linear equation (2.47) will be handled by the second order Diagonally Implicit Runge-Kutta (DIRK) scheme given in Section 2.2.3. Equation (2.48) will be solved by an implicit/explicit conservative/non-conservative scheme, see Section 2.2.2. We highlight here that the split into (2.47) and (2.48) is motivated by the need to treat linear and nonlinear terms separately. This fractional step scheme converges to the solution of equation (2.3) and this can be shown following a similar analysis as given by Holden et al. [30]. For the specific case when δ = 0, we will combine only equations (2.47) and (2.48) to obtain the solution of the CH equation in (2.49). First, we discuss the solution of equation (2.3) for the case when δ = 0 for different choices of α. All the simulations were run on a Windows XP, Intel Core 2 Duo, 2GB RAM desktop and all the solutions were achieved within 10 seconds of computing time. Example 2.5 For the convergence analysis, we consider ( ut + (αu3 + u + γ 2 uxx )xx = 0, x ∈ (0, 6) u(x, 0) = cos(πx/6), (2.53) and boundary conditions (4.27). It is obvious that the solution operator of this equation can be split into the solution operator of equation (2.48) and the solution operator of equation (2.47), and the Algorithm in Section 2.2 can be used. We determine the spatial rate of convergence of this scheme for the Cahn-Hilliard equation (4.24) when γ = α = 1 using the refined grid as the exact solution. The rate of convergence was computed by employing equation (2.32) and the results are presented in Table 2.6. In addition, we note that the evolution of equation (4.24), for α = 1/3 and γ 2 = 0.02, shown in Figure 2.9 compares well with the work of [18]. 2.3.5.1 Solution property of the Cahn-Hilliard equation 4.24 It has been documented that the solution provided by any numerical scheme that solves the Cahn-Hilliard equation must satisfy two main properties [13, 14, 18, 62]: 1. Conservation of mass: Z u0(x)dx = Ω Z udx = ML, t > 0, Ω where M is the total mass of the system and © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 36 Table 2.6: Convergence rate of the fractional step for the Cahn-Hilliard equation Grid points 20 40 80 160 maximum absolute Conservative CN Explicit 98.8 102 45.9(1.1) 47.5(1.1) 19.6(1.21) 20.3(1.2) 6.52(1.6) 6.74(1.6) 2 2 t=0 t=2 t=4 t=6 t=8 t = 10 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −2 0 1 2 3 x 4 5 initial solution conser. impl. nonconser. impl. conser. expl. nonconser. expl. 1.5 u u error×103 (p) at T = 1 Non Conservative Schemes CN Explicit 102 101 47.5(1.1) 47.2(1.1) 20.4(1.2) 20.2(1.2) 6.78(1.6) 6.73(1.6) 6 −2 0 (a) Evolution in time 1 2 3 x 4 5 6 (b) Schemes comparison Figure 2.9: Solution of the Cahn-Hilliard (4.24) at time T = 10. Profiles in (a) are given at equally spaced time intervals. 2. Dissipation of the Ginzburg-Landau free energy functional (equation (2.51)) as time passes. Hence, this section is dedicated to validate these two main properties using Algorithm 2.9 and Example 2.5. The profiles in Figure 2.10(a) verifies the conservation of mass. We expect Z d d udx = ML = 0. The decay in the profiles in Figure change in mass to be zero since dt Ω dt 2.10(b) validates the dissipation of energy as expected (see for example, [14, 18]). Note that most of the profiles for the different schemes overlap and hence cannot be distinguished from each other. The mass profile when the conservative implicit scheme is applied to the nonlinear subproblem shows some deviation from the other schemes. 2.3.6 Convective Cahn-Hilliard equation In this section we give the solution of the cCH equation (2.3) using the splitting Algorithm 2.9. © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 37 −0.3 0.5 conser. impl. nonconser. impl. conser. expl. nonconser. expl. 0.3 conser. impl. nonconser. impl. conser. expl. nonconser. expl. −0.4 −0.5 −0.7 0.1 Energy ∆ Mass −0.6 −0.1 −0.8 −0.9 −1 −0.3 −1.1 −1.2 −0.5 0 1 2 3 4 t 5 −1.3 0 1 2 3 4 5 t (a) Conservation of Mass ×10−4 (b) Dissipation of Energy Figure 2.10: Conservation of mass and dissipation of energy. Example 2.6 We consider ( ut − δuux + (u − αu3 + γ 2 uxx )xx = 0, u(x, 0) = − sin(x/6), (2.54) and boundary conditions (2.45). For the purpose of comparison with results in the literature, we choose γ = α = 1 and δ = 0.1 – see for example [21, 86]. Following the same refined-grid method as discussed for the Cahn-Hilliard equation, the convergence rate was computed and the results are shown in Tables 2.7 to 2.9. Approximately, all the schemes predict a second order convergence in space. Table 2.7: Convergence rate of the fractional step for the cCH equation with the NSTG scheme. Grid points 20 40 80 160 L∞ error×103 (convergence rate) at T = 1 Conservative Non Conservative Schemes CN Explicit CN Explicit 116 116 116 116 53.6 (1.08) 53.5 (1.08) 53.6 (1.07) 53.4 (1.08) 22.9 (1.21) 22.8 (1.21) 22.8 (1.21) 22.8 (1.20) 7.72 (1.55) 7.70 (1.55) 7.72 (1.55) 7.70 (1.55) The profiles of the solution as compared with the exact steady state solution are also given in Figures 2.11 and 2.12. For all the schemes, the CFL number is taken between 0.1 and 0.5 which also gives a time step within the stability restriction of the explicit schemes for © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 38 Table 2.8: Convergence rate of the fractional step for the cCH equation with the WENO scheme. Grid points 20 40 80 160 L∞ error ×103 (convergence rate) at T = 1 Conservative Non Conservative Schemes CN Explicit CN Explicit 110 110 110 110 51.5 (1.05) 51.4 (1.06) 51.5 (1.05) 51.3 (1.06) 22.2 (1.20) 22.1 (1.20) 22.2 (1.20) 22.1 (1.20) 7.58 (1.54) 7.55 (1.54) 7.57 (1.54) 7.55 (1.54) Table 2.9: Convergence rate of the fractional step for the cCH equation with the FD scheme. Grid points 20 40 80 160 L∞ error×103 (convergence rate) at T = 1 Conservative Non Conservative Schemes CN Explicit CN Explicit 110 110 110 110 51.6 (1.06) 51.5 (1.06) 51.6 (1.05) 51.4 (1.06) 22.2 (1.20) 22.1 (1.20) 22.2 (1.19) 22.1 (1.20) 7.58 (1.54) 7.56 (1.54) 7.57 (1.54) 7.56 (1.54) the nonlinear equation. Figures 2.12 highlights the poor performance of explicit schemes (for the nonlinear equation) when coupled with the NSTG scheme (for the advection equation). 1 1 NSTG Steady WENO FD 0.8 0.6 0.4 0.4 0.2 0.2 0 u u 0.6 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −10 −5 0 x 5 10 (a) Conservative implicit schemes NSTG Steady WENO FD 0.8 −1 −10 −5 0 x 5 10 (b) Non-Conservative implicit schemes Figure 2.11: Numerical solution of the convective Cahn-Hilliard equation. © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 1 1 NSTG Steady WENO FD 0.8 0.6 0.6 0.2 0 0 u 0.4 0.2 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −10 −5 0 x 5 NSTG Steady WENO FD 0.8 0.4 u 39 −1 10 (a) Conservative explicit schemes −10 −5 0 x 5 10 (b) Non-Conservative explicit schemes Figure 2.12: Numerical solution of the convective Cahn-Hilliard equation. 2.3.6.1 Solution properties of the convective Cahn-Hilliard equation (4.26) We will conclude this work by showing the conservation of mass and the coarsening property for the convective Cahn-Hilliard equation (2.3). Coarsening is a major property since it is the behavior of the physical process (spinodal decomposition) that the equation models [25, 69, 86] as time progresses. Example 2.7 We solve equation (4.26) with u(x, 0) = 0.5 cos x subject to periodic boundary conditions, for δ = 0.1 using 160 grid points and Algorithm 2.9 (see for example [17]). 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 u u The integration continues until the solution coarsens and below we present a sequence of figures (i.e., Figures 2.13 and 2.14) showing the behavior of the solution over time. −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −10 −5 0 x 5 10 −1 −10 (a) T = 1 (Formation of periodic structure) −5 (b) T = 50 Figure 2.13: Coarsening process. © University of Pretoria 0 x 5 10 Fractional splitting method for higher order equations 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 u u Chapter 2 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −10 −5 0 x 5 −1 10 (a) T = 320 40 −10 −5 0 x 5 10 (b) T = 640 (Kink-anti-kink pair) Figure 2.14: Coarsening process. For the conservation of mass, we have: Z Z 4π d 4π udx = ut dx, dt −4π −4π Z 4π u2 = (δ + (−γ 2 uxx + φ(u))x )x dx, 2 −4π2 u 4π 2 = 0. = δ + (−γ uxx + φ(u))x 2 −4π (2.55) The last equality is due to periodicity. 0.5 0.5 NSTG WENO FD 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 −0.4 −0.5 0 10 20 30 40 NSTG WENO FD 0.4 ∆ Mass ∆ Mass 0.4 50 −0.5 0 10 t 20 30 40 50 t (a) Conservative Implicit Schemes (b) Non-Conservative Implicit Schemes Figure 2.15: Conservation of mass . Figures 2.15 and 2.16 shows the conservation of the mass for all the schemes as expected. Another important property of the cCH equation (2.3) is transition from roughening to an orderless pattern. This is due to the fact that as the value of the driving force, δ, increases, © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 0.5 0.5 NSTG WENO FD 0.4 0.2 0.1 0.1 ∆ Mass 0.2 ∆ Mass 0.3 0 0 −0.1 −0.1 −0.2 −0.2 −0.3 −0.3 −0.4 −0.4 10 20 30 40 NSTG WENO FD 0.4 0.3 −0.5 0 41 −0.5 0 50 10 20 t 30 40 50 t (a) Conservative Explicit Schemes (b) Non-Conservative Explicit Schemes Figure 2.16: Conservation of mass. the convective term dominates the equation, consequently it behaves exactly like the famous Kuramoto-Sivashinsky equation. This can be verified by taking u → u/δ in equation (2.3), then take δ → ∞. This was noted by Golovin et al. [26] and Watson [86] among others. The former verified this property using the stationary and the traveling wave solution of equation (2.3) while the latter just remarked this behavior. Here, Figure 2.17, the solution profile of Example 2.7 at final computation time T = 80 with δ = 5, clearly shows this behavior. 30 25 x 20 15 10 5 0 0 20 40 t 60 80 Figure 2.17: Rough behavior of the convective Cahn-Hilliard equation. 