A Multi-Adaptive ODE-Solver Anders Logg Master of Science Thesis in Engineering Physics Examensarbete för civilingenjörsexamen i teknisk fysik med inriktning teknisk matematik Chalmers Finite Element Center Department of Mathematics Chalmers University of Technology Göteborg, Sweden 1998 c Anders Logg 1998 The front page shows three different discretizations for a multidimensional system of equations. The first one is a non-adaptive discretization, the second one is an adaptive discretization and the third one is a multi-adaptive discretization. This document was generated with LATEX on Solaris 2.6 at dd.chalmers.se. The font is Palatino 10pt. This report, as well as the code implementing the method proposed in it, are available for download at http://www.dd.chalmers.se/˜f95logg/Tanganyika/. Computations have been made with the Tanganyika multi-adaptive ODE-solver library (available for download at http://www.dd.chalmers.se/˜f95logg/Tanganyika/) on Linux 2.0 (Intel Pentium 200MHz) and on Solaris 2.6 (Sun Ultra 1 Model 170). Göteborg, October 3, 1998 Abstract In this work I present a multi-adaptive finite element method for initial value problems for ordinary differential equations, including an a posteriori estimate of the error. The method is multi-adaptive in the sense that the resolution of the time discretization is chosen individually for each component of the system of ordinary differential equations, based on an estimation of the error. The method has been successfully implemented in the Tanganyika library, available for download. Included are a few example computations made with this library, as well as instructions for downloading and using the package. Acknowledgements I wish to thank Claes Johnson, my advisor, for his continuous encouragement and support in the making of this project; Rickard Lind, Mathias Brossard and Andreas Brinck for beta-testing the programs; Jim Tilander for his expertise help with C++; Greger Cronquist for some useful hints on the typesetting; Göran Christiansson for proof-reading the manuscript; Anna for letting me do this all summer. Contents 1 Introduction 1.1 Quantitative Error Control . . . . . . . . . . . . . . . . . . . . . . 1.2 Multi-Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Method – Multi-Adaptive Galerkin 2.1 Equation . . . . . . . . . . . . . . . . . . . . 2.2 Finite Element Formulation . . . . . . . . . 2.2.1 Details . . . . . . . . . . . . . . . . . 2.2.2 Even more Flexibility . . . . . . . . . 2.3 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Constant 2.3.2 A Correction of the Error Estimate . 2.3.3 Other Error Contributions . . . . . . 2.4 Adaptivity . . . . . . . . . . . . . . . . . . . 2.4.1 Moderating the Choice of Timesteps 2.4.2 Choosing Data for the Dual Problem 7 8 9 . . . . . . . . . . . 10 10 11 12 13 13 14 15 16 16 18 18 3 The Implementation – Tanganyika 3.1 Individual Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Organization, Book-Keeping . . . . . . . . . . . . . . . . 3.2 Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 19 20 20 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Multi-Adaptive ODE-Solver 3.3 The Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results 4.1 A First Simple Example . . . . . . . . . 4.2 Wave Propagation in an Elastic Medium 4.3 Gravitation . . . . . . . . . . . . . . . . . 4.4 The Lorenz System . . . . . . . . . . . . 4.5 True Error vs. the Error Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 21 21 22 22 24 26 29 31 5 Conclusion 35 6 Download 36 Bibliography 37 Appendix 39 A Notation 39 B Tanganyika User Manual B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . B.2 Download . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Installation . . . . . . . . . . . . . . . . . . . . . . . . . B.4 The Tanganyika Library . . . . . . . . . . . . . . . . . B.4.1 What it does . . . . . . . . . . . . . . . . . . . . B.4.2 How to use it . . . . . . . . . . . . . . . . . . . B.4.3 InitializeSolution() . . . . . . . . . . . B.4.4 ClearSolution() . . . . . . . . . . . . . . . B.4.5 Solve() . . . . . . . . . . . . . . . . . . . . . . B.4.6 Save() . . . . . . . . . . . . . . . . . . . . . . B.5 The Tanganyika X-interface, Antananarive . . . . . . B.5.1 Introduction - What is this program anyway? . B.5.2 Using the program - Step by Step . . . . . . . . B.5.3 Settings . . . . . . . . . . . . . . . . . . . . . . B.5.4 Options . . . . . . . . . . . . . . . . . . . . . . B.5.5 .xt-file syntax . . . . . . . . . . . . . . . . . . B.5.6 Download, Updates, Further Information . . . B.6 GNU General Public License . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5 5 5 8 8 8 9 12 12 12 12 12 13 14 14 14 16 17 CHAPTER 1 Introduction Numerical methods for solving initial value problems for ordinary differential equations have been around for a long time and the number of methods is almost as large as the number of equations. Common methods, such as the ones supplied with Matlab (ode45(), ode23(), ode113(), ode-whatever()), are often fast, meaning that they terminate in a short time. These methods often provide some sort of local error control, where the error is controlled in some way in each integration step. This, however does not mean control of the global error. Although a tolerance is specified, it is not related – otherwise than by some (hopefully) monotonically increasing, and otherwise unknown, function – to the global error of the solution. The program is thus not concerned with the actual value of the error, leaving the user unaware of the quality of the computed solution. In fact, it was wrong. Bill Clinton (1998). Using such a classical numerical solver usually means solving the problem at a number of different tolerance levels for the local error, and comparisons between these solutions. Error control is thus (perhaps) obtained manually. 7 A Multi-Adaptive ODE-Solver This manual effort should also be taken into account when comparing the efficiencies of different solvers. 