2 A second-order accurate in time IMplicit-EXplicit (IMEX) integration scheme for sea ice dynamics 3 Jean-François Lemieuxa,⇤, Dana A. Knollb , Martin Loschc , Claude Girardd 1 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 a Recherche en Prévision Numérique environnementale/Environnement Canada, 2121 route Transcanadienne, Dorval, Qc H9P 1J3, Canada b Los Alamos National Laboratory, P.O. Box 1663, Los Alamos, NM 87545, USA c Alfred-Wegener-Institut, Helmholtz-Zentrum für Polar-und Meeresforschung, Postfach 120161, 27515, Germany d Recherche en Prévision Numérique atmosphérique/Environnement Canada, 2121 route Transcanadienne, Dorval, Qc H9P 1J3, Canada Abstract Current sea ice models use numerical schemes based on a splitting in time between the momentum and continuity equations. Because the ice strength is explicit when solving the momentum equation, this can create unrealistic ice stress gradients when using a large time step. As a consequence, noise develops in the numerical solution and these models can even become numerically unstable at high resolution. To resolve this issue, we have implemented an iterated IMplicit-EXplicit (IMEX) time integration method. This IMEX method was developed in the framework of an already implemented Jacobian-free Newton-Krylov solver. The basic idea of this IMEX approach is to move the explicit calculation of the sea ice thickness and concentration inside the Newton loop such that these tracers evolve during the implicit integration. To obtain second-order accuracy in time, we have also modified the explicit time integration to a second-order Runge-Kutta approach and by introducing a second-order backward di↵erence method for the implicit integration of the momentum equation. These modifications to the code are minor and straightforward. By comparing results with a reference solution obtained with a very small time step, it is shown that the approximate solution is second-order accurate in time. The new method permits to obtain the same accuracy as the splitting in time but by using a time step that is 10 times larger. Results show that the second-order scheme is more than five times more computationally efficient than the splitting in time approach for an equivalent level of error. ⇤ Corresponding author Preprint submitted to Journal of computational physics January 8, 2014 Email address: [email protected] (Jean-François Lemieux) 34 35 36 1 2 3 4 5 6 7 8 9 Keywords: sea ice, IMEX method, backward di↵erence, Newton-Krylov method, numerical accuracy 1. Introduction Various mechanisms associated with sea ice dynamics play a key role in shaping the ice cover of the polar oceans. To properly model the processes of lead and pressure ridge formation, sea ice models require a sophisticated representation of sea ice rheology, i.e. the relation between internal stresses, material properties (ice strength) and deformations of the ice cover. Most current sea ice models use the Viscous-Plastic (VP) formulation of Hibler [1] to represent these ice interactions. The VP formulation leads to a very nonlinear problem which is known to be difficult to solve. 10 11 12 13 14 15 16 17 18 19 20 21 22 To the best of our knowledge, all sea ice model time integration schemes are based on a splitting in time between the momentum and the continuity equations (e.g., [1, 2, 3, 4, 5]). This means that when solving the momentum equation, the thickness distribution (including the amount of open water) is held constant at the previous time level (it, however, varies spatially). Once the velocity field is obtained, the thickness distribution is advanced to the next time level. Furthermore, an operator splitting approach is generally used to separate the change of the thickness distribution associated with advection and the growth/melt related to thermodynamic processes (e.g., [2, 3]). This paper focuses on dynamics and we therefore only discuss the solution of the momentum equation and of the continuity equation without the thermodynamic source terms. 