2.4 Conclusion We have employed the fractional time splitting method to approximate the solutions of the K-S and cCH equations through the finite volume discretization. For the K-S equation, the deviation of each of the experimental solvers from the traveling wave solution as shown in © University of Pretoria Chapter 2 Fractional splitting method for higher order equations 42 Figure 2.4 reveals that all the schemes approximates the K-S solution. The NSTG is seen to be the closest to the traveling wave solution. The chaotic behaviour of the solution as expected was also replicated through this method (see Figure 2.5) and agrees with the observation of Yan and Chi-Wang [89]. We observed the exponential growth of the mean energy of the linear equation (2.35) in Figure 2.2(b) and the preservation of the mean energy of the hyperbolic equation (2.30) before the onset of shock in Figure 2.1. The poor performance of the Godunov and the implicit schemes are evident here. The performance of the Implicit schemes was also observed also in [91]. The behaviour of the mean energy of these two equations leads to the dissipation of the mean energy of the K-S equation shown in Figure 2.6 [24]. We have shown numerically the observations documented in [24, 63] in Figures 2.6 and 2.8. Our numerical experiments also reveal that the fractional splitting schemes’ approximation of the CH and cCH equations compares well both with existing result in the literatures as well as the exact steady state solution. This is obvious from the Figures 2.9 to 2.12 and Tables 2.6 to 2.9. In all these experiments, the consistency of the WENO scheme in approximating the solution of equation (2.3) is evident. We also validated the dissipation of the Ginzburg Landau energy functional and conservation of mass for the CH equation as well as conservation of mass for the cCH equation, shown in Figures 2.10, 2.15 and 2.16 respectively. In addition to these, Figures 2.13 and 2.14 demonstrates what happens when the mixture of substances in hot temperature is allowed to cool. The solution coarsens because the grains separates. In the presence of high driving force, convection dominates, hence the patterns form becomes rough. This is shown in Figure 2.17. These observations agrees with the physics of the equations as documented in [25]. © University of Pretoria Chapter 3 Nonclassical methods for singularly perturbed equations In this chapter, we employ the matched asymptotic expansion method and the nonstandard finite difference method to design two different finite volume schemes for the Schrödinger equation. Results from these schemes are reported in [4]. 3.1 Introduction We investigated second order singularly perturbed equations with the goal of designing accurate numerical schemes. Generally, these are equations of the type − ǫ2 u′′ (x) + b(x)u′ (x) + c(x)u(x) = f (x), (3.1) with ǫ ≪ 1. Equation (3.1) is a convection-diffusion equation if c = 0 or a diffusion-reaction equation if b = 0. The application of equations of type (3.1) cuts across fluid dynamics, investigating water quality in river networks, oil extraction simulations, thermodynamics, semiconductor device simulation, among others. A quick look at the equation below will open our eyes to what happens with these type of equations. Consider − ǫ2 u′′ (x) + u(x) = 1, x ∈ (0, 1), u(0) = u(1) = 0. (3.2) Setting ǫ = 0 gives u(x) = 1 which completely disagrees with the boundary conditions. The reduced problem therefore has no solution in C 2 [0, 1]. This suggests that the solution of equation (3.2) is not well behaved when ǫ becomes very small. Differential equations depending on small parameters whose solutions approach discontinuous limit as ǫ → 0 like the one above are termed singularly perturbed equations. Due to their applications, their analysis and numerical solution has attracted the attention of many researchers. As expected, 43 © University of Pretoria Chapter 3 Singularly perturbed equations 44 any discrete model applied to this equation yields ǫ-dependent solution. Hence, classical methods fail to compute the properties of the solution as ǫ approaches some critical value. Of practical interest, their solutions often contain layers; interior or boundary layers. The special case when b = f = 0 in equation (3.1) with non-zero Dirichlet boundary condition is of practical interest because of the oscillatory behaviour of the solution. This is evidence from the investigation of the equation of the harmonic oscillator in which c in equation (3.1) is constant. Obviously, the solution of such an equation is trigonometric with wavelength r ǫ2 . λ = 2π c Any difference-type scheme that will well approximate the solution to such an equation therefore, must be with grid size less than the wavelength. Consider a case when the parameter ǫ becomes very small, then such scheme will need much more smaller grid size, consequently it becomes computationally costly. Hence, recent attraction of research activities into providing both analytical and numerical approximation of singularly perturbed equations. Here, we will focus on the robustness of two schemes discussed in succeeding sections. 3.1.1 Non classical finite volume method This method combines the knowledge of analytical methods, like perturbation and matched asymptotic expansion, with the strength of numerical approximation. It is referred to as semi-analytical method in [37]. Our discussion here is based on the works of [36, 38, 70]. We will give a detailed description in Section 3.2.2. Jung [36] designed a finite element scheme for singularly perturbed reaction-diffusion equations which has boundary layer elements embedded in it to reduce the effect of the small parameter. Later, Jung and Temam [38] introduced some correctors based on asymptotic expansion into the finite volume scheme to obtain a second order scheme to solve stiff convection-diffusion models. They observed that their finite volume schemes are flexible to incorporate into the finite volume spaces. Recently, Jung and Nguyen [37] introduced some perturbation techniques into the finite volume scheme to resolve the oscillation due to the small parameter in convection-diffusion equations with turning points. 3.1.2 Nonstandard finite difference scheme Unlike the non classical method, this method derives its strength from the fact that the exact difference scheme of the differential equation is first derived and solved. The idea from the solution of the difference equation is then employed to correct the finite difference scheme of the differential equation. Through this, the denominator is usually changed to a special function of space and/or time step instead of just the grid size or time step. We will follow the results in [53, 54, 57, 58] and the literatures therein. Nonstandard finite difference scheme has been a very useful scheme in investigating differential equations because they © University of Pretoria Chapter 3 Singularly perturbed equations 45 are always designed to give exact schemes of the differential equations. Hence, their design for singularly perturbed equation always incorporate the small parameter into the step size. Mickens [56] applied the nonstandard finite difference scheme to solve the unsteady Schrödinger equation. He gave a comprehensive discussion on the derivation and application including the challenges faced when designing the scheme to solve several models of ordinary and partial differential equations. Recently, Lubuma and Patidar [53] employed the scheme to solve self-adjoint singularly perturbation problems. Later, Patidar and Sharma [65] extended the scheme to handle second order singularly perturbed differential-difference equations with delay. Lubuma and Patidar [54] considered some singularly perturbed problems with oscillatory solutions. They constructed and analysed an exact scheme based on the the works of Mickens. Additional advantages of this method includes preservation of positivity and monotonicity of solutions of differential models with uniform convergence. We will discuss the derivation and application of this method briefly in Section 3.2.3. 3.2 Numerical approach In this section we design three different finite volume schemes to approximate the solution of the equation ǫ2 u′′ + αu = 0, x ∈ (0, 1), (3.3) u(0) = α1 , u(1) = α2 . 3.2.1 Classical finite volume scheme Here we design the traditional classical finite volume (CFV) scheme for equation (3.3). Recall that in finite volume method, given the spatial domain x ∈ [0, 1] with xj = jh, j = 0, 1, 2, 3, · · · , m, h = ∆x numerical approximations are implemented at cell interfaces xj± 1 = xj ± h2 , j = 1, 2, 3, · · · , m − 1, and 2 uj+1 + uj . 2 2 Now, we multiply equation (3.3) by a test function χ(xj− 1 ,xj+ 1 ) (x) and integrate with respect 2 2 to x. By this we have Z x 1 j+ 2 2 ′ xj+ 12 uχ(xj− 1 ,xj+ 1 ) (x)dx = 0, ǫ u x 1 +α uj± 1 = j− 2 xj− 1 2 2 2 giving us x 1 ǫ2 u′ xj+ 21 + αhuj = 0. j− 2 We employ the interpolation functions [38] for discrete values uh (x) = m X j=0 uj χ(xj− 1 ,xj+ 1 ) (x), 2 2 © University of Pretoria (3.4) Chapter 3 Singularly perturbed equations and ′ u (x) ∼ ∇h uh = so that 46 m X uj+1 − uj h j=0 χ(xj ,xj+1 ) (x), (3.5) m−1 X uj+1 − uj um+1 − um u1 − u0 χ(x 1 ,x1 ) (x) + χ(xj ,xj+1 ) (x) + χ(xm ,xm+ 1 ) (x). u (x) ∼ 2 2 h h h j=1 ′ The boundary values of the variable u are approximated and ghost points are eliminated using the given boundary conditions. Bearing these in mind, the equations above can be written as σj+1 uj+1 + σj uj + σj−1 uj−1 = zj , where σj are some coefficients of discrete unknowns uj and zj = 0, j = 1, 2, 3, · · · , m − 1. Therefore, we have a system of linear equations of the form σu = Z to solve where σ is a tridiagonal matrix with entries σj−1 = σj = −2ǫ2 h σj+1 = ǫ2 , h + αh, ǫ2 h (3.6) , and Z is a modification of zj via the boundary correction. Remark 3.1 The boundary correction in the above scheme is direct if the boundary condition is Dirichlet. It becomes much involved when dealing with Neumann boundary condition or 2 mixture of the two type of boundary condition. For the case above Z1 = z1 − α1 ǫh while 2 Zm−1 = zm−1 − α2 ǫh . Remark 3.2 The above scheme as documented in literatures is deficient in handling singularly perturbed differential equations most especially in the presence of boundary layer, interior layer or when the solution is strongly oscillatory, see [8, 37, 38]. Hence, the introduction of the semi analytical schemes. 