1.1 Quantitative Error Control Using a posteriori estimates of the error, i.e. error estimates based on the computed solution, it is possible to accurately control the size of the global error. Finite elements present a general framework for solving differential equations, such as e.g. initial value problems for ordinary differential equations, considered in this report. Depending on the choice of basis functions, normally piecewise polynomials of different kinds, the result is a new step method , ,. . . , for solving the initial value problem. These methods include , ,. . . . Efficieny is obtained by adaptivity, putting the computational effort where it is most needed. For initial value problems this usually means adjusting the size of the timestep, thus choosing the timestep to be small where the solution is especially sensitive to errors in the numerical method. Proper a posteriori error control requires knowledge of the stability of the problem. Stability properties are in general obtained by solving a so-called dual problem. Thus, error control requires some extra effort from the solver, which in some cases is comparable to the effort of solving the problem itself. Work on quantitative error-control during the last ten years (see references [1]-[10]) has resulted not only in extensive theoretical results, but also in working implementations of the methods, such as e.g. CARDS (solver of initial value problems for ordinary differential equations – see [8]) and FEMLAB (solver of partial differential equations). The current approach to quantitative error control was originated with the article by Johnson ([9]) in 1988, discussing error estimation for the and methods. Error estimation for these methods are further discussed by Estep in [5]. The method, which is the basis for the multi-adaptive method presented in this report, is discussed at length in [7]. A more classical approach to error analysis can be found in [11]. A comprehensive and major article on adaptive methods for differential equations is [3]. A general and non-technical discussion on error control and adaptivity is [6]. 8 A Multi-Adaptive ODE-Solver 1.2 Multi-Adaptivity It is desirable, in short, that in things which do not primarily concern others, individuality should assert itself. John Stuart Mill, On Liberty (1909). If we view a system of ODE:s as the representation of a mechanical system and notice that different parts, components, of such a system may behave very differently – some parts oscillating very rapidly and others slowly, perhaps undergoing even uniform motion – we realize that different components of an ODE-system may be differently sensitive to the resolution of the discretization. There is obviously a need for multi-adaptivity, allowing individual components of an ODE-system to use individual timesteps. Normally, the same timestep is used for all components of an ODE-system. The novelty of multi-adaptivity is thus allowing individual adaption of the timesteps for the different components. PSfrag replacements Figure 1.1: These are the actual timesteps used for an example computation on a simple two-dimensional system. 9 CHAPTER 2 The Method – Multi-Adaptive Galerkin This chapter describes the multi-adaptive method, complete with an a posteriori error estimate. The basis for the multi-adaptive method is a generalization of the continuous Galerkin method, , described in e.g. [4]. 2.1 Equation The equation to be solved is $ ! ! % " ! !#" where (2.1) is some function1 depending on the solution and , which may represent time. 1 In order to guarantee the existence of a unique solution, it may be good to know that Lipschitz continuous. 10 & is A Multi-Adaptive ODE-Solver 2.2 Finite Element Formulation $ " " " " 0 / !%#" ! % %$& ' )( ! !%%*,+. ! % # " ! % % $ ( ! % 1 , * + " 23+5164 " The weak (variational) formulation of equation (2.1) reads Find such that and for all test functions denotes the usual -inner product. where To define the multi-adaptive method, we introduce the trial space, and the test space, , of functions on , where (2.2) , is continuous Thus means that all its components are continuous and piecewise polynomial on the intervals , and means that all its components are in general discontinuous and piecewise polynomial (of one degree less) on the same intervals as the corresponding trial function. The multi-adaptive method is then Find 7 " 7 and 1 98: " 7 such that (2.3) <;! + 6 such that = = @8: 0 / ! %A" !! %%$ ( ! !%%* (2.4) ?> 7 ?> 7 ;! + and 7 is continuous, are the.parameters where the determining the piecewise polynomials 7 + . Note B parameters that there are determining a polynomial of degree , so the index is from zero to . ; 3 + in agreement with eq. (2.4) yields the deFinding the parameters < ! + , i.e. the sired solution. What remains is to find the proper discretization, ! + timesteps . To choose the timesteps, we need an error estimate, which will The discontinuity of the test functions means we may rewrite this as Find be the basis for adaptivity. By means of this error estimate, the discretization will be chosen in a way to give a resulting final error smaller than the specified tolerance. 11 A Multi-Adaptive ODE-Solver 2.2.1 Details ; ! + ! The parameters may e.g. be the nodal values for a subdivision of the intervals into subintervals. For an interval , let the nodal points of an equipartition of this interval be . The corresponding nodal (Lagrange) basis functions, , are then defined on for , by ! + 6 + ! ! % 3 ! 3 ! !: ! 33 // 3 ! ! : : ! ! ! 3 On the interval , 7 may then be written (uniquely) as 7 6 ; ! ! ; ! + . for some values (2.5) (2.6) Inserting this into eq. (2.4), computing a few integrals (simple but tedious) and solving the resulting system of linear algebraic equations, yields ;3 ;3 . ;! .. ? > / / > >!! ! $ ;! B 3 ! 7 ;! B 3 ! 7 ;! B 3 ! 5 7 + 6 # *) # &% (' $ + #"!, .-' B ' (2.7) are polynomial weight functions. " . # where and the These are given in table 2.1 for # % *) # + ' B 0/ (' 0 Table 2.1: Weight functions for the cG(2 ) integrals, 2*35476896;: . 12 1) ) A Multi-Adaptive ODE-Solver 2.2.2 Even more Flexibility Note that we could have allowed each component to be piecewise polynomial without beforehand fixing the deqree of the polynomial on the whole of the discretization. We could thus have allowed the polynomial degree to change from one interval to the next. The method would then be even -adaptive, choosing the (in some sense) best degree of the polynomials for every single interval . For simplicity, though, the polynomial degrees have been chosen to be rather than . The difference would be an extra index on . 3 ! + 2.3 " + Error Estimation The error estimate is obtained starting the same way as in references [1], [4] and [10]. To estimate the error at final time in the -norm, the dual problem of eq. (2.1) is introduced. The dual problem is 7 where 7 5 is the error, # 7 (2.8) -norm and is defined as B 7 is the i.e. is the transpose (or more generally, the adjoint) of the Jacobian of mean value of and . Note now that by the chain rule, 7 7 7 B 7 B % 7 7 We may thus write 13 7 (2.9) at a (2.10) A Multi-Adaptive ODE-Solver %% % where )B 7 . 7 . 7 7 77 7 7 7 ! is the residual, i.e. % 7 7 " " " 6 = " 6 164 = !> = 6 164 ?> = ?> = " 6 164 ! ?> ?> Using the finite element formulation for , we continue to get 2.3.1 % + (2.13) are constants. 3 B / 3 Choosing the test as the function around on yields The Constant (2.12) where the (2.11) :th-order Taylor-expansion of / The proof is simple. Noting that, with / B B ! B 14 (2.14) / / (2.15) A Multi-Adaptive ODE-Solver we have / / B / / / / / / B / and thus, with (2.16) , ! (2.17) Another useful estimate (see the section on adaptivity below) is / = ! > (2.18) where B (2.19) / to be the :th-order Taylor which is obtained as above, choosing expansion around the midpoint. 