23 24 25 26 27 28 29 30 31 32 Current sea ice model numerical schemes su↵er from significant numerical issues. First, as explained by Lipscomb et al. [2], the splitting in time approach leads to noise in the numerical solution and can even make the model numerically unstable. As an illustrative example, consider ice converging toward a coast due to an onshore wind; a stress gradient, associated with an ice strength gradient, develops to oppose the wind stress. When using a large time step with the splitting in time approach, an unrealistically large ice strength gradient can occur. The stress gradient force can then overcompensate the wind stress and cause an unrealistic reversal of the flow (the ice 2 33 34 35 36 then diverges at the coast). This instability, fundamentally numerical, can be cured by reducing the time step. Unfortunately, this obviously increases the total computational time. Lipscomb et al. [2] proposed a modification to the ridging scheme in order to mitigate this problem. 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 A second numerical issue is related to the solution of the momentum equation. The rheology term, which determines the deformations of the ice cover based on the internal ice stresses, causes the momentum equation to be very nonlinear. Indeed, the VP rheology leads to a large change in the internal stresses when going from a slightly convergent flow to a slightly divergent one (same idea for shear stresses). The current numerical solvers for the momentum equation, however, have difficulties in finding the solution of this very nonlinear problem. There are two main classes of schemes to solve the momentum equation: the implicit solvers, which involve an outer loop iteration (sometimes referred to as Picard iteration, [5, 6, 7]) and the ones based on the explicit solution of the momentum equation using the ElasticVP approach [8, 9]. Both of these approaches, however, lead to a very slow convergence rate [7, 9] if they converge at all [9, 10]. Because of this slow convergence rate, it is typical to perform a small number of Picard iterations or of subcycling iterations. The approximate solution therefore contains residual errors which are carried on in the time integration. 54 55 56 57 58 59 60 61 62 63 To resolve this slow convergence rate issue, Lemieux et al. [4] developed a Jacobian-free Newton-Krylov (JFNK) implicit solver. They showed that the JFNK solver leads to a more accurate solution than the EVP solver [10] and that it is significantly more computationally efficient than a Picard approach [4]. Following the work of Lemieux et al. [4], Losch et al. [11] have recently developed a parallel JFNK solver for the MIT general circulation model with sea ice [12]. The numerical approaches of Lemieux et al. [4] and Losch et al. [11], however, still rely on the splitting in time scheme and are therefore susceptible to exhibit the numerical instability issue. 64 65 66 67 68 69 70 It is the purpose of this paper to introduce a fast and accurate time integration scheme that resolves the instability associated with the splitting in time approach. One possibility would be to solve fully implicitly the momentum and continuity equations. This avenue would imply significant modifications to the code and would be quite complex to implement. Instead, the splitting in time issue is cured by using an iterated IMplicit-EXplicit 3 71 72 73 74 75 76 77 78 79 80 (IMEX) approach when solving the momentum and continuity equations. This approach is built around our existing JFNK solver. Basically, the idea is to move the explicit calculation of the thickness distribution inside the implicit Newton loop. We take this approach one step further by modifying the time integration in order to get second-order accuracy in time for the full system. To do so, we introduce a second-order Runge-Kutta scheme for the advection operation and discretize in time the momentum equation using a second-order backward di↵erence (as in [13]). This paper is inspired by the work of [14, 15] on an iterated IMEX method for radiation hydrodynamics problems. 