3.2.2 Non classical finite volume method We discuss the derivation of the non classical finite volume method. Firstly we briefly introduce the known analytical methods for some singularly perturbed differential equations. By so doing we will familiarize ourselves with the tools that are essential for the method under discussion. In particular, we discuss the following: 1. Perturbation method, 2. Matched asymptotic expansion (MAE) and 3. Wentzel-Kramers-Brillouin (WKB) method. © University of Pretoria Chapter 3 3.2.2.1 Singularly perturbed equations 47 Perturbation method Even though this method is mainly designed for regular perturbation equation, it is a good point to start for better understanding of the new method. Recall that a regular perturbation problem is one in which the solution remains valid even if the small parameter ǫ → 0. A possible example is the eigenvalue problem for the vertical displacement, u(x), of an elastic string with variable density u′′ + γ 2 [1 + ǫµ]u = 0, (3.7) where µ(x) is positive and continuous. The solution u(x) and the eigenvalues γ are taken as functions of ǫ. The regular perturbation assumes u ∼ u0 + ǫu1 , γ ∼ γ0 + ǫγ1 . We can put these ansatz into equation (3.7) and equate terms with the same power of ǫ. Inserting the ansatz into the equation, we have u′′0 + ǫu′′1 + (γ02 + 2ǫγ0 γ1 )(1 + ǫµ(x))(u0 + ǫu1 ) = 0, which after expansion and equating the coefficients of the powers of ǫ gives O(1) → u′′0 + γ02 u0 = 0, u0 (0) = u0 (1) = 0, O(ǫ) → u′′1 + γ02 u1 = 0, u1 (0) = u1 (1) = 0. (3.8) From the equation above, it is obvious that u0 = A sin(γ0 x), γ0 = jπ where j ∈ Z. Remark 3.3 Regular perturbation technique is not always uniformly valid for singularly perturbed equation most especially near boundary layer regions. 3.2.2.2 Matched asymptotic expansion The origin of this method is credited to the works of Prandtl on boundary layer problems, see [33] and the literatures therein. It was a source of breakthrough in the study of singularly perturbed differential equations. The algorithm follows three major steps. 1. Obtain the outer expansion (at least the leading order term). 2. Determine the boundary layer region and obtain the inner expansion. 3. Match the two approximations. This is illustrated with the following example. © University of Pretoria Chapter 3 Singularly perturbed equations 48 Example 3.1 Consider the boundary value problem ǫu′′ + 2u′ + u3 = 0, x ∈ (0, 1), u(0) = 0, u(1) = 12 . (3.9) We will determine the composite expansion of equation (3.9) by the matched asymptotic expansion method [33]. In this regard, assume the two term expansion u = u0 + ǫu1 and substitute this into (3.9). This leads to ǫ(u0 + ǫu1 )′′ + 2(u0 + ǫu1 )′ + (u0 + ǫu1 )3 = ǫ(u0 + ǫu1 )′′ + 2(u0 + ǫu1 )′ + u30 + 3ǫu20 u1 = 0. This yields, O(1) → 2u′0 + u30 = 0, O(ǫ) → 2u′1 + 3ǫu20 u1 = −u′′0 , u0 (0) = 0, u0 (1) = 12 , u1 (0) = 0, u1 (1) = 0. (3.10) Equation (3.10)1 is a first order differential equation which only needs one boundary condition. The fact that we have two boundary conditions means there will be issues (say, boundary layer) at one of the boundaries. We will use the right hand boundary condition and assume there is boundary layer at x = 0. With this, the leading order solution is u0 = √ 1 . 2x + 2 Since we assumed that there is a boundary layer at x = 0, we introduce the transformation x̃ = ǫxα at this boundary, where α > 0 will be determined in the succeeding analysis. The transformation above is referred to as stretching transformation. By this transformation, we have dx̃ d d = . dx dx dx̃ Suppose, U(x̃) is the solution we sought for at the boundary layer, then we have ǫ1−2α U′′ (x̃) + 2ǫ−α U′ (x̃) + U3 (x̃) = 0, U(0) = 0. (3.11) Also, we assume U ∼ U0 (x̃) + ǫγ U1 (x̃), · · · for γ > 0. Putting this into equation (3.11) we have ǫ1−2α (U0 + · · · )′′ + 2ǫ−α (U0 + · · · )′ + (U0 + · · · )3 = 0. (3.12) If we assume that the first and the second term are of the same order of ǫ while the third term is of higher order, then α = 1 which implies that the first and the second terms are of order O(ǫ−1 ) while third term is of order O(1). Hence, we need to solve O( 1ǫ ) → U0′′ + 2U′0 = 0, U0 (0) = 0. © University of Pretoria Chapter 3 Singularly perturbed equations 49 The solution of this equation is A(1 − exp(−2x̃)) where A is a constant. Before matching these solutions, we need to find the value of A. This is achieved by assuming that the solution of the boundary layer equation outside the layer and the value of the outer solution as one approaches the boundary layer should be equal, i.e. u0 (0) = U0 (∞). Hence, A = √12 . Now, we can determine the composite expansion which is the required solution by adding the outer and the inner solution and subtracting the region common to both, 1 2x 1 . (3.13) − √ exp − u = u0 (x) + U0 (x̃) − u0 (0) = √ ǫ 2x + 2 2 Even though this method is outlined for a boundary layer problem, it is also applicable to interior layer, boundary layer or corner layer problems. Remark 3.4 It is note worthy here that most of the solutions by the match asymptotic expansion method ended up having an exponential dependence on the boundary layer coordinate. 3.2.2.3 WKB method This method which supposedly originated from the work of the trio Wentzel, Kramers and Brillouin assumes the exponential dependence on the boundary layer coordinate from the beginning, see [33]. It has been a successful tool in solving problems arising from quantum and solid mechanics, for instance, the Schrödinger equation. As done for the previous methods, we will outline the method by employing it to solve the example below. Example 3.2 Approximate the solution of ǫ2 u′′ − f (x)u = 0, (3.14) analytically by the WKB method. Suppose that f (x) = f˜ is a constant function, we know that equation (3.14) will admit a general solution q q 1 ˜ 1 ˜ f x + β exp fx . (3.15) u(x) = α exp − ǫ ǫ In the WKB method, the solution (3.15) is generalized to solve equation (3.14). The ansatz here is θ(x) u(x) ∼ e ǫα (u0(x) + ǫα u1 (x) + · · · ) . (3.16) We will need the second derivative of the assumed solution given above and then substitute it into equation (3.14), equate the coefficients of the powers of ǫ, solve reduced equations and hence, the entire equation. By ideas from calculus, θ (3.17) u′ ∼ e ǫα ǫ−α θ′ u0 + u′0 + θ′ u1 + · · · , and θ u′′ ∼ e ǫα ǫ−2α θ′2 u0 + e−α (θ′′ u0 + 2θ′ u′0 + θ′2 u1 ) + · · · . © University of Pretoria (3.18) Chapter 3 Singularly perturbed equations 50 Putting equations (3.16) and (3.18) into equation (3.14) we have ǫ2 ǫ−2α θ′2 u0 + e−α (θ′′ u0 + 2θ′ u′0 + θ′2 u1 ) + · · · − f (x) (ǫα (u0 (x) + ǫα u1(x) + · · · )) = 0. (3.19) Interestingly, the exponential term dropped out. Following the same procedure as we did in the last section, we determine that α = 1. With this, we have the eikonia equation O(1) → (θ′ )2 = f (x), which have the solution θ(x) = ± Z x p (3.20) f (s)ds. (3.21) The first term in the expansion is determined by solving the transport equation O(ǫ) → θ′′ u0 + 2θ′ u′0 + θ′2 u1 = f (x)u1 . (3.22) Equation (3.21) reduces this to O(ǫ) → θ′′ u0 + 2θ′ u′0 = 0, (3.23) which has the solution c u0(x) = √ . θ′ Therefore, the solution of equation (3.14) is Z Z 1 1 xp − 41 f (s)ds + β exp u ∼ f (x) α exp − ǫ ǫ x p f (s)ds . (3.24) The methods outlined above are all analytical and easy to apply when treating linear equations with O(ǫ) or greater. A more realistic approximation will be obtained if one can consider terms with O(ǫ2 ) or less. But this may prove really difficult and handling nonlinear equations may present more difficult challenges. This is one of the reasons we may have to appeal to numerical studies of these equations. This is non trivial since, we already know that it is always costly employing standard numerical schemes for this type of equations. Therefore, we will employ our ideas of analytical methods when constructing the numerical schemes, see [37]. To this extent, we will dedicate the next section to the study of corrector methods as introduced and discussed in [70]. The non classical method builds on the idea of the method of matched asymptotic expansion that was explained in Section 3.2.2.2. We will motivate the method by an example also. Example 3.3 Investigate the solution of the equation L := −ǫu′′ + b(x)u′ + c(x)u = f (x), x ∈ (0, 1), (3.25) u(0) = u(1) = 0, c(x) ≥ 0, b′ (x) 6= 0, x ∈ [0, 1], (3.26) given that b, c, f are smooth functions and 0 < ǫ ≪ 1. © University of Pretoria Chapter 3 Singularly perturbed equations 51 It is evident that we can approximate the solution of equations (3.25) and (3.26) by the method of MAE as outlined above in which case we need to first find the outer expansion, say uas . Except in the presence of boundary layer, this will be a well behaved solution of the equation. So let, m X uas ∼ ǫi ui (x), (3.27) i=0 ui is to be determined. Following the same procedure as in Section 3.2.2.1 and putting the ansatz (3.27) into (3.25) leads to O(1) → bu′0 + cu0 = f, O(ǫi ) → bu′i + cui = u′′i−1 , f or i = 1, · · · , m. (3.28) Now, given some other conditions, we will be able to determine the values of the unknowns [u0 , ui ] from these equations. To find u0 we need only one boundary condition, but we have two, so we are faced with the challenge of choice making. To be able to make the correct choice of the