2.3.2 A Correction of the Error Estimate The method to be used is not, because of the difficulty involved with solving method, as will be described further in eq. (2.7), the true multi-adaptive chapter 3. Not solving the equations properly will introduce the discrete residual, which should be zero if the discrete equations, i.e. (2.7), were solved properly. The following analysis will result in an extra term in the error estimate (2.13), including the discrete residual together with its proper stability factor, accounting for accumulation of errors due to a non-zero discrete residual. Defining the discrete residual to be 15 A Multi-Adaptive ODE-Solver ! = ? > ; ! ; ! %:" ! ! %%$ ) ( ! !%%* (2.20) / ! and some 3 ! , = ! = ! ! (2.21) ?> ?> 7 = ! > Thus, differs from zero and we get an additional term in our error we get for estimate, continuing from eq (2.11): " " 6 = " 6 164 = !> " 6 164 !> )= 6 164 ! = ?> " 6 164 ! ?> if we choose 2.3.3 = ?> ! = ?> = B ?> B 3 3 ! = ! !> = ! !> (2.22) close to . Other Error Contributions Other error contributions that are not dealt with here are quadrature errors and numerical errors due the finite precision arithmetic. 2.4 Adaptivity Introducing the stability function, defined by ! = ?> 3 " ! %%$ ( ! ! %1* and the stability factor, defined by 16 (2.23) A Multi-Adaptive ODE-Solver '( ! !1* "" 6 164 3 6 (2.24) = ! !> the error estimate (2.13) may be written in two alternative ways as (2.25) The stability properties are obtained by numerical approximation (by the multi-adaptive method) of the solution of the dual problem. Notice that the error contribution from the non-zero discrete residual is not included in these expressions, since I have chosen to base the adaptivity on the Galerkin discretizational error alone. However, the contribution from the non-zero discrete residual is of course included in the computation of the error estimate and thus, indirectly, also in the adaptive procedure. Adaptivity is then based on the expression (2.26) where TOL is a given tolerance for the error of the solution at time . The discretization is now chosen by equidistribution of the error, both onto the different components and onto the different intervals, i.e. ! ! = ! > * TOL $ ( Alternatively, we may whish to do * TOL (2.27) (2.28) Knowing thus the residuals and the stability functions (or factors) we may choose the proper timesteps. This is done in a way that is iterative in two respects. Firstly, the timestep for an interval is chosen based on the residual in the previous interval. Secondly, the , are not known until the end of the computation. The values are then a more or less clever guess based on a previous computation. Of course, having computed the solution, we don’t have to guess these values to compute an error estimate. 2$ + 2$ + 17 A Multi-Adaptive ODE-Solver 2.4.1 Moderating the Choice of Timesteps Choosing timesteps as described in the previous section without any extra moderation may cause problems. If the residual in one interval is small, the timestep of the next interval will be large. A large timestep will (often) result in a large residual, which in turn in the same way means the timestep of the next interval will be small. There is thus a chance the timestep will oscillate if it is only based on the residual of the last interval. What needs to be done is to make sure the timesteps (and thus also the residuals) don’t differ too much between adjacent intervals. This may be done in a lot of different ways, e.g. by choosing the (harmonic) mean of the previous timestep and the value of the new timestep, as based on the residual. (The Tanganyika library uses a somewhat more sophisticated moderation of the timesteps.) 2.4.2 Choosing Data for the Dual Problem According to eq. (2.8), we need to know the true error in order to solve the dual problem. If we indeed knew the true error, we would not have to bother with any of this, and since the true error is unknown, we have to make a clever guess. We now discover another benefit of multi-adaptivity – it makes it easier for us to estimate the data for the dual problem! Since we equidistribute the error onto the different components, an estimation of the proper data for the dual problem should be , being the dimension, for the different components. The signs for the different components may be obtained by solving at different tolerance levels. Since, however, we don’t know the stability properties of the problem until the computation is done, we cannot expect the errors of an initial computation to be fully equidistributed onto the different components. Hence, we cannot expect to always work as data for the components of the dual problem. Again, proper data is obtained by e.g. solving at different tolerance levels. * * * 18 CHAPTER 3 The Implementation – Tanganyika This section describes the actual implementation of the method described in the previous section. 3.1 Individual Stepping The individual stepping is done according to eq. (2.7). This requires knowledge about , including the values of all other components. These values are evaluated by interpolation (or extrapolation), according to the order of the method, of the nearest known values of the other components. The solution of the integral equation is done iteratively for every component. The order of the stepping follows one simple principle; 7 the last component steps first. It is the fact that the equations are not solved simultaneously that results in non-zero discrete residuals. 19 A Multi-Adaptive ODE-Solver 3.1.1 Organization, Book-Keeping Doing the stepping individually rather than stepping all components together requires some book-keeping, keeping track of the positions of all components and which one is to step next. The individual stepping is done according to figure 3.1 below. The implementation pretty much follows this scetch. positions information interaction Figure 3.1: This is how the individual stepping is done. The different components tell/send their respective positions and in turn they get their interactions with (forces from) the other components. Thus, just as in nature itself, progress is made by the exchange of information, small pieces of information (gravitons or perhaps femions). 