81 82 83 84 85 86 87 The main contribution of this paper is the development and demonstration of a first-of-a-kind second-order accurate in time iterated IMEX integration scheme for sea ice dynamics. This manuscript also shows the gain in accuracy and computational time of the second-order IMEX method compared to the common first-order integration scheme based on the splitting in time. 88 89 90 91 92 93 94 95 96 97 98 It is worth mentioning that some authors have recently questioned the validity of the VP rheology. Sea ice models based on a VP rheology do not capture the largest deformations events [16] and statistics of simulated deformations do no match observations [16] in both space and time [17]. While some authors propose new and very di↵erent formulations of ice interactions [18, 19], others claim that a VP rheology with modified yield curve and flow rule can adequately represent the sea ice deformations [20]. These new physical parameterizations, under evaluation, also lead to very nonlinear problems which would also clearly benefit from the availability of reliable and efficient numerical schemes. 99 100 101 102 103 104 105 106 This paper is structured as follows. Section 2 describes the sea ice momentum equation with a VP formulation and the continuity equation. In section 3, the discretization of the momentum and continuity equations and the descriptions of the standard splitting in time and new IMEX integration schemes are presented. In section 4, more information about the model is given. The description of the experiments and the results are outlined in section 5. A discussion and concluding remarks are provided in section 6. 107 4 108 109 110 111 112 113 2. Sea ice momentum and continuity equations As the ratio between the horizontal and the vertical scales is O(1000 km/10 m) = O(105 ), sea ice dynamics is often considered to be a twodimensional problem [21]. The two-dimensional sea ice momentum equation is obtained by integrating in the vertical the momentum equation. It is given by Du2 = ⇢hf k ⇥ u2 + ⌧a ⌧w + r · ⇢hgrHd , (1) Dt where ⇢ is the density of the ice, h is the ice volume per unit area (or the D mean thickness and just referred to as thickness in this paper), Dt is the total derivative, f the Coriolis parameter, u2 = ui + vj the horizontal sea ice velocity vector, i, j and k are unit vectors aligned with the x, y and z axis of our Cartesian coordinates, ⌧a is the wind stress, ⌧w the water stress, the internal ice stress tensor (r · is defined as the rheology term), g the gravity and Hd the sea surface height. The subscript in u2 indicates that it is a 2-D vector and it is used to distinguish u2 from the vector u obtained from the spatial discretization (explained in section 3). ⇢h 114 115 116 117 118 119 120 121 122 123 124 125 126 As in Tremblay and Mysak [3], the sea surface tilt is expressed in terms of the geostrophic ocean current. Using a quadratic law and constant turning angles ✓a and ✓w , ⌧a and ⌧w are expressed as [22] ⌧a = ⇢a Cda |uga |(uga cos ✓a + k ⇥ uga sin ✓a ), ⌧w = ⇢w Cdw |u2 127 128 129 130 ugw |[(u2 ugw ) cos ✓w + k ⇥ (u2 ugw ) sin ✓w ], (2) (3) where ⇢a and ⇢w are the air and water densities, Cda and Cdw are the air and water drag coefficients, and uga and ugw are the geostrophic wind and ocean current. As u2 is much smaller than uga , it is neglected in the expression for the wind stress. 131 132 133 The VP constitutive law, that relates the internal stresses and the strain rates, can be written as [1] ij = 2⌘ ✏˙ij + [⇣ ⌘]✏˙kk ij 5 P ij /2, i, j = 1, 2, (4) 134 135 136 137 where ij are the components of the ice stress tensor, ij is the Kronecker @v delta, ✏˙ij are the strain rates defined by ✏˙11 = @u , ✏˙22 = @y and ✏˙12 = @x 1 @u @v ( + @x ), ✏˙kk = ✏˙11 + ✏˙22 , ⇣ is the bulk viscosity, ⌘ is the shear viscos2 @y ity and P is a pressure-like term which is a function of the ice strength. 138 With a two-thickness category model, the ice strength Pp is parameterized 139 140 as Pp = P ⇤ h exp[ C(1 141 142 143 A)], (5) where P ⇤ is the ice strength parameter, A is the sea ice concentration and C is the ice concentration parameter, an empirical constant characterizing the strong dependence of the compressive strength on sea ice concentration [1]. 144 145 146 147 148 The formulation of the bulk and shear viscosities depends on the yield curve and the flow rule. In the following, the elliptical yield curve with a normal flow rule [1] is used. In this case, the bulk and shear viscosities are given by ⇣= Pp , 24 (6) ⌘ = ⇣e 2 , 149 150 151 (7) 1 2 where 4 = [(✏˙211 + ✏˙222 )(1 + e 2 ) + 4e 2 ✏˙212 + 2✏˙11 ✏˙22 (1 e 2 )] , and e is the aspect ratio of the ellipse, i.e. the ratio of the long and short axes of the elliptical yield curve. 