boundary condition, an appeal is made to cancelation law [70]. Remark 3.5 The location of the boundary layer depends on the behaviour of b. It will be towards the right hand boundary given a positive b and at the left hand boundary otherwise. For our case, because of the definition of the problem, it will be located at x = 1. Therefore, we employ only the boundary value at x = 0 to define u0 . Hence, uas does not approximate the solution of equation (3.25) because it is deficient at x = 1. Hence we focus on determining the reduced solution, u0 , from the equations O(1) → bu′0 + cu0 = f, u0 (0) = 0, i ′ ′′ O(ǫ ) → bui + cui = ui−1, ui (0) = 0 f or i = 1, · · · , m. (3.29) There is need for a local corrector of the solution uas as we approach x = 1. To this end, we introduce a stretching transformation at x = 1, y= 1−x , 0 < δ ≪ 1. δ Suppose L = ǫL1 + L0 where L0 v := bv ′ + cv, then we want to choose δ so that ǫL1 and L0 will have the same order with respect to ǫ after transformation from x to y , i.e., ǫδ −2 = δ −1 , which gives δ = ǫ. To continue, we have to expand the functions b(x) = b(1 − ǫy) and c(x) = c(1 − ǫy) by the Taylor’s expansion thus, ∞ X bi ǫi y i , with, b0 = b(1), i=0 © University of Pretoria Chapter 3 Singularly perturbed equations ∞ X 52 ci ǫi y i , with, c0 = c(1). i=0 Based on these expansions, the boundary layer equation to be solved is ∞ ∞ 1 d2 u 1 X i i du X i j − + − bi ǫ y ci ǫ y u = 0. ǫ dy 2 ǫ i=0 dy i=0 (3.30) Suppose, we have a local expansion for u(y) uloc (y) = m+1 X ǫl θl (y). (3.31) l=0 Substituting this into equation (3.30) and equating the powers of ǫ we have T0 θl = − where l X i=1 T0 θ0 = 0, Ti θl−i , l = 1, · · · , m + 1, (3.32) (3.33) d d2 − b , 0 dy 2 dy d T1 := −b1 y + c0 . dy It appears now that we have to solve second order differential equations which require two boundary conditions each for uniqueness. Since we are focussing on the boundary at x = 1, we need a boundary condition here so that the error of our approximation will be zero, then θl (0) = −ul (1). The second boundary condition will be based on the fact that we need the boundary layer solution localized to this boundary alone, hence we need limy→∞ θl (y) = 0. The MAE approximation is m m X X 1−x i k . (3.34) uas = ǫ ui (x) + ǫ ul ǫ i=0 T0 := − l=0 Remark 3.6 It is the idea of the corrector here that was employed by Jung [36] and Jung and Nguyen [37] in their work. The same procedure will be employed here to solve two equations. 3.2.3 Non standard finite volume method In this section we will investigate some singular perturbed second order ordinary differential equations (ODEs) in order to design more accurate scheme to solve them. As stated in Section 3.2, we will investigate equations of the form ( −ǫ2 u′′ (x) − g(x)u = 0, x ∈ [0, 1], (3.35) u(0) = α, u(b) = β. Our discussion in this section is based on the works that were referred to in Section 3.1.2. © University of Pretoria Chapter 3 3.2.3.1 Singularly perturbed equations 53 Nonstandard finite difference schemes for first order differential equations For better understanding, we derive the scheme for a simple first order decay differential equation du = −γu. (3.36) dx Integrating equation (3.36), we have its general solution as u = u0 e−γx , u0 = u(0). This means the corresponding difference equation to (3.36) is uj+1 = uj e−γh . Hence, uj+1 − uj = uj (e−γh − 1), and the nonstandard scheme for (3.36) is uj+1 − uj 1 − e−γh = −γuj , φ(h) = . φ(h) γ (3.37) The replacement of the space width h in the denominator of the finite difference scheme by the function φ(h) as in equation (3.37) such that lim φ(h) ∼ h, h→0 gives a significant difference between the classical difference schemes and the NSFD. 3.2.3.2 Nonstandard finite difference schemes for second order differential equations Here, we derive the denominator function φ(h) for equation (3.35) in the standard form u′′ ± g(x)ǫ u = 0, g(x)ǫ = g(x) . ǫ2 (3.38) Assuming g is constant, for simplicity, we know that the two linearly independent solution of (3.38) are √ (3.39) u1 = exp(ıgx), u2 = exp(−ıgx), ı = −1. √ √ φ exp(−ı gǫ hj) exp(ı gǫ hj) j √ √ (3.40) φj+1 exp(ı gǫ h(j + 1)) exp(−ı gǫ h(j + 1)) = 0. φj+2 exp(ı√gǫ h(j + 2)) exp(−ı√gǫ h(j + 2)) © University of Pretoria Chapter 3 Singularly perturbed equations 54 From equation (3.40), it is evident that √ uj (eı gǫ h √ − e−ı gǫ h √ ) − uj+1(e2ı gǫ h √ − e−2ı gǫ h √ ) + uj+2(eı gǫ h √ − e−ı gǫ h ) = 0, which by the trigonometric identities 2 sin(θ) = eıθ − e−ıθ , gives √ uj − 2uj+1 cos( gǫ h) + uj+2 = 0. (3.41) The interesting feature here is the common behaviour of the solution νj = Aeıγxj + Be−ıγxj of both difference equation (3.41) and the constant coefficient equation (3.38) at the grid point xj . Recalling that θ 2 , cos(θ) = 1 − 2 sin 2 after an index shift, equation (3.41) finally gives the nonstandard finite difference scheme for the equation (3.39) as √ g uj−1 − 2uj + uj+1 2ǫ + g (x)u = sin h . (3.42) 0, φ(h) = √ ǫ j φ(h)2 g 2ǫ Now, for g = g(x), equations (3.42) gives uj−1 − 2uj + uj+1 2ǫ + gǫ (xj )uj = 0, φ(h) = p sin 2 φ(h) g(xj ) p ! g(xj ) h . 2ǫ (3.43) On a domain discretized into m + 1 grid points j = 0, 1, 2, 3, · · · , m, the above scheme will result into a system of linear equations of the form Au = Z, where the sparse matrix A has the entries 1 , Aj−1 = − φ(h)2 2 Aj = − gǫ (xj ), j = 1, 2, · · · , m, φ(h)2 1 , Aj+1 = − φ(h)2 (3.44) (3.45) (3.46) and Z1,2,··· ,m = [− u0 um , 0, 0, 0, · · · , − ], φ(h) φ(h) where u0 and um are boundary values. It has been shown that the equation (3.64) when solved subject to Dirichlet boundary condition such that u(0), u(1) ≥ 0 satisfies the maximum principle and that the schemes (3.43) is qualitatively stable [54]. That is, the difference equation or its solution replicates the properties of the original equation. © University of Pretoria Chapter 3 3.2.3.3 Singularly perturbed equations 55 Non standard finite finite volume scheme The new finite volume scheme to be designed will be an adaptation of the nonstandard finite difference scheme discussed above into the classical finite volume scheme. Comparing the classical finite volume scheme with (3.43), the denominator of the nonstandard finite volume scheme is the function ! p Z x 1 j+ 2 fǫ (xj ) 1 2 sin h , fǫ = 2 g(x)dx. (3.47) φ(h) = p 2 hǫ x 1 fǫ (xj ) j− 2 This makes sense since, limh→0 φ(h) ∼ h. Then, we will be solving system of linear equations in the form giving in equation (3.44) where 1 , φ(h)2 2 + fǫ (xj ), j = 1, 2, · · · , m Aj,j = − φ(h)2 1 , Aj,j+1 = φ(h)2 Aj,j−1 = (3.48) (3.49) and Z1,2,··· ,m = [− um+1 u0 , 0, 0, 0, · · · , − ]. φ(h) φ(h) We discuss the application of these method to three different problems in the sequel. 3.3 Numerical experiments Here we apply the three schemes designed above to equation (3.50) below Example 3.1 ǫ2 u′′ + αu = 0, x ∈ (0, 1), u(0) = 1, u(1) = 0. (3.50) The boundaries are corrected through u 1 = 1, um+ 1 = 0, 2 2 and the interpolation functions. The numerical approximations are compared with exact solution ) sin( 1−x ǫ . u(x) = 1 sin( ǫ ) © University of Pretoria Chapter 3 Singularly perturbed equations 56 The non classical method follows some algorithms to correct the solutions at the points where the classical method fails using the idea of analytical methods. Therefore, we study the formal perturbation approximation of the equation under investigation. Put uper ∼ ∞ X ǫi ui , i=0 into equation (3.50) and equate the orders of powers of ǫ. This leads to the equations O(1) → O(ǫ) → u0 = 0, u0 (0) = 1, u0(1) = 0, (3.51) u1 = 0, u1 (0) = 0, u1(1) = 0, (3.52) ui (0) = 0, ui (1) = 0. (3.53) ui = ′′ , − α1 ui−2 Let u∼ then the boundary layer equation is with the leading order equation ∞ X ǫi θi (y), i=0 X ∞ d2 +α ǫi θi = 0, 2 dy i=0 d2 θ0 + αθ0 = 0. (3.54) dy 2 This being a second order equation requires two boundary conditions. As discussed in Section 3.2.2, if α = 1, the solution is oscillatory, hence we solve this equation subject to θ0 (0) = −1, θ0 (1) = 0. The solution of equation (3.54) is x x 1 − cos . (3.55) sin θ0 = cot ǫ ǫ ǫ Now, we will outline the design of the non-classical finite volume scheme. We seek the solution of the target equation in the form u = uc + γθ0 , where θ0 is the corrector just discussed above and uc will be discretized with finite volumes, while γ ∈ R will be determined alongside the unknown solution, see [38]. Therefore, the desired solution is ũ = γθ0 + uh , uh = m X uj χ(xj− 1 ,xj+ 1 ) (x), (3.56) ′′ (3.57) 2 j=1 2 ǫ2 uc + αuc = 0. The boundary condition for this new equation is dependent on the original boundary and the behaviour of the corrector at the boundary. Therefore, uc (0) = u(0) − γθ0 (0) = 1 + γ, uc (1) = u(1) − γθ0 (1) = 0. © University of Pretoria (3.58) Chapter 3 Singularly perturbed equations 57 Remark 3.1 It is evident here that if γ = 0, then we will recover the classical finite volume approximation. Obviously, we need an additional equation to cater for the unknown parameter γ for the existence of a unique solution to the expected system of discretized equations. To this extent, we will multiply equation (3.50) by a test function θ0 χ[0,x 3 ) and integrate by part 2 over (0, 1). This leads to x 3 ǫ u θ0 0 2 − ǫ2 2 c′ Z x3 2 c′ θ0 u dx + α 0 Z x3 2 uc dx = 0. (3.59) 0 At this point, we introduce the discrete interpolation functions for u and u′ as discussed above to replace uc and recall that by the boundary condition uc0 = 1 + γ. Then, ′ uc (x 3 ) = 2 uc2 − uc1 , h and uc − 1 − γ uc1 − uc0 = . u = h h Putting all these together we have a system of linear equations c′ σ0,0 γ + σ0,1 uc1 + σ0,2 uc2 = 0, (3.60) to solve. Here, σ0,0 σ0,1 σ0,2 Z x1 ǫ2 ′ θ0 (0) − θ0 dx , = h 0 Z x1 Z x2 ǫ2 ′ −(θ0 (x 3 ) + θ0 (0)) − ( = − )θ0 dx + hα, 2 h 0 x1 Z x2 Z x3 ǫ2 ′ θ0 (x 3 ) + ( = − )θ0 dx . 