3.2 Quadrature The integrals of eq. (2.7) are evaluated by Gaussian (Gauss-Legendre) quadrature. Since the order of the weight functions for the integrals of a method are , we expect the total order of the integrands to be of order B 20 A Multi-Adaptive ODE-Solver (and even more if is of quadratic or higher order). It would thus be wise to use quadrature that is exact at least for polynomials of order , which is exactly the case for Gaussian quadrature with nodal points. Thus, midpoint quadrature for , two-point Gaussian quadrature for and so on. 3.3 The Program The method has been implemented as a library, called Tanganyika. To use the library functions, all one needs to do is to #include <tanganyika.h> in one’s C/C++ program. For more details, refer to the Tanganyika User Manual included in Appendix B. For even more details (all!) download the source code – see chapter 6. 3.3.1 Language The language of the Tanganyika library is C++, although its interface is pure C. An object-oriented programming language such as C++ is obviously wellsuited for such a program like the Tanganyika library, viewing the different objects as classes; Solution, Component, Element, etc. 3.3.2 Modularity A nice feature of the C++ programming language is the use of class derivation and inheritance, enabling a modular implementation of the different methods. Implemented in the current version (1.0) of the library are , and #" , but the implementation of another method, such as e.g. , would require only the implementation of a new subclass, specifying only what differs from the already existing methods. (This would in reality mean perhaps 50 lines of code.) 21 CHAPTER 4 Results In this chapter I present the results from a few computations made with the Tanganyika library. 4.1 A First Simple Example % / / As a first simple example, consider the following system of equations: (4.1) / . The equations are% solved The solution is of course $ by the multi-adaptive method with tolerance and . (The tolerance was actually chosen to be . The resulting error estimate was, $ $ ' however, .) The true error is, according to figure 4.1, and the $ $ % " / component errors are and respectively. Note the behaviour of the multi-adaptive method, choosing different timesteps for the two methods. The timesteps are chosen on basis of the residuals and stability functions. These are shown, together with the resulting / 22 / A Multi-Adaptive ODE-Solver timesteps, in figure 4.1. Note also the approximate equidistribution of the error. Solution 1 0.5 0 −0.5 −1 PSfrag replacements 0 −4 x 10 5 10 15 20 0 5 10 15 20 0 5 10 15 20 25 30 35 40 45 50 25 30 35 40 45 50 25 30 35 40 45 50 Error 5 0 −5 Timesteps 0.04 0.03 0.02 0.01 0 Figure 4.1: The solution of the simple harmonic oscillator problem, the errors and the timesteps respectively. 23 A Multi-Adaptive ODE-Solver PSfrag replacements Solution Error Timesteps 0.015 0.015 0.01 0.01 0.005 0.005 0 0 5 10 15 0 20 1 1 0.5 0.5 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 0 5 10 15 0 20 0.03 0.03 0.02 0.02 0.01 0.01 0 0 5 10 15 0 20 Figure 4.2: Residuals, stability functions and timesteps for the two components of the harmonic oscillator problem, shown for the interval 068 . 4.2 Wave Propagation in an Elastic Medium As a second example, consider wave propagation in an elastic medium, represented by a number of masses connected with springs according to figure 4.3. Figure 4.3: A system of masses and springs. 24 A Multi-Adaptive ODE-Solver The proper equations are easily obtained from Newton’s second law of motion. where .. .. . .. . . (4.2) This may also be thought of as a FEM space discretization of the wave equation. With initial conditions corresponding to all but one masses being at rest , we expect a propagation of the timesteps. At the beginning all but at one mass are at rest, so the timesteps for these masses may be large. As the oscillations of a mass increase, the corresponding timesteps should decrease and oscillate. This is also the case according to figure 4.4. 25 A Multi-Adaptive ODE-Solver 0.6 0.4 0.4 0.2 0.2 0.2 5 10 0.6 0.4 1 0.6 0 0 −0.2 −0.2 −0.2 −0.4 −0.4 −0.4 −0.6 −0.6 −0.6 −0.8 −0.8 0 5 10 15 0 5 10 15 0 −0.8 0 5 0 5 10 15 10 15 PSfrag replacements 0.05 0.05 0.05 0.04 0.04 0.04 0.03 10 5 0.03 1 0.03 0.02 0.02 0.02 0.01 0.01 0.01 0 0 5 10 15 0 0 5 10 15 0 Figure 4.4: Solutions for components 1,5 and 10 of a system consisting of 10 masses and 11 springs, together with their respective timesteps, solved at 3 4 with the multi-adaptive 4 method. 4.3 Gravitation As a third example, consider a system of three bodies (planets) in a somewhat complicated situation where one of the planets is in orbit around a larger one, and a third even smaller planet comes in making sort of a weird sling-shot around the smaller planet. and for a certainchoice of initial conditions, The forces involved are the solution is as depicted in figure 4.5 below for , solved with the multi-adaptive method. 26 A Multi-Adaptive ODE-Solver 1 0.5 0 PSfrag replacements −0.5 −1 −0.5 0 0.5 Figure 4.5: Orbits for the three planets. The circles drawn represent the planets at time . 3 27 A Multi-Adaptive ODE-Solver 10000 5000 0 10 0 8 x 10 0.5 1 1.5 2 2.5 3 3.5 0 8 x 10 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 5 0 10 PSfrag replacements 5 0 Figure 4.6: Stability functions for the -components of the three planets. As one might expect, the three bodies are differently sensitive to the resolution of the discretization. This is also evident in figure 4.7, where are drawn the timesteps for the components corresponding to the -coordinates of the three planets. (The problem is in two dimensions so there is a total number of 12 components.) In this figure are also the number of timesteps used for the different components. The larger planet, corresponding to components 1,2,7 and 8, obviously doesn’t require as many steps as the two smaller ones. The largest number of steps is, according to this figure, needed to resolve the -velocities of the smallest planet, which is not too strange, considering the main acceleration is in the -direction at the critical point. 28 A Multi-Adaptive ODE-Solver 9000 0.03 8000 0.025 7000 6000 0.02 5000 0.015 4000 3000 0.01 2000 PSfrag replacements 0.005 1000 0 0 1 2 0 3 1 2 3 4 5 6 7 8 9 10 11 12 Figure 4.7: Timesteps (left) and the number of timesteps (right) for the 12 different components of the three-body problem. It is obviously crucial for the timesteps (of the involved components) to be small just when the smallest planet makes the sling-shot. This is realized in the adaptive algorithm by an extremely large value of the stability functions for the involved components, as was shown in figure 4.6. 4.4 The Lorenz System As a fourth and final example, consider the Lorenz system given by the equations 29 A Multi-Adaptive ODE-Solver # A / # 9" (4.3) , and , and . where % The solution at and is shown in figure 4.8, together with the timesteps used for the computation. The “chaotic”, flipping, behaviour of the Lorenz system is not evident in this figure, since is too small. The purpose of this example is however not to illustrate certain characteristics of the Lorenz system, but to illustrate the use of multi-adaptivity for the three components. −4 32 4 x 10 31 3 30 2 29 28 1 27 0 26 25 0 2 4 6 8 10 −5 x 10 16 24 14 23 12 PSfrag replacements 22 0 8 −5 10 6 4 −10 −15 −12 −10 2 −8 −6 −4 5 5.5 6 Figure 4.8: At the left is the solution of the Lorenz system, solved with the mulitadaptive 4 method at 3 8 94 and with final time 3 4 . At the right are the timesteps used for the computation. 