152 153 154 When 4 tends toward zero, equations (6) and (7) become singular. To avoid this problem, ⇣ is capped using an hyperbolic tangent [7] ⇣ = ⇣max tanh( 155 156 157 158 Pp ). 24⇣max (8) As in equation (7), ⌘ = ⇣e 2 . The coefficient ⇣max is set to the value proposed by Hibler [1]: 2.5 ⇥ 108 Pp (this is equivalent to limiting 4 to a minimum value of 2 ⇥ 10 9 s 1 ). As opposed to the regularization introduced by Hibler [1], this formulation for ⇣ is continuously di↵erentiable. 159 6 160 161 We use a replacement closure similar to the one presented in Kreyscher et al. [23]. The pressure term is given by P = 2⇣4. The continuity equations for the thickness and the concentration are given 162 163 (9) by 164 @h + r · (u2 h) = Sh , @t 165 166 167 168 169 (10) @A + r · (u2 A) = SA , (11) @t where Sh and SA are thermodynamic source terms. Note that A is limited above to 1.0. This does not a↵ect the conservation of mass as the mass per m2 is given by ⇢h. The source terms in equations (10) and (11) are set to zero in the simulations for this paper (unless otherwise stated) as we concentrate on matters related to the dynamics. 170 171 172 173 174 175 176 3. Numerical approaches 3.1. Temporal discretization The advection of momentum is neglected as it is small compared to the other terms in the momentum equation (as done in [6, 8]). The momentum and continuity equations are solved at time levels t, 2 t, 3 t, . . . where t is the time step and the index n = 1, 2, 3, . . . refers to these time levels. 177 178 179 180 181 182 183 The standard numerical approach involves a Splitting In Time (SIT) between the implicit momentum and explicit continuity equations. This splitting implies that h and A (and therefore Pp ) are considered to be known in the momentum equation as they are held at the previous time level. Using a backward Euler approach for the acceleration term, the u and v momentum equations at time level n are written as 184 ⇢hn 1 (u n un 1 ) n = ⇢hn 1 f v n + ⌧au t n ⌧wu + 7 @ n n 11 (Pp @x 1 ) + @ n n 12 (Pp @y 1 ) , (12) ⇢h 185 186 187 188 189 n 1 (v n vn 1) = t ⇢h n 1 fu n n + ⌧av n ⌧wv + @ n n 22 (Pp 1 ) @y + @ n n 12 (Pp 1 ) @x , (13) where the sea surface tilt term is ignored here to simplify the presentation. As the water drag and the rheology term are written in terms of the velocity field, the only unknowns in equations (12) and (13) are un and v n . Once these equations are solved for un and v n everywhere on the grid, the thickness and concentration fields are advanced in time according to (hn hn 1 ) + r · (un2 hn 1 ) = 0, t (14) (An 190 191 An 1 ) + r · (un2 An 1 ) = 0, (15) t for which we use a first-order (in space) upstream scheme (as in [3, 23, 24]). We introduce the operator L given by hn = L(hn 1 , un2 ), 192 193 194 195 (16) which allows one to write concisely the explicit calculation of hn based on the upstream scheme (same idea for An ). This scheme is stable if the CourantFriedrichs-Lewy (CFL) condition max(u, v) < xt is respected, with x being the spatial resolution. 196 197 198 199 200 201 202 This scheme for the integration of the momentum and continuity equations is first-order accurate in time as a consequence of the first-order treatment in both the momentum and continuity equations, and as a result of the SIT splitting error which is not iterated. We here introduce a few straightforward modifications that allows one to solve simultaneously these equations with second-order accuracy in time. 203 204 205 206 First, we introduce a second-order backward di↵erence (BDF2, [13]) approach for the momentum equation. Hence, the u and v equations are written as ⇢hn 3 n ( u t 2 2un 1 1 n + un 2 ) = ⇢hn f v n + ⌧au 2 8 n ⌧wu + @ n n 11 (Pp ) @x + @ n n 12 (Pp ) @y , (17) ⇢hn 3 n ( v 2v n t 2 207 208 1 1 + vn 2) = 2 n n ⇢hn f un +⌧av ⌧wv + @ n n 22 (Pp ) @y + @ n n 12 (Pp ) @x , (18) where h, A and Pp are at time level n because BDF2 is used along with IMEX (as explained below). 