2 h x1 x2 (3.61) We will move straightforward to design scheme for equations (3.57) putting in mind the interpolation functions earlier discussed. Of course, we will solve a system of linear equations in the form σu = Z, where the sparse matrix σ has the entries ǫ2 σj,j−1 = − , h 2ǫ2 σj,j = − hα, j = 1, · · · , m, h ǫ2 σj,j+1 = − , h © University of Pretoria (3.62) Chapter 3 Singularly perturbed equations 58 2 and Zj = 0 for j = 2, · · · , m while after boundary correction Z1 = − ǫh . Finally, the system to be solved is Z x1 ǫ2 ′ θ0 (0) − σ0,0 = θ0 dx , h 0 Z x1 Z x2 ǫ2 ′ −(θ0 (x 3 ) + θ0 (0)) − ( − )θ0 dx + hα, σ0,1 = 2 h 0 x1 Z x2 Z x3 ǫ2 ′ θ0 (x 3 ) + ( σ0,2 = − )θ0 dx , 2 h x1 x2 ǫ2 σj,j−1 = , h 2ǫ2 σj,j = − + hα, j = 1, · · · , m, (3.63) h ǫ2 σj,j+1 = , h The performance of the two schemes (3.6) and (3.63) are compared in Figures 3.1 and 3.2. 1 2 Exact NCFV CFV 0.8 Exact NCFV CFV 1.5 1 0.5 0.6 0 0.4 −0.5 −1 0.2 −1.5 0 0 0.2 0.4 0.6 0.8 1 −2 0 (a) Simulation at ǫ = 1 0.2 0.4 0.6 0.8 1 (b) Simulation at ǫ = 0.1 Figure 3.1: Comparison of the solution of the equation (3.50) as computed by the classical scheme and the new scheme on 10 grids and compared with the exact solution on 200 grids. We also compare the approximation of the classical and the nonstandard finite volume schemes. The error computations are shown in Tables 3.1 and 3.2. The profile comparison are given in Figures 3.3 and 3.4. 3.3.1 Schrodinger equation In this section we will apply the three schemes discussed in Section 3.2 to approximate the solution of the Schrodinger equation. © University of Pretoria Chapter 3 Singularly perturbed equations 59 2 Exact NCFV CFV 1.5 1 0.5 0 −0.5 −1 −1.5 −2 0 0.2 0.4 0.6 0.8 1 Figure 3.2: Comparison of the solution of the equation (3.50) as computed by the classical scheme and the new scheme on 40 grids and compared with the exact solution on 200 grids when ǫ = 0.01. Table 3.1: Error values of the nonstandard scheme and the classical scheme for different values of ǫ Grid points L∞ Norm error ǫ=1 ǫ = 0.1 ǫ = 0.01 5 16 15 CFV×10 NSFV×10 CFV NSFV×10 CFV NSFV×1013 10 6.577 3.330 0.8186 4.4408 1.9626 910.3 20 1.650 5.551 0.2938 3.8580 1.9743 14.21 40 0.412 15.54 0.0806 18.20 1.9743 1.043 80 0.103 18.87 0.0208 37.74 3.0325 22.64 Table 3.2: Error values of the nonstandard scheme for different values of ǫ Grid points L∞ Norm error −3 13 ǫ = 10 (×10 ) ǫ = 10−4 (×1012 ) ǫ = 10−5 (×108 ) 10 1.165 2.695 1.137 20 1.150 2.819 1.138 40 94.48 2.684 1.137 80 1.066 2.864 1.138 Example 3.2 Solve the Schrödinger equation 2 ′′ −ǫ u (x) − q(x)u(x) = 0, 0 < x < 1, ǫu′ (0) + ıp(0)u(0) = 2ıp(0), ǫu′ (1) = ıp(1)u(1), where p(x) = p q(x) > 0 and 0 < ǫ ≪ 1. © University of Pretoria (3.64) Chapter 3 Singularly perturbed equations 2 2 NCFV Exact CFV 1.5 60 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −2 0 0.2 0.4 0.6 0.8 NCFV Exact CFV 1.5 −2 0 1 (a) Simulation on 20 grids with ǫ = 0.1 0.2 0.4 0.6 0.8 1 (b) Simulation on 80 grids with ǫ = 0.1 Figure 3.3: Comparison of the solution of equation (3.50) by the schemes (3.6) and (3.47) 5 2 NCFV Exact CFV NCFV Exact 1.5 1 0.5 0 0 −0.5 −1 −1.5 −5 0 0.2 0.4 0.6 0.8 −2 0 1 (a) Simulation on 20 grids with ǫ = 0.45 × 10 −1 0.2 0.4 0.6 0.8 1 (b) Simulation on 20 grids with ǫ = 1 × 10−2 Figure 3.4: Comparison of the solution of equation (3.50) by the schemes (3.6) and (3.47) It is well known that equation (3.64) is strongly oscillatory and even though appears simple, the presence of the small parameter ǫ has made it a very interesting problem over decades. Also, the relevance of the equation in quantum physics modeling is another reason for gaining attention over the years. Therefore, a lot of numerical analysis and simulation have been done on (3.64) based on finite difference (both classical and adaptive mesh), finite volume, finite element and WKB methods (for example see [8] and literatures therein). Perturbation methods have been very useful in the analysis of equations of this type, most especially, the WKB method, see [33]. Lately, numerical methods have been combined with such perturbation methods. For instance, WKB combined with or used as basis for finite element methods [59, 68]. It should be noted that effort to solve this equation are directed towards the design of cost effective scheme in term of computing memory usage and computing time. © University of Pretoria Chapter 3 Singularly perturbed equations 61 Here, we will employ the classical, non classical and the nonstandard finite volume method as described above to solve (3.64). 3.3.1.1 Classical finite volume scheme for equation (3.64) We follow the same approach as listed for the classical scheme in Section 3.2. Hence, for all j ′ s, the classical scheme for equation (3.81) is Z x 1 j+ 2 uj−1 − 2uj + uj+1 bdx = 0. − uj (3.65) −ǫ h x 1 j− 2 The ghost points are taken cared of through the boundary conditions. The discretized form of the boundary conditions are ǫ(u 1 − u− 1 ) + ıp(0)u0 = 2ıp(0), (3.66) ǫ(um+ 1 − um− 1 ) − ıp(1)um = 0, (3.67) 2 2 at x = 0 and 2 2 where the cell averages are approximated via the interpolation functions given in Section 3.2. 3.3.1.2 Non classical finite volume scheme for equation (3.64) In order to resolve the in accurateness in the approximation given by the classical scheme, we design here a non classical scheme. as we did for the first example we appeal to the perturbation analysis of equation (3.64). The formal outer expansion based on the ansatz u= ∞ X ǫl ul , l=0 gives on substitution into (3.64) ( u0(x) = 0, q(x)ul (x) = −u′′l−1 (x), l > 0. (3.68) At x = 0 we have that ∞ X l+1 ǫ ul (0) + ıp(0) l=0 ∞ X l ǫl ul (0) = 2ıp(0), l=0 ǫ ul−1 (0) + ıp(0) l=1 ∞ X ∞ X ǫl ul (0) + ıp(0)u0 (0) = 2ıp(0), l=1 ( u0 (0) = 2, 1 ′ ul (0) = − ıp(0) ul−1(0), l > 0. © University of Pretoria (3.69) Chapter 3 Singularly perturbed equations Also, at x = 1 ∞ X l=0 ∞ X 62 ǫl+1 ul (1) − ıp(1) ǫl ul−1 (1) + ıp(1) l=1 ∞ X ∞ X ǫk ul (1) = 0, l=0 ǫk ul (1) + ıp(1)u0(1) = 0, l=1 ( u0 (1) = ul (1) = 0, 1 u′ (1), ıp(1) l−1 l > 0. (3.70) From equations (3.68), (3.69) and (3.70) it is observed that u0 does not agree with the boundary condition u0(0) = 2 while it satisfies u0 (1) = 0. This suggests there will be boundary layer issue at the boundary x = 0. In order to resolve this, following [38], we carry out an inner expansion at x = 0. We employ a formal asymptotic expansion at this boundary layer and Taylor expansion for q(x) at x = 0, u(x) ∼ and P∞ l=0 q(x) ∼ th ǫl θl (x̄), x̄ = xǫ , ∞ X ql x̄l l=0 l! , (3.71) (3.72) where ql is the l order derivative of q. Putting both (3.71) and (3.72) into (3.64), we have the leading order equation as − θ0′′ (x̄) − q0 θ0 (x̄) = 0. (3.73) Remark 3.2 The equation (3.73) is also a leading order equation near x = 0 for ǫ2 θ′′ (x) + ql θ(x) = 0. (3.74) We now apply the WKB method to obtain the solution of (3.74). Note that our usage of the WKB approximation here is not the same as it was used in [1, 8, 59]. Here, the solution will only be employed as a corrector at the boundary. Using the ideas from [33], the first order WKB approximation to equation (3.74) is R √ R √ 1 x 1 x 1 q(s)ds q(s)ds + α1 eı ǫ . (3.75) θ(x) ∼ −q(x) 4 α0 e−ı ǫ We determine the values of α0 and α1 via boundary conditions. Now, following the same perturbation in the solution as done in the last problem, the boundary conditions of the Schrödinger equation becomes ( ǫu′ (0) + γθx (0) + ıp(0)u(0) + ıp(0)γθ(0) = 2ıp(0), (3.76) ǫu′ (1) + γθx (1) = ıp(1)u(1) + ıp(1)γθ(1). © University of Pretoria Chapter 3 Singularly perturbed equations 63 Now, we will multiply the Schrödinger equation by the function θχ[0, x 3 ] and integrate over 2 (0, 1). This leads to the equation Z x3 Z x3 x3 2 2 2 2 2 ux θx dx − θqdx = 0, − ǫ ux θ |0 +ǫ (3.77) 0 0 which we write in the form σ00 γ + σ01 u1 + σ02 u2 = 0, (3.78) where, employing interpolation functions (3.4) and (3.5), we have θx (a)h ıp(0)θ(0)h ǫ2 , + σ00 = θ(x1 ) h αǫ αǫ ! Z x3 αǫ + 1 + ıp(a)h ǫ2 2 2ǫ σ01 = θ(x 3 ) + θ(x1 ) − qθdx, 2 h αǫ 0 Z x5 ǫ2 2 θ(x 3 ) − 2θ(x2 ) + θ(x1 ) − qθdx, σ02 = − 2 h x3 2 where ıp(a)h − 1. 2ǫ This is done in order to find an extra equation through which the value of the extra parameter γ will be determined. Note that we have made use of the interpolation functions defined earlier in the last problem. Substituting α= ũ = uh + γθ, (3.79) into the Schrödinger equation (3.64) and integrating over each cell, we have σj,j−1uj−1 + σj,j uj + σj,j+1 uj+1 = Z̃j , © University of Pretoria (3.80) Chapter 3 Singularly perturbed equations 64 where Z̃ is zero everywhere except at j = 1 with Z̃1 = σ1,0 ǫ2 =− h σ1,1 = ǫ2 h 2ıp(0)hǫ2 h( ıp(0)h −1)ǫ 2ǫ and h ′ h θ (0) + θ(0) , ǫ ǫ ! ıp(0)h 1 + 2ǫ , 2 + ıp(0)h −1 2ǫ ǫ2 σ1,2 = − , h ǫ2 σj,j−1 = − , hZ xj+ 1 2 2ǫ 2 σj,j = − q(x), h x 1 j− 2 ǫ2 σj,j+1 = − , h ǫ2 σm,m−1 = − , h ! 2 1 + ıp(1)h ǫ 2 + ıp(1)h 2ǫ σm,m = . h −1 2ǫ © University of Pretoria (3.81) Chapter 3 Singularly perturbed equations 65 Finally, we will solve a linear system σu = Z̄ for u = [γ, u1 , u2 , · · · , um ] given Z̄ = [0, Z̃j ], where θx (a)h ip(0)θ(0)h ǫ2 , + σ00 = θ(x1 ) h αǫ αǫ ! Z x3 αǫ + 1 + ıp(0)h ǫ2 2 2ǫ σ01 = θ(x 3 ) + θ(x1 ) − qθdx, 2 h αǫ 0 Z x5 ǫ2 2 θ(x 3 ) − 2θ(x2 ) + θ(x1 ) − qθdx, σ02 = − 2 h x3 2 h ǫ2 h ′ θ (0) + θ(0) , σ1,0 = − h ǫ ǫ ! 