30 A Multi-Adaptive ODE-Solver Below in figure 4.9 is given the behaviour of one of the stability functions. 60 50 40 30 20 10 PSfrag replacements 0 0 1 2 3 4 5 6 7 8 9 10 Figure 4.9: The figure shows the stability function 3 for the component of the Lorenz system. The other two components are similar to this one. 4.5 True Error vs. the Error Estimate In this section, we return to the first simple example, the harmonic oscillator, and compare the true error to the error estimate. Ideally the true error is smaller than and close to the error estimate. Is this the case for the multi-adaptive cG method proposed in this work? To check the reliability of the solver, the solution of eq. (4.1) was computed with at a large number of tolerances. The results are given " for cG in figure 4.10. 31 A Multi-Adaptive ODE-Solver −3 True Error 1 x 10 −4 1 x 10 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0.5 1 0 0 0.5 −3 0.97 0 1 x 10 0 &" 0.5 1 −4 x 10 True Error / Error Estimate −6 1 −6 x 10 0.8 x 10 0.9 0.8 0.965 PSfrag replacements 0.75 0.7 0.96 0.6 0.7 0.955 0.95 0.5 0 0.5 Error Estimatex 10 1 −3 0.65 0 0.5 Error Estimatex 10 1 0.4 −4 0 0.5 1 Error Estimatex 10−6 Figure 4.10: True error vs. error estimate for multi-adaptive cG 4 , cG #8 and cG : respectively. Solid lines indicate the ideal maximum size of the true error. As can be seen the true error is smaller than and close to the error estimate for the three methods. For this specific problem at these specific tolerance levels, the error for the cG method is mostly discretizational error#" (arising from the finite element discretization of the error), whereas for the cG method the error is mostly mostly computational (arising from a non-zero discrete residual). For the cG method the situation is somewhere in between. This explains the different variances in error–tolerance correlations for the three methods. Notice also how sharp the error estimate is, especially for the cG method. Again, this is due to the fact that at this tolerance level, most of the error is the usual finite element discretizational error for the method. For comparison, the same computations were performed with the often used M ATLAB ODE-solver, ode45(). As can be expected with a solver lacking 32 A Multi-Adaptive ODE-Solver global error control, the tolerance is only nominal, in the sense that its correlation to the true error is unknown. 3000 True Error / “Tolerance” 2500 2000 1500 1000 500 PSfrag replacements True Error 100 0 0 0.2 0.4 0.6 “Tolerance” 0.8 1 −4 x 10 Figure 4.11: True error / tolerance vs. tolerance for M ATLAB:s ODE-solver ode45(). The above comparisons between true error and error estimate were made for a simple 2-component linear system. We conclude this section by showing the results for a computation on the following nonlinear problem: ) The solution is (obviously) ) $ B BB B B B ) $ (4.4) ) $ 33 ) $ $ $ . A Multi-Adaptive ODE-Solver A comparison between true error and error estimate is given in figure 4.12 for the multi-adaptive -method. Also for this nonlinear problem, the true error is smaller than and close to the error estimate, as desired. −4 1 x 10 1 0.9 0.8 0.8 0.7 0.7 True Error / Error Estimate 0.9 True Error 0.6 0.5 0.4 PSfrag replacements 0.6 0.5 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.2 0.4 0.6 Error Estimate 0.8 0 1 −4 x 10 0 0.2 0.4 0.6 Error Estimate 0.8 1 −4 x 10 Figure 4.12: True error vs. error estimate for the multi-adaptive cG 4 -method. The solid lines indicate the ideal maximum size of the true error. Finally, notice that these results were all obtained automatically, the only data specified being the equation (including initial data) and the tolerance. The equations were then solved automatically, including the solution of the dual problem – which was automatically generated by numerical differentiation of the given equation – and error estimation, giving a resulting final error smaller than the given tolerance. 34 CHAPTER 5 Conclusion As was shown in the previous section, the correlation between error and error estimate is as desired for the three methods – at least for simple model problems. Multi-adaptivity is thus a reality and the method is already implemented – in the Tanganyika multi-adaptive ODE-solver library. This library (at least the current version, 1.0) was written primarily with the intention to be a working implementation of the multi-adaptive method, secondarily with the intention to be a general, fast and reliable ODE-solver. Although the current implementation is indeed general and reliable, it is still not fast and effective enough, mainly because of the large amount of work needed to solve the dual problem. This has nothing to do with the multi-adaptivity itself. It is a consequence of the generation and full solution of the dual problem. There are cures for this and in future versions, more focus will be on speed and effectivity. The main focus, however, will always be on proper error control. The facts all contribute only to setting the problem, not to its solution. Ludwig Wittgentstein, Tractatus Logico-Philosophicus (1909). 35 CHAPTER 6 Download The program is available for download – as is this report – at http://www.dd.chalmers.se/˜f95logg/Tanganyika/ Included in the package is the Tanganyika library containing the actual solver together with Antananarive, an X-interface for the library. The program will run under any (not too antique) UNIX system, such as Linux, SunOS, Solaris, . . . . You will also need GTK, the Gimp ToolKit, for the X-interface. GTK is available for download at http://www.gtk.org/ The program is distributed under the GNU General Public License (GPL). (See Appendix A.) 36 A Multi-Adaptive ODE-Solver Bibliography [1] E. B URMAN, Adaptive Finite Element Methods for Compressible Two-Phase Flow, Phd thesis, Chalmers University of Technology 1998. [2] N. E RICSSON, A Study of Transition to Turbulence for Incompressible Flow using a Spectral Finite Element Method, Lic thesis, Chalmers University of Technology 1998. [3] K. E RIKSSON , D. E STEP, P. H ANSBO , C. J OHNSON, Introduction to Adaptive Methods for Differential Equations, Acta Numerica (1995), 105-158. [4] K. E RIKSSON , D. E STEP, P. H ANSBO , C. J OHNSON, Computational Differential Equations, Studentlitteratur, 1996. [5] D. E STEP, A Posteriori Error Bounds and Global Error Control for Approximations of Ordinary Differential Equations, SIAM J. Numer. Anal. vol 32 (1995), 1-48. [6] D. E STEP, S. V ERDUYN L UNEL , R. W ILLIAMS, Error Estimation for Numerical Differential Equations, (1995), http://www.cacr.caltech.edu/publications/techpubs/ [980930]. [7] D. E STEP, D. F RENCH, Global Error Control for the Continuous Galerkin Finite Element Method for Ordinary Differential Equations, M AN vol 28 (1994), 815-852. 