209 210 211 212 213 We note in passing that a second-order Crank-Nicolson scheme for the momentum equation was not successful because the water stress term leads to an an undamped oscillation. For more details, the reader is referred to Appendix A. 214 215 216 217 218 Secondly, to obtain second-order accuracy in time for the continuity equations, we use a second-order Runge-Kutta (RK2) predictor-corrector approach to obtain hn and An . Hence, they are obtained in two steps by doing (h⇤ (hn n 219 220 221 222 223 224 hn 1 ) + r · (un2 1 hn 1 ) = 0, t/2 hn 1 ) n + r · (u2 t 1 2 h⇤ ) = 0, (19) (20) 1 where u2 2 = (un2 1 + un2 )/2. h⇤ is centered in time as t/2 is used to perform the advection for the predictor step. Both steps use the upstream scheme. We introduce the operator hn = LRK2 (hn 1 , un2 1 , un2 ), similar to the one in equation (16), in order to denote the two-step calculation of hn . The RK2 approach with the upstream scheme has the same CFL condition than the first-order scheme. 225 226 227 228 Before we introduce our third modification and explain how these equations can be solved simultaneously for un , v n , hn and An , we need to present the JFNK solver. 229 230 231 232 233 3.2. Spatial discretization and boundary conditions The components of the velocity (u and v) are positioned on the Arakawa C-grid. A Dirichlet boundary condition is applied at an ocean-land boundary (u = 0, v = 0) and a Neumann condition at an open boundary (i.e., 9 234 235 236 237 238 239 240 241 242 243 244 245 246 the spatial derivatives of the components of velocity in the normal direction with the open boundary are chosen to be zero). Gradients of h and A are also set to zero at an open boundary. For stability, the ice strength Pp is set to zero at the open boundaries [25]. A f-plane approximation is used with f = 1.46 ⇥ 10 4 s 1 . Spatial derivatives (in the rheology term) are discretized using centered finite di↵erences except close to land boundaries where second order accurate Taylor series expansions are used. As opposed to our work in [4] and [10], the viscous coefficients are calculated following the method described in Bouillon et al. [9]. The spatial discretization (with nx tracer points in one direction and ny in the other one) leads to a system of N = (ny(nx + 1) + nx(ny + 1)) nonlinear equations for the velocity components and (nx + 2)(ny + 2) equations for each h and A (this includes the boundary conditions). 247 248 249 250 251 3.3. The JFNK solver We give a brief overview of the JFNK implementation. More details can be found in [4, 10, 26]. The u and v equations to be solved at time level n for each grid cell can be written as 252 ⇢hlu (↵un + un t 1 + u n 2 )= ⇢hlv (↵v n + v n 1 + v n 2 ) = t 253 254 255 256 257 258 n n ⇢hlu f vavg +⌧au n n @ 11 (Ppl ) @ 12 (Ppl ) n ⌧wu + + , n n ⇢hlv f unavg +⌧av ⌧wv + @x @ n l 22 (Pp ) @y @y + @ n l 12 (Pp ) @x (21) , (22) where hu is the thickness evaluated at the u location on the C-grid and vavg is the average of the four v components surrounding the u location (similar idea for hv and uavg ). The parameters ↵, and are respectively equal to 1, -1 and 0 for the SIT approach and to 23 , -2 and 12 for the BDF2 scheme. The superscript l is n 1 for the SIT method while it is n with the IMEX method (explained below). 259 260 261 262 From both approaches, we obtain equations that are functions of un and v n . The spatial discretization of equations (21) and (22) leads to a system of N nonlinear equations with N unknowns that can be concisely written as 10 Am (un )un = b(un ), 263 264 265 266 267 268 269 270 271 272 (23) where Am is an N ⇥ N matrix. We added a subscript m to distinguish the system matrix from the ice concentration vector A. The vector un , of size N , is formed by stacking first the u components followed by the v components. The vector b is a function of the velocity vector un because of the water stress term. Note that the system of equations also depends on the vectors hn and An for IMEX and on hn 1 and An 1 when using the SIT approach. The systems of equations to be solved are di↵erent whether the SIT or BDF2 approach is used (the two methods lead to di↵erent system matrix, vector b and solution). We drop the superscript n knowing that we wish to find the solution u = un . We introduce the residual vector F(u): 273 F(u) = Am (u)u 274 275 b(u). (24) The residual vector F(u) is useful as it allows one to evaluate the quality of the approximate solution as F(u) = 0 if the solution is fully converged. 