1 + ıp(0)h ǫ2 2 + ıp(0)h 2ǫ , σ1,1 = h −1 2ǫ ǫ2 σ1,2 = − , h ǫ2 σj,j−1 = − , hZ xj+ 1 2ǫ2 2 σj,j = − q(x), h x 1 (3.82) j− 2 2 ǫ σj,j+1 = − , h ǫ2 σm,m−1 = − , h ! ıp(1)h 2 1 + ǫ σm,m = 2 + ıp(1)h 2ǫ . h −1 2ǫ Remark 3.3 The Scheme (3.81) gives the classical finite volume when γ = 0 otherwise, NCFV scheme. We ran the numerical experiment on this with q(x) = (x+1/2)2 ∀x ∈ [0, 1]. The convergence and the behavior of the solution for different values of ǫ are shown in the Figures 3.5 and 3.6. Remark 3.4 The rate of convergence is ∼ O(h1.56 ) for both schemes. When ǫ ≤ 0.02 error due to the classical scheme becomes significant and the rate of convergence becomes nonuniform. Remark 3.5 We observe uniform convergence rate of O(h1.56 ) with the new scheme while the convergence rate due to the classical scheme is not uniform, in fact it is ǫ−dependent. © University of Pretoria Chapter 3 Singularly perturbed equations 66 −2 10 log(error) log(error) 10 NCFV−num NCFV−best fit CFV−num CFV−best fit −3 10 10 10 0 −2 NCFV−num NCFV−best fit CFV−num CFV−best fit −4 −2 10 log(h) 10 −2 10 −1 log(h) (a) Simulation at ǫ = 2 × 10−1 (b) Simulation at ǫ = 2 × 10−2 Figure 3.5: The slope of the lines say r = ∆(log(error)) represents the convergence rate due ∆(log(h)) to each scheme. The approximation on 32, 64, 128, 256 elements grid are compared with a reference solution on 512 element grid. 1 NCFV CFV refined grid real(u) 0.5 0 −0.5 −1 0 0.2 0.4 0.6 0.8 1 x Figure 3.6: Comparison of the solution of the equation (3.64) between the classical scheme and the new scheme. Solution was computed with ǫ = 2 × 10−2 on 32 element grid. 3.3.1.3 Nonstandard finite volume scheme for equation (3.64) Here, we design the nonstandard finite volume schemes designed in the previous section to equation (3.64). We need to derive an exact scheme to handle the boundary equations. We therefore, follow the same procedure that was highlighted above for first order equations at the boundary. At the boundary x = 0, u′ + 2ıp(0) ıp(0) u= . ǫ ǫ © University of Pretoria (3.83) Chapter 3 Singularly perturbed equations 67 x), and integrate. This yields Multiply by the integrating factor, exp( ıp(0) ǫ 2ıp(0) ıp(0) ıp(0) d u exp( x) = exp( x), dx ǫ ǫ ǫ ıp(0) ıp(0) x) = 2 exp( x) + c, ǫ ǫ where c is an arbitrary constant of integration. This gives u exp( u = 2 + c exp(− ıp(0) x). ǫ c = φ(0) − 2 given that φ(0) is the initial condition. By this, the difference scheme will be uj = 2 + (uj−1 − 2) exp(− ıp(0) h). ǫ Then, ıp(0) h) − uj−1, ǫ ıp(0) ıp(0) uj − uj−1 = 2(1 − exp(− h)) − uj−1(1 − exp(− h)). ǫ ǫ Hence, the scheme at the boundary x = 0 is, uj − uj−1 = 2 + (uj−1 − 2) exp(− ǫ ǫ (1 ıp(0) uj − uj−1 + ıp(0)uj−1 = 2ıp(0). h)) − exp(− ıp(0) ǫ Following the same procedure at the boundary x = 1, we have ıp(1) u′ = , u ǫ ıp(1) x + c, ǫ c is just an arbitrary constant of integration. This gives ln(u) = u = Ae ıp(1)x ǫ , A is determined through the initial condition and we have u = u0 e ıp(1)x ǫ . Therefore, a corresponding general difference solution at this boundary will be uj+1 = uk e ıp(1)h ǫ Then, uj+1 − uj = uj (e . ıp(1)h ǫ − 1), © University of Pretoria (3.84) Chapter 3 Singularly perturbed equations 68 and exact difference scheme for the boundary condition at x = 1 is ǫ uj+1 − uj ıp(1)h ǫ (e ǫ ıp(1) − 1) = ıp(1)uj . (3.85) From the equation (3.84) and (3.85) the values of u−1 and um+1 are estimated. Now we design the scheme for x ∈ (0, 1). If we integrate the Schrödinger equation (3.64) and let Z x 1 j+ 2 q(x)dx, Fj = xj− 1 2 then the nonstandard finite volume scheme follows directly from Section 3.2. The entire scheme results into solving system of linear algebraic equations. The entries of the sparse tridiagonal matrix constructed is given below: exp( ıp(a) h) 2 ǫ + F1 , A1,1 = 2 − φ̃1 φ̃1 1 A1,2 = − 2 , φ̃1 1 Aj,j−1 = − 2 , φ̃j−1 2 Aj,j = 2 − Fj , φ̃j 1 Aj,j+1 = − 2 , φ̃j+1 1 Am,m−1 = − , φ̃2m Am,m (3.86) exp( ıp(b) h) 2 ǫ − + Fm , = 2 φ̃m φ̃m (3.87) where, r ! Fj 2ǫ . φ̃j = p sin h 4ǫ2 Fj Our numerical experiments and observations are shown in Table 3.3 and Figures 3.7 to 3.8. 3.4 Conclusion We investigated singularly perturbed second order ordinary differential equations via two different methods. Firstly, we employed the semi-analytical method as proposed in [37, 70] © University of Pretoria Chapter 3 Singularly perturbed equations 64 grids 128 grids 256 grids 32 grids 1 1 0.5 0.5 Real(u) Real(u) 32 grids 69 0 −0.5 −1 0 64 grids 128 grids 256 grids 0 −0.5 0.2 0.4 0.6 0.8 −1 0 1 0.2 0.4 x 0.6 0.8 1 x (a) NSFD Simulation (b) Classical Simulation 1 1 0.5 0.5 Real(u) Real(u) Figure 3.7: Convergence of the simulations with, ǫ = 1 × 10−2 and several grid choices. 0 −0.5 −1 0 0 −0.5 0.2 0.4 0.6 0.8 1 −1 0 0.2 x 0.4 0.6 0.8 1 x (a) NSFD Simulation (b) Classical Simulation Figure 3.8: Comparison of the solution of the Schrödinger equation (3.64) on 40 grid nodes (dots) and 512 grid nodes (solid line) with ǫ = 2 × 10−2 . Table 3.3: Error between the solution at different grid points solution at 256 grid points Grid points L∞ Norm error ǫ = 0.2 ǫ = 0.1 CFV NSFV CFV NSFV 16 0.1702 0.0407 0.4025 0.0412 32 0.0814 0.0204 0.2213 0.0278 64 0.0343 0.0091 0.1104 0.0151 128 0.0100 0.0031 0.0441 0.0056 © University of Pretoria as compared with a reference ǫ = 0.02 CFV NSFV 2.2942 1.9891 1.5878 0.1154 0.1578 0.0146 0.0207 0.0037 Chapter 3 Singularly perturbed equations 70 to solve equation (3.50). Figures 3.1 and 3.2 show the better performance of the non classical schemes even when ǫ is small with relatively small number of grids. This motivates us to extend the scheme to approximate the Schrödinger equation which is more complicated because the boundary condition is defined on a complex plane. Figure 3.5 shows the convergence rate obtained for different grids when ǫ = 0.2 and 0.02 using the solution at 128 grid points as reference solution. It is clearly seen that the new scheme converges better than the classical scheme. Also, the approximation at 16 grids compared with the reference solution when ǫ = 0.02 proves a better performance of the new scheme as shown in Figure 3.6. The second part of the chapter employed a non standard finite volume scheme to the same equations as above. The table of errors, Tables 3.1 and 3.2, reveals a better performance of this new scheme as compared with the classical scheme for very small ǫ. This is also verified in Figures 3.3 and 3.4. This non standard scheme approximates the solution almost exactly. When this scheme is applied to the Schrödinger equation, Figure 3.7 shows the convergence of approximations at different grid points to a reference solution. The new scheme is shown to perform better at less number of grids than the classical one. This is also justified in table of errors, Table 3.3. © University of Pretoria Chapter 4 Nonclassical methods for higher order equations In this chapter, we design a nonstandard finite volume scheme for fourth order equations and test the application of the schemes by seeking numerical solution of the Kuramoto-Sivashinsky (K-S) and the Cahn-Hilliard (CH) equations. 4.1 Introduction We design an explicit nonstandard finite volume (NSFV) schemes each for the K-S and CH equations investigated in Chapter 2 by the fractional splitting method. The schemes follow the idea of nonstandard finite volume method discussed in Chapter 3. Our numerical algorithm makes use of the a finite volume scheme for the hyperbolic equation and nonstandard schemes for the nonlinear diffusion and the fourth order linear equations. The nonstandard schemes were designed following the rules in [57]. As discussed in Chapter 3, the nonstandard finite difference methods has the advantage of preserving the properties of the physical models in that it preserves positivity and monotonicity of the solution. It has been employed in approximating various differential equations of orders not more than 3. Here, we extend it to approximate the solution of fourth order PDEs. We achieve this by designing nonstandard schemes for the sub-equations that make up these higher order equations. This idea was employed earlier in solving the viscous Burger’s equation and advection-logistic equation among others, see [57]. The sub-equations of the CH and K-S equations are the inviscid Burger’s equation , the porous media equation and the linear fourth order equation. Therefore, we design nonstandard schemes for each of these equations. Denoting the solution operators Hk , N k and Lk for the inviscid Burger’s, porous media and the linear fourth order equations respectively, we apply the fractional time splitting algorithm to approximate the solution of the K-S and the CH equations. Giving v 0 = u0 for 71 © University of Pretoria Chapter 4 Nonclassical methods for higher order equations 72 n = 0; for n > 0, we obtain v n+1 from v n via the solution of vk (x, nk) = [Lk ◦ Hk ]n v0 (x), (4.1) vk (x, nk) = [Lk ◦ N k ]n v0 (x), (4.2) vk (x, nk) = [Lk ◦ N k ◦ Hk ]n v0 (x), (4.3) for the K-S equation, for the CH equation and convective CH equation. In Section 4.2, we use the subequation approach to design the NSFV scheme for each of the equations and in Section 4.3, the performance of all the schemes were tested and our observations were discussed in Section 4.4. 4.2 Numerical approach We are interested in designing nonstandard finite volume schemes for the inviscid Burger’s equation, porous media equation (PME) and the linear fourth equations which are components of either the K-S or CH equations. Recall that for h = ∆x = xj+ 1 − xj− 1 , j = 2 2 0, 1, · · · , m and k = tn − tn−1 , n = 1, 23, · · · , the intercell average Z x 1 j+ 2 1 n u(x, tn )dx. v ∼ ∆x x 1 j− 2 4.2.1 The inviscid Burger’s equation In this section, we present the nonstandard scheme for the inviscid Burger’s equation 2 u = 0. (4.4) ut + δ 2 x Integrating this first, with respect to x in I = [xj− 1 , xj+ 1 ] and later with respect to time 2 2 and employing the interpolation functions discussed in preceding chapters, we have n n n n vjn+1 − vjn vj+1 vj+1 − vj−1 + vj−1 +δ = 0. (4.5) ∆t 2 2∆x An alternative approximation is the 3-point averaging used in [92] for the numerical approx of the KdV equation. Here the schemes is given by n n n n vjn+1 − vjn vj+1 + vjn + vj−1 vj+1 − vj−1 +δ = 0. (4.6) ∆t 3 2∆x © University of Pretoria Chapter 4 Nonclassical methods for higher order equations 4.2.2 73 The porous media equation Here, we design a nonstandard scheme for the porous media equation ut = α(up)xx in one-dimension.The nonstandard scheme for this equation is given as vjn+1 − vjn 1 n n n n+1 n n n v + f )v v , = rα 2 fj+ − (f + f 1 j+1 j− 12 j j+ 12 j− 21 j−1 2 ∆t ∆x (4.7) where the average fj+1 + fj , 2 and fjn = (unj )r−1 . The scheme (4.7) is non consistent but can be made conditionally consistent. This is achieved as fj+ 1 = 2 rα vjn+1 − vjn rα n n n n+1 n n n = f (f n 1 +f n 1 )(v n+1 −vjn ), + f 1 vj+1 − (f 1 )vj 1 + f 1 vj−1 + j− 2 j+ 2 j− 2 ∆t ∆x2 j+ 2 ∆x2 j+ 2 j− 2 j which implies that n n+1 n n n+1 n n n n n n + f ) (v v + f )v v , 1 − rαλ(fj+ − v ) = rαλ f − (f + f 1 j j j− 21 j− 21 j j+ 21 j+1 j+ 21 j− 21 j−1 2 (4.8) which gives n n n n n n n , vjn+1 = vjn + rαλ fj+ 1 )vj + f 1 vj−1 1 + f 1 vj+1 + (f j− j− j+ 2 where λ = ∆t . ∆x2 2 2 2 By this, we have to re-scale the time derivative of the target equation by n n ), 1 − rαλ(fj+ 1 + f j− 1 2 2 for consistency. We employ this scheme to solve porous media equation when r = 3, ut = α(u3)xx = 3α(u2ux )x = 3α(f (u)ux)x = 0. 