37 A Multi-Adaptive ODE-Solver [8] D. E STEP, R. W ILLIAMS, Accurate Parallel Integration of Large Sparse Systems of Differential Equations, Math. Models Meth. Appl. Sci. (to appear). [9] C. J OHNSON, Error Estimates and Adaptive Error Time-Step Control for a Class of One-Step Methods for Stiff Ordinary Differential Equations, SIAM J. Numer. Anal. vol 25 (1988) no 4, 908-926. [10] R. S ANDBOGE, Adaptive Finite Element Methods for Reactive Flow Problems, Phd thesis, Chalmers University of Technology 1996. [11] G. D AHL Q UIST, Error Analysis for a Class of Methods for Stiff Nonlinear Initial Value Problems, Lecture Notes in Mathematics 506, Springer-Verlag (1976). 38 A APPENDIX Notation In this chapter, I explain the notation used in this report. Unfamiliar expressions should in general be explained when first introduced. Since, however, it is not always clear which expressions are familiar and which are not, I include the following list of notation: the finite element method, which is the basis for the multi-adaptive method proposed in this report a Galerkin method with continuous piecewise polynomials of order multi-adaptivity adaptive error control, where the discretizations are chosen individually for the different components [of and ODE-system] Tanganyika besides being a geographical location in the south of Africa, Tanganyika is the name of the multi-adaptive ODE-solver library, based on this report Antananarive this is the X-Windows interface for the Tanganyika library 39 A Multi-Adaptive ODE-Solver dual problem an auxiliary problem that has to be solved in order to get an estimation of the error the solution, in this case of the initial value problem (2.1) 7 the finite element approximation of the solution independent variable, often thought of as the time the end-value of * $ " the number of dimensions (components) of the ODE-system the number of intervals for the subpartition of " for component ( the trial space for our finite element formulation the test space for our finite element formulation the solution of the dual problem the error of our approximate solution, i.e. in eq. (2.1) the residual, i.e. 7 7 the Jacobian of ! ! " 7 ( the size of the :th timestep for component , i.e. the length of the interval 40 A Multi-Adaptive ODE-Solver numerical constants appearing in the error estimates the discrete residual, i.e. the residual of the discrete equations obtained from the finite element formulation of the continuous problem ( the stability function for component , a function obtained from the solution of the dual problem, describing the local stability properties for component ( ( the stability factor for component , a number obtained from the solution of the dual problem, describing the the global stability properties for component ( the tolerance, i.e. a beforehand specified upper bound for the error of the solution 41 APPENDIX B Tanganyika User Manual 43 U SER M ANUAL TANGANYIKA L IBRARY 1.0 TANGANYIKA X- INTERFACE 1.0 (A NTANANARIVE ) Anders Logg Chalmers Finite Element Center Department of Mathematics Chalmers University of Technology Göteborg, Sweden 1998 Jag har en syster i Tanganyika. A Multi-Adaptive ODE-Solver B.1 Introduction The Tanganyika library is a multi-adaptive solver of initial value problems for ordinary differential equations. " The method used for solving the equations is a variant of the , finite element method. The solver is adaptive in the sense that the size of the timesteps is chosen small enough to give an error smaller than the given tolerance, equidistributing the error onto the different intervals. The solver is multi-adaptive in the sense that the timesteps are chosen individually for the different components. For further details on the solver, download the report A Multi-Adaptive ODE-Solver from http://www.dd.chalmers.se/˜f95logg/Tanganyika/ The Tanganyika X-interface, Antananarive, is just that, an X-windows interface for the Tanganyika Library. B.2 Download To download the Tanganyika library and X-interface, goto http://www.dd.chalmers.se/˜f95logg/Tanganyika/, click the link named Download and follow further instructions on this page. You will then receive the whole package, containing everything you need – almost. In addition you must also have GTK, the Gimp ToolKit, installed on your system. (GTK is used by the X-interface for drawing the buttons.) If you just want to use the library and if you can do without the buttons, you don’t need GTK. However, if you do want the X-interface and you don’t have GTK, download GTK from http://www.gtk.org/ and install it according to the instructions. B.3 Installation If you haven’t realized that until now, you should be on a Unix system (Linux, SunOS, Solaris,. . . ). For the following instructions, it is assumed that commands are typed to a shell ( /bin/bash, /bin/sh, /bin/tcsh, . . . ), i.e. probably in an xterm. Commands are written as 5 A Multi-Adaptive ODE-Solver >> command Note that you shouldn’t type the ’>> ’! Oh well, you probably know all this but just in case you’re one of our sysadmins at dd.chalmers.se. . . ;) 1. Unpacking. The first thing you need to do is to unpack the Tanganyika source code. To do this, type >> unzip tanganyika-1.0.zip or the corresponding command for uncompression if you choose to download another format. This will create a directory (with a couple of sub-directories) named Tanganyika-1.0/ 2. Configuring. Edit the file defs in the Tanganyika-1.0 library for the variables to match your system. It should probably look something like this: CC LINK =g++ =g++ INCLUDE_PATH =-I/usr/include -I/usr/include/g++ LIBRARY_PATH =-L/usr/lib as it does on my Linux 2.0 or CC LINK INCLUDE_PATH LIBRARY_PATH =g++ =g++ =-I/opt/gnu/include =-L/opt/gnu/lib -R/opt/gnu/lib as it does on Sun Solaris 2.6 at dd.chalmers.se. 3. Compiling. Compile the library and the X-interface by typing >> make 6 A Multi-Adaptive ODE-Solver in the Tanganyika-1.0/ directory. The library and the X-interface will now compile. (If not, something went wrong and hopefully you know how to deal with it.) This will also generate the file .antananariverc in your home directory. 4. Running the demo. Check if you managed to compile the library correctly by typing >> ./demo in the Tanganyika-1.0/bin/ directory. This should result in some text output ending with something like Message: Computing error estimate... Message: ...done! Message: Error estimate: 7.526e-04 <= TOL = 1.000e-03 Message: Error estimate small enough, so I’m done. Message: Saving... Message: ...done! and data stored in the file tst.data, together with a M ATLAB .m file. You may also want to run the X-interface by typing >> ./antananarive in the same directory. 