276 277 278 279 280 281 The Newton method is used to solve the nonlinear system of equations given in (23). The iterates obtained during the Newton method are referred to as uk where the superscript k corresponds to the Newton iteration number. This nonlinear method is based on a multivariate Taylor expansion around a previous iterate uk 1 : F(uk 282 283 284 1 0 + uk ) ⇡ F(uk 1 ) + F (uk 1 ) uk . The higher order terms are neglected in the expression above. Setting F(uk 1 + uk ) = 0, uk = uk uk 1 can be obtained by solving the linear system of N equations: J(uk 1 ) uk = 285 286 287 288 289 (25) 0 F(uk 1 ), (26) where the system matrix J ⌘ F is the Jacobian, an N ⇥ N matrix whose entries are Jqr = @Fq (uk 1 )/@(ukr 1 ) (where q = 1, N and r = 1, N ). For k = 1, an initial iterate u0 needs to be provided. The initial iterate here is the previous time level solution un 1 . Once the linear system of equations (26) is solved, the next iterate is given by 11 290 291 292 293 294 ⇥ ⇤ 1 uk = uk 1 + uk , (27) where = 1, 12 , 14 , 8 is iteratively reduced until ||F(uk )|| < ||F(uk 1 )|| or until = 18 . The symbol || || denotes the L2-norm. This linesearch approach is an addition compared to the previous model versions described in Lemieux et al. [4] and Lemieux et al. [10] (see also Losch et al. [11]). This method greatly improves the robustness of the nonlinear solver. 295 296 297 298 299 300 301 302 303 The linear system of equations in (26) is solved using the Flexible Generalized Minimum RESidual (FGMRES, [27]) method. Krylov methods such as FGMRES approximates the solution in a subspace of small dimension. When creating the subspace, Krylov methods only need the product of J times certain vectors (see Knoll and Keyes [28] for details). The Jacobian matrix therefore does not need to be formed per se but only its action on a vector is required. Given a certain vector w formed during the Krylov process, the product of J times w can be approximated by J(uk 1 )w ⇠ 304 F(uk 1 + ✏w) ✏ F(uk 1 ) , (28) where ✏ is a small perturbation. 305 306 307 308 309 310 311 312 313 To speed up convergence of the linear solution, the system of equations is transformed using right preconditioning. The preconditioning operator is based on the matrix Am linearized with the previous iterate and involves 10 iterations of a Line Successive Over Relaxation (LSOR) scheme [4, 26]. The preconditioning operator is slightly di↵erent whether the SIT or the BDF2 method is used. This is a consequence of the di↵erent formulation of the inertial term which just leads to a multiplying factor of 32 for BDF2 and of 1 for SIT. 314 315 316 317 318 319 320 321 To improve robustness and computational efficiency, an inexact Newton method [29] is employed. With this approach, a loose tolerance is used in early Newton iterations and it is progressively tighten up as the nonlinear solution is approached. The preconditioned FGMRES method solves the linear system of equations until the linear residual is smaller than (k) k F(uk 1 ) k where (k) is the tolerance of the linear solver at iteration k (a value smaller than 1). The tolerance of the linear solver with this inexact 12 322 Newton approach is given by ( h ini , (k) = ||F(uk ||F(uk 323 324 325 326 327 328 1 )|| 2 )|| i↵ if ||F(uk , if ||F(uk 1 )|| 1 )|| < r. r, (29) The tolerance ini for the initial stage is set to 0.99. The exponent ↵ is set to 1.5 and r = 23 ||F(u0 )||. Because of the linesearch approach, a more aggressive evolution of the linear tolerance is used compared to the settings in [4, 10]. The tolerance (k) is also forced to be larger than 0.1 to prevent excessive use of the linear solver which tends to slow down the nonlinear solver. We will get back to this issue later in the paper. 329 330 331 332 333 Finally, a termination criterion (defined by nl ) for solving the nonlinear system of equations is also needed. The JFNK solver stops iterating after the L2-norm of the residual is lower than nl ||F(u0 )||. JFNK fails to converge when the termination criterion is not reached in kmax =100 iterations. 334 335 336 The JFNK algorithm with the SIT approach and the first-order upstream scheme is: 337 338 339 340 341 