4.2.3 (4.9) The linear fourth order equation Our focus here is to derive a nonstandard scheme for the equation ut (x, t) + (u(x, t) + u(x, t)xx )xx = 0, u(x, 0) = u0 (x). (4.10) In doing this we will follow the rules and the procedures highlighted in [57, 58]. The steady state of equation (4.10) is the linear homogeneous equation (u + γ 2 uxx )xx = 0, © University of Pretoria (4.11) Chapter 4 Nonclassical methods for higher order equations which admits four linearly independent solutions γ, that h u hj eıj γ j 1 h uj+1 1 h(j + 1) eı γ (j+1) uj+2 1 h(j + 2) eı hγ (j+2) h uj+3 1 h(j + 3) eı γ (j+3) uj+4 1 h(j + 4) eı hγ (j+4) x This can be written as uj uj+1 uj+2 uj+3 uj+4 1 1 1 1 1 1 1 1 1 1 uj uj+1 uj+2 uj+3 uj+4 x eıh γ and e−ıh γ . From this we know x , γ h e−ıj γ h e−ı γ (j+1) h e−ı γ (j+2) h e−ı γ (j+3) h e−ı γ (j+4) This simplifies to give h 74 (4.12) = 0. j 1 1 h −ı ıh j+1 e γ e γ h h j + 2 e2ı γ e−2ı γ h h j + 3 e3ı γ e−3ı γ h h j + 4 e4ı γ e−4ı γ j 1 1 h −ı ıh j+1 e γ e γ h h j + 2 e2ı γ e−2ı γ h h j + 3 e3ı γ e−3ı γ h h j + 4 e4ı γ e−4ı γ = 0. = 0. Factorization leads to 2uj+1 − uj+2 − uj 2eı hγ − e2ı hγ − 1 2e−ı hγ − e−2ı hγ − 1 3ı h −3ı h ıh −ı h γ γ γ γ − 3uj+1 − uj+3 − 2uj 3e − e − 2 3e −e − 2 = 0. 4uj+1 − uj+4 − 3uj 4eı hγ − e4ı hγ − 3 4e−ı hγ − e−4ı hγ − 3 The determinant can be simplified to h h h h (2uj+1 − uj+2 − uj )(3eı γ − e3ı γ − 2)(4e−ı γ − e−4ı γ − 3) h h h h −(2uj+1 − uj+2 − uj )(3e−ı γ − e−3ı γ − 2)(4eı γ − e4ı γ − 3) h h h h +(3uj+1 − uj+3 − 2uj )(2e−ı γ − e−2ı γ − 1)(4eı γ − e4ı γ − 3) h h h h −(3uj+1 − uj+3 − 2uj )(2eı γ − e2ı γ − 1)(4e−ı γ − e−4ı γ − 3) h h h h +(4uj+1 − uj+4 − 3uj )(2eı γ − e2ı γ − 1)(3e−ı γ − e−3ı γ − 2) h h h h −(4uj+1 − uj+4 − 3uj )(2e−ı γ − e−2ı γ − 1)(3eı γ − e3ı γ − 2). Appealing to the identity eıθ − e−ıθ , 2ı the equation when simplified, after shifting the index j, gives sin(θ) = uj−2 − 4uj−1 + 6uj − 4uj+1 + uj+2 uj−1 − 2uj + uj+1 + = 0, φ(h)4 φ(h)2 © University of Pretoria (4.13) Chapter 4 Nonclassical methods for higher order equations where φ(h) = 2γ sin h 2γ 75 . Equation (4.13) is the nonstandard finite difference scheme for equation (4.11). The nonstandard scheme for the fourth order time dependent equation (4.10) is therefore, using Euler method to integrate with respect to time un+1 − unj unj−2 − 4unj−1 + 6unj − 4unj+1 + unj+2 unj−1 − 2unj + unj+1 j + γ2 + = 0. ∆t φ(h)4 φ(h)2 (4.14) Equation (4.14), leads to an explicit scheme γ 2 ∆t n ∆t n n n n n n n u − 4u + u − 2u + u = − 6u − 4u + u − . j−2 j−1 j−1 j j+1 j j+1 j+2 φ(h)4 φ(h)2 (4.15) Now, we will adapt the NSFD scheme discussed above to a finite volume scheme for equation (4.10). As usual, we multiply equation (4.10) by a characteristic function χ(xj− 1 ,xj+ 1 ) (x) 2 2 and then integrate within each cell. This gives us un+1 j unj n n n n n n vjn+1 − vjn vj−2 − 4vj−1 + 6vjn − 4vj+1 vj−1 − 2vjn + vj+1 + vj+2 + γ2 + = 0, ∆t φ(h)3 φ(h) (4.16) for j = 2, 3, · · · , m − 2, where we have used the interpolation functions u ∼ uh = ux ∼ ∇uh = and uxxx ∼ ∇3 uh = m X uj χ(xj− 1 ,xj+ 1 ) (x), 2 j=0 m X uj+1 − uj j=0 h 2 χ(xj− 1 ,xj+ 1 ) (x), 2 2 m X −uj−1 + 3uj − 3uj+1 + uj+2 j=0 h3 χ(xj− 1 ,xj+ 1 ) (x). 2 2 The solution approximations at j = 0, 1 and j = m − 1, m are addressed by employing the given boundary conditions. The replacement of h by φ(h) makes sense since lim φ(h) → h. h→0 4.3 Numerical experiments We will demonstrate the performance of all the schemes designed and discussed above. We will start with the nonlinear porous media, the linear fourth order, the K-S, the CH and the convective CH equations. © University of Pretoria Chapter 4 Nonclassical methods for higher order equations 4.3.1 76 The porous media equation We carried out the following experiment in order to test the performance of the scheme (4.7). Example 4.1 Solve the porous media equation 3 ut = (u )xx , x ∈ [−6, 6], u(−6, t) = u(6, t) = 0, u(x, 1) = u (x). 0 (4.17) We compare our results with the famous Barenblatt solution of the porous media equation s 1 |x|2 u3 (x, t) = t 4 [1 − √ ]+ , t > 0, 12 t at time T = 2, where [.]+ = max(., 0). Taking λ = the scheme (4.7) as k → 0. k , h2 Figure 4.1 shows the convergence of 1 0.8 0.6 u λ = 0.01 exact λ = 0.05 λ = 0.1 0.4 0.2 0 −6 −4 −2 0 x 2 4 6 Figure 4.1: Solution profile of equation (4.17) Remark 4.1 Here, we did not solve Example 4.1 by (4.8). We observe that as long as 0 < unj < 1, then fjn ≪ 1 by which it is obvious that equation (4.7)∼(4.8). Also, we note that the solution profile when λ = 0.01 coincides with the exact solution profile. 4.3.2 The linear fourth order equation In order for us to check the efficiency of the scheme (4.16) we perform the experiment below. © University of Pretoria Chapter 4 Nonclassical methods for higher order equations 77 Example 4.2 Solve the equation 2 ut (x, t) + (u(x, t) + γ uxx (x, t))xx = 0, u(x + L, t) = u(x, t), u(x, 0) = sin(x), (4.18) where we have chosen γ = 1. The error computation for the classical finite volume (where φ is replaced by h), the BDF2 and nonstandard schemes (4.16) are compared in Table 4.1. The numerical solution is compared with the steady state exact solution u(x, t) = sin(x). Grid points 10 20 40 80 Table 4.1: Error computations for Example 4.2 L∞ Norm error 4 Classical (error × 10 ) NSFV (error × 1016 ) BDF 150.1 2.220 40.71 1.665 10.26 6.661 2.568 4.441 (error × 102 ) Example 4.3 Apply the scheme (4.16) to approximate the solution of 2 ut (x, t) + (u(x, t) + γ uxx (x, t))xx = 0, u(x + L, t) = u(x, t), u(x, 0) = exp(−x2 ), 15.91 7.923 3.951 (4.19) and show the behaviour of the mean energy 1 E(t) = L in time. Z L u(x, t)2 dx, (4.20) 0 Solving the above problem, we observe the exponential growth of the mean energy in time as expected [24]. This is shown in Figure 4.3(a). 4.3.3 Kuramoto-Sivashinsky equation In this section we consider the Kuramoto-Sivashinsky equation ( ut + uux + uxx + uxxxx = 0, ∀ (x, t) ∈ (−L, L) × (0, T ], u(x, 0) = g(x, 0). (4.21) Throughout this section, we employ the fractional splitting scheme (4.1) and use schemes (4.6) and (4.16) to approximate the solutions of the inviscid Burger’s and the linear subproblems respectively. © University of Pretoria Chapter 4 Nonclassical methods for higher order equations 78 Example 4.4 Solve equation (4.21) by the fractional splitting scheme (4.1). 1. u(−L, t) = g(−L, t), u(L, t) = g(L, t), ux (−L, t) = gx (−L, t), ux (L, t) = gx (L, t), (4.22) where g(x, t) is the exact solution given by r 15 11 (−9 tanh[l(x − ct − x0 )] + 11 tanh3 [l(x − ct − x0 )]), g(x, t) = c + 19 19 (4.23) and c, l and x0 are constants. In the computations we take L = 30, x0 = −12, c = 5 p and l = 11/19/2 as documented in [7, 89]. 2. periodic boundary condition and initial condition u(x, 0) = exp(−x2 ). The errors due to the scheme (4.1) are shown in Table 4.2 while the solution profile for the boundary condition (4.22) and the chaotic property of equation (4.21) due to periodic boundary condition are shown in Figures 4.2(a) and 4.2(b) respectively. Table 4.2: Error computations for Example 4.41 Grids Errors×102 rate of convergence 40 474.3 80 99.87 2.2 160 12.84 2.9 320 2.891 2.1 Remark 4.2 The major contribution to this error is coming from the solver for the hyperbolic sub-problem. The error due to the scheme for the linear equation is shown in Table 4.1. Example 4.5 Solve equation (4.21) subject to periodic boundary condition and initial condition u(x, 0) = sin(x) by the fractional splitting scheme (4.1). 1 Here, we verify the bounds on the mean energy (LE) 2 as was done in Chapter 2. The scheme reproduces the bound that has been established in the literatures on the KuramotoSivashinsky equation (4.21) as discussed in Chapter 2, this is shown in Figure 4.3(b). We note that the slope of this graph is 0.978. © University of Pretoria Chapter 4 Nonclassical methods for higher order equations 7 79 NSFV Exact 6.5 6 5.5 u 5 4.5 4 3.5 3 −30 −20 −10 0 x 10 20 30 (a) Solution on a non periodic domain (b) Chaotic Solution on a periodic domain Figure 4.2: Solution of the Kuramoto-Sivashinsky equation by the NSFV scheme. 1.6 1.4 0 1 10 2 log(||u|| ) mean energy 1.2 0.8 0.6 0.4 numerical least squares fit 0.2 0 0 2 4 6 8 1 10 10 log(L) t (a) Mean energy bound of equation (4.10) (b) Energy bound of the K-S equation. Figure 4.3: Bounds verified for the linear and the Kuramoto-Sivasinsky (K-S) equation by the NSFV scheme. 4.3.4 Cahn-Hilliard equation In this section we approximate the solution of the Cahn-Hilliard equation by the fractional step scheme (4.2) employing the nonstandard solution operators for each of the subproblems involved. That is, we make use of the schemes (4.7) and (4.16) for the Cahn-Hilliard equation. Example 4.6 Solve the CH equation ( ut + (αu3 + u + γ 2 uxx )xx = 0, u(x, 0) = cos(πx/6), © University of Pretoria x ∈ (0, 6) (4.24) Chapter 4 Nonclassical methods for higher order equations 80 subject to boundary conditions ux = γ 2 uxxx − φ(u)x = 0, x ∈ ∂ Ω. (4.25) We highlight here that these schemes perform comparably well with the schemes discussed in Chapter 2. The evolution of the solution of the Cahn-Hilliard equation is shown in Figure 4.5(a). We also solve Example 4.6 and demonstrate the conservation of mass and the dissipation of the Ginzburg-Landau energy as expected. Figure 4.4 shows that the new schemes preserves these properties. 