5. Completing the Installation. Complete the installation by putting the generated files wherever you want them. You may want to do the following (assuming the current directory is the Tanganyika-1.0/ directory): Place the X-interface. Type e.g. >> cp bin/antananarive /usr/local/bin or >> cp bin/antananarive /usr/bin 7 A Multi-Adaptive ODE-Solver Place the library header file. Type e.g. >> cp include/tanganyika.h /usr/include Place the library. Type e.g. >> cp lib/libtanganyika.a /usr/lib Notice that you probably cannot do this otherwise than as superuser (root)! B.4 The Tanganyika Library This is a tutorial for the Tanganyika library, version 1.0. B.4.1 What it does This library provides functions for solving initial value problems for systems of ordinary differential equations. The method used is a multi-adaptive finite element method, which is described in detail in the report A Multi-Adaptive ODE-solver. B.4.2 How to use it In your C(++) program, include the library by doing #include <tanganyika.h> What will then be included is the following: #ifndef TANGANYIKA_H #define TANGANYIKA_H #define TAN_METHOD_CG1 1 #define TAN_METHOD_CG2 2 #define TAN_METHOD_CG3 3 #define #define #define #define TAN_OUTPUT_DEVNULL TAN_OUTPUT_COUT TAN_OUTPUT_CERR TAN_OUTPUT_COM 0 1 2 3 #define TAN_FORWARD_PROBLEM 1 #define TAN_DUAL_PROBLEM 2 #define TAN_ERROR_ESTIMATE 3 8 A Multi-Adaptive ODE-Solver bool InitializeSolution (double *dInitialData, double dStartTime, double dEndTime, double dTolerance, double (*fFunction) (double *U, double t, int iIndex), int *iMethods, int iSizeOfSystem, int iMessageOutput, void (*Progress) (double dProgress), bool bErrorEstimation); void ClearSolution(); bool Solve (); bool Save(const char *cFileName); #endif What the different functions do should be quite clear from their names. Below follows a description of the different functions. B.4.3 InitializeSolution() Use this function to tell the library what to solve. The data passed to this function are described below. 1. dInitialData should be a valid pointer to a block of doubles, specifying the initial data for the problem, i.e. e.g. double *dInitialData = new double[2]; dInitialData[0] = 0.0; dInitialData[1] = 1.0; 2. dStartTime should be a double specifying the time, , at the beginning of the solution. (You probably want to pass 0 for this argument.) 3. dEndTime should be a double specifying the time, , at the end of the solution, such as e.g. 10. 0 4. dTolerance should be a double specifying the tolerance for the -norm of the error at the end of the solution. The library will try to solve the equations with an error that is smaller than this tolerance. 9 A Multi-Adaptive ODE-Solver 5. fFunction should be a pointer to a function specifying the equations, being " (B.1) i.e. e.g. double f(double *U, double t, int iIndex) { switch(iIndex){ case 0: return ( U[1] ); case 1: return ( sqrt(U[1]) + U[0] ); default: return 0.0; } } In this example, the name of the function (that must be declared with the parameter list as above) is f, so the reference that should be passed to InitializeSolution() is simply the name of the function, i.e. f. 6. iMethods should be a valid pointer to a block of ints, specifying the methods to be used for the different components. Valid values are TAN METHOD CG1, TAN METHOD CG2 and TAN METHOD CG3. 7. iSizeOfSystem should be an integer specifying the size of the system, i.e. the number of equations. 8. iMessageOutput should be an integer specifying the desired type of output from the library during solution. Valid values are TAN TAN TAN TAN OUTPUT OUTPUT OUTPUT OUTPUT DEVNULL, COUT, CERR, COM. 10 A Multi-Adaptive ODE-Solver These will set the adress of output from the program. TAN OUTPUT DEVNULL means no output will be written. TAN OUTPUT COUT means output will be to standard output. TAN OUTPUT CERR means output will be to standard error. TAN OUTPUT COM means output will be to standard output, in a special format that may be interpreted by e.g. the Tanganyika X-interface. With this output set, the current status of the program will be written to standard output as STAT iStatus, where iStatus is one of TAN FORWARD PROBLEM, TAN DUAL PROBLEM or TAN ERROR ESTIMATE, indicating what is going on. 9. Progress should be a pointer to a function that will be passed the progress of the computation, the progress being a number between 0 and 1. This might be useful for updating e.g. progress bars. (The Tanganyika X-interface does not use this for updating the progress bars. Instead the progress is parsed from the output.) The function should be declared as void FunctionName(double dProgress) { // Code goes here } 10. bErrorEstimation should be true or false, telling whether or not an error estimate should be computed. If false, no dual problem will be solved and the given tolerance will only be nominal in the sense that it won’t (necessarily) be close to the true error. However, a smaller nominal tolerance will (probably) mean a smaller error. InitializeSolution() will return true upon successful initialization of the solution and false if something went wrong, i.e. if e.g. the data passed was illegal. (A negative tolerance or whatever.) 11 A Multi-Adaptive ODE-Solver B.4.4 ClearSolution() Call this function to free all memory used by the library. B.4.5 Solve() Call this function to solve the equations, after having done InitializeSolution(). The return value will be true or false depending on whether or not the computation was successful. B.4.6 Save() Call this function to save data from the computation in M ATLAB format. This will generate two files. One filename.m file and one filename.data file, where filename is the filepath specified by cFileName. The first of these is called from M ATLAB by typing the name of the file (excluding the .m extension), which will load the data stored in the second one into the proper variables. B.5 The Tanganyika X-interface, Antananarive This is a tutorial for Antananarive, the Tanganyika X-interface, version 1.0. B.5.1 Introduction - What is this program anyway? This program is an interface for the Tanganyika multi-adaptive ODE-solver library. All it does is to call the library functions to generate a program from given user data. This program is then compiled using g++ or whichever compiler you prefer. (This may be changed in the “settings...” menu or in the .antananariverc file in your home directory.) The compiled program will output data to files (file path specified in the ”options...” menu) in Matlab format. Two files will be generated; one .data and one .m. Data from the solution is stored (ASCII) in the first of these files. Typing the filename in Matlab will call the .m-file, reading all data properly from the .data-file. The compiled program may be run either from this program (the Tanganyika X-interface, version 1.0) or manually from a shell. If you run the compiled program from this program, you get the benefit of parsed output as messages and 12 A Multi-Adaptive ODE-Solver progress bars. (The Tanganyika X-interface will read output from the generated program at standard output.) B.5.2 Using the program - Step by Step All you have to do is to press open - (edit) - (save) - make - solve in that order. Below follows a more detailed description. 1. Open a .xt-file, specifying the system of ordinary differential equations, by pressing the ”Open...” button and then choosing a .xt-file. (The suffix is not really important so there may be xt-files without the .xt-suffix.) 2. Edit the equations by pressing the ”edit” button. The contents of the opened file will then be editable in the text window. Of course you don’t have to edit the file if you don’t wanna change anything, but remember to save the file before moving on to compiling the program, as the program will be generated from the file and not from the contents of the text window. For information on the syntax, see the section below, ”.xt-file syntax”. 3. Save the changes if you made any by pressing the ”save...” button and then typing/choosing a file name. Note that the .xt-suffix will not be added automatically. 4. Generate the program and compile it by pressing the ”make” button. A .C-file will then be generated in the working directory specified in the ”settings...” menu. This file is then compiled and the output program will be filename.bin, also in the working directory. 5. Solve the equations by pressing the ”solve” button. This will run the generated program and parse its output to update the progress bars and typing status of the solution. By the way, the upper of the two progress bars is for the forward solution (the solution of the equations you specified) and the one below is for the solution of the dual (backward) problem that is solved to estimate the error of the solution. 13 A Multi-Adaptive ODE-Solver B.5.3 Settings Working directory This is where the .C and .bin files will be generated. Compiler This is the name of the (C++) compiler present in your system. CFLAGS These flags are passed to the compiler, specifying e.g. code optimizations. INC PATH This is where the compiler will look for include files. LIB PATH This is where the compiler will look for libraries (lib*-files). B.5.4 Options Start Time Well,..., this is the start time, the value of at the beginning. End Time And this would then be , the value of at the end of the solution. Tolerance This is the value of the tolerance for the . # -norm error of the solution at Output filename This is the file in the current working directory where the generated program will store the solution. B.5.5 .xt-file syntax You have to specify four things: The size of the system: Initial data: Equations: Methods: N = ? U[i] = ? F[i] = ? M[i] = ? 14 A Multi-Adaptive ODE-Solver All data must end with a semicolon (;). % at the beginning of a line means a comment, i.e. this line will not be interpreted. Indices begin with 0! Specification of equations must be C syntax. You may thus not write F[5] = U[2] * U[1]ˆ2 + sqrt(abs(U[3])); Instead you must write F[5] = U[2] * pow(U[1],2) + sqrt(fabs(U[3])); Methods are specified as integers 1, 2 or 3 for cG(1): continuous first-order Galerkin, cG(2): continuous second-order Galerkin and cG(3): continuous third-order Galerkin, respectively. Here follows a simple example (for a simple harmonic oscillator): % % This is an example % % size of system N = 2; % initial data U[0] = 0; U[1] = 1; % equations F[0] = U[1]; F[1] = -U[0]; % methods M[0] = 1; M[1] = 2; 15 A Multi-Adaptive ODE-Solver B.5.6 Download, Updates, Further Information This program is available for download at http://www.dd.chalmers.se/˜f95logg/Tanganyika/ together with the Tanganyika library. At this site is also available in postscript format the report A Multi-Adaptive ODE-Solver. 16 A Multi-Adaptive ODE-Solver B.6 GNU General Public License These programs (both the Tanganyika library 1.0 and the Tanganyika X-interface) are distributed under the GNU General Public license, GPL, included below. GNU GENERAL PUBLIC LICENSE Version 2, June 1991 Copyright (C) 1989, 1991 Free Software Foundation, Inc. 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software--to make sure the software is free for all its users. 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You may not copy, modify, sublicense, or distribute the Program except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense or distribute the Program is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance. 5. You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to modify or distribute the Program or its derivative works. These actions are prohibited by law if you do not accept this License. Therefore, by modifying or distributing the Program (or any work based on the Program), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Program or works based on it. 6. Each time you redistribute the Program (or any work based on the Program), the recipient automatically receives a license from the original licensor to copy, distribute or modify the Program subject to these terms and conditions. You may not impose any further restrictions on the recipients’ exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties to this License. 7. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Program at all. For example, if a patent license would not permit royalty-free redistribution of the Program by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program. If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances. It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system, which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice. This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License. 8. If the distribution and/or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License. 9. The Free Software Foundation may publish revised and/or new versions of the General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. Each version is given a distinguishing version number. If the Program specifies a version number of this License which applies to it and "any later version", you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of this License, you may choose any version ever published by the Free Software Foundation. 10. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally. 19 NO WARRANTY 11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. END OF TERMS AND CONDITIONS fin

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