0.1 −0.4 −0.6 Energy ∆ Mass 0.05 0 −0.8 −1 −0.05 −1.2 −0.1 0 1 2 3 4 5 0 1 2 3 4 5 t t (a) Conservation of mass (b) Dissipation of the Ginzburg Landau energy Figure 4.4: Properties of the Cahn-Hilliard equation verified by the nonstandard schemes with γ 2 = 0.03 Remark 4.3 We highlight that this properties are resolved better here when compared with the standard schemes discussed in Chapter 2. 4.3.5 Convective Cahn-Hilliard equation In this section we approximate the solution of the convective Cahn-Hilliard equation by the fractional step scheme (4.3) employing the nonstandard solution operators for each of the subproblems involved. That is, we make use of the schemes (4.6), (4.7) and (4.16) for the Burger’s, porous media and the linear sub-problems respectively. We solve the problems given below. Example 4.7 Solve ( ut − δuux + (u + αu3 + γ 2 uxx )xx = 0, u(x, 0) = − sin(x/6), (4.26) subject to boundary conditions ux = γ 2 uxxx − φ(u)x = 0, © University of Pretoria x ∈ ∂ Ω. (4.27) Chapter 4 Nonclassical methods for higher order equations 81 The comparison of the numerical solution with the steady state solution for the convective Cahn-Hilliard equation is shown in Figure 4.5(b). These nonstandard schemes are employed 1 0.5 NSFV Exact 0.5 0 u u 1 t0 t1 t2 t3 t4 t5 t6 −0.5 0 −0.5 −1 1 2 3 x 4 −1 −15 5 (a) Evolution of the CH equation −10 −5 0 x 5 10 15 (b) Solution of the cCH equation Figure 4.5: Solutions of the Cahn-Hilliard and the convective Cahn-Hilliard equation by the nonstandard schemes. (a)γ 2 = 0.02, (b) γ = 1. to solve Example 4.7 subject to periodic boundary condition and initial condition 0.5 cos(x) for all x ∈ [−4π, 4π]. The conservation of mass is shown in Figure 4.6. 0.1 ∆ Mass 0.05 0 −0.05 −0.1 0 1 2 3 4 5 t Figure 4.6: Conservation of Mass of the cCH equation We also verify the coarsening process of the cCH equation. To achieve this, we investigate the problem below. Example 4.8 Solve the Example 4.7 subject to periodic boundary condition and initial condition 0.5 cos(x) for all x ∈ [−4π, 4π] by the fractional splitting scheme (4.3). Figure 4.8 shows that the solution coarsens as we advance in time from integration time T = 1 to T = 640. We have taken the value of γ = 1 in this experiment. © University of Pretoria Chapter 4 Nonclassical methods for higher order equations 82 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 u u 0 0 −0.1 −0.2 −0.2 −0.4 −0.6 −0.3 −0.8 −0.4 −10 −5 0 x 5 10 −10 (a) Formation of periodic structure, T = 1 −5 0 x 5 10 (b) Coarsening process, T = 50 Figure 4.7: Coarsening properties of the convective Cahn-Hilliard equation verified by the nonstandard schemes 1 u 0.5 0 −0.5 −1 −10 −5 0 x 5 10 Figure 4.8: Coarsening process T = 640 4.4 Conclusion We have proposed a fractional splitting scheme each for the the K-S and the CH equations where each of the sub equations are approximated by nonstandard schemes. Hence, the schemes are explicit. The nonstandard finite volume scheme (4.16) approximates the solution of the fourth order time dependent equation more accurate that the classical scheme as shown in Table 4.1. The exponential growth of the mean energy of its solution as expected is shown in Figure 4.3(a). All the properties of the K-S equation were reproduced when this scheme is combined with the NSFV scheme (4.6) by the fractional step method. The chaotic behaviour of the solution on a periodic domain is shown in Figure 4.2(b) while the profile for the mean energy bound is shown in Figure 4.3(b). Of particular significance, almost all the properties of the solution of the CH and cCH © University of Pretoria Chapter 4 Nonclassical methods for higher order equations 83 equations are reproduced by these new schemes. The dissipation of the Ginzburg-Landau energy and the conservation of mass of the CH equation are shown in Figure 4.4. The conservation of mass of the cCH equation is observed in Figure 4.6. The coarsening progress are shown in Figure 4.7 and 4.8. © University of Pretoria Chapter 5 Conclusion and future perspective This thesis is dedicated to the design and implementation of finite volume schemes to differential equations arising in mathematical physics. In Chapter 1 we introduced the equations under investigation, i.e. the K-S equation, the CH equation and the Schrödinger equation. We also introduced the numerical recipes used throughout this work. Our observations and conclusions are highlighted below. In Chapter 2, we proposed a fractional splitting method for each of the K-S and CH equations. The hyperbolic sub-equation was handled by the shock capturing schemes. We have compared the performance of some of the explicit schemes such as the semi-discrete, the fully discrete, the non-staggered finite difference scheme and the WENO scheme for this equation. The nonlinear diffusion equation was handled by a θ-method while the linear fourth order by either of the backward differentiation formula or the diagonally implicit Runge-Kutta scheme. This relaxes the stability restriction since the explicit schemes were employed only for the hyperbolic equation and the schemes for the fourth order equation are A-stable. The solution of the simplified equation are combined by the fractional step splitting method. The complexity in the two equations were simplified without losing the physical properties of the equations. We observed that all the known solution properties of these two equations are reproduced. For the K-S equation, the traveling wave solution, chaotic behaviour, preservation of periodicity, energy bounds are shown in Figures 2.4, 2.5, 2.7, 2.6 and 2.8 respectively. For the CH and cCH equation, the dissipation of the GinzburgLandau energy, the conservation of mass, the coarsening process and the transition from coarsening to roughening of the cCH equation are respectively observed in Figures 2.10, 2.13 and 2.14 and 2.17. In Chapter 3, our quest for optimal schemes to handle singularly perturbed fourth order equation encountered in Chapter 2 led us to the study of singularly perturbed second order ordinary differential equations. One of the interesting example of these types of equations is the Schrödinger equation. The complex boundary condition of the Schrödinger equation makes its approximation by these two schemes non trivial. We proposed two new schemes for the Schrödinger equation. The semi-analytical scheme is based on some perturbation 84 © University of Pretoria Chapter 5 Conclusion 85 technique while the nonstandard finite volume scheme is based on the existing nonstandard finite difference scheme. The semi-analytical scheme followed three main steps namely, • perturbation analysis of the given equation which leads to • boundary layer analysis through which boundary layer equations are derived, solved and employed and lastly • correcting of classical schemes. The nonstandard finite volume scheme follows from deriving an exact scheme by the idea of the difference equation of the given equation. The merit of this method is that it preserves positivity, monotonicity and in fact the properties of the physical model. We design this nonstandard scheme both for the equation and its complicated boundary conditions. To test the performance of these schemes, they were employed to solve some singularly perturbed equations. The approximation compares well with the exact solution of the tested equations. This can be seen in Tables 3.1 to 3.2 and Figures 3.3 to 3.4. The two methods reduce the computational cost known with classical approximation of singularly perturbed equations. In addition, these two schemes reproduces the the expected oscillatory behaviour of the Schrödinger equation with relatively little numbers of grids. Figures 3.5, 3.6, 3.7, 3.8 and Table 3.3 attests to the optimal performance of these new schemes as compared with classical schemes. In Chapter 4, we employed the nonstandard finite volume scheme discussed in Chapter 3 to design robust schemes for the singularly perturbed fourth order equation. We design nonstandard schemes for the hyperbolic equation, the nonlinear diffusion equation and the linear fourth order parabolic equation. We showed that the scheme gave a better approximation of the linear fourth order PDE than any classical scheme. This is shown in Table 4.1. The exponential growth of the mean energy of its solution as expected is shown in Figure 4.3(a). The nonstandard scheme for each of the sub-equations are combined via the fractional splitting algorithm to yield nonstandard schemes for each of the K-S equation, the CH equation and the cCH equation. We highlight here, that this is the first time such a nonstandard scheme is designed for a higher order equation and suggest that this can be extended to other higher order equations. For the K-S equation, Figures 4.2(b) and 4.3(b) shows the chaotic behaviour and the profile for the mean energy bound respectively. For the CH and cCH equations, the dissipation of the Ginzburg-Landau energy and the conservation of mass of the CH equation are shown in Figure 4.4. The conservation of mass of the cCH equation is observed in Figure 4.6 while the coarsening progress is shown in Figure 4.8. The results in this work can be employed as tools to investigate many other practically relevant models. Some of the obvious area of further research from this thesis are listed below. © University of Pretoria Chapter 5 Conclusion 86 1. Numerical analysis of the schemes is possible even though non trivial. It is interesting to investigate the convergence of operator splitting schemes to fourth order equations [32]. 2. Extension to higher order splitting is non-trivial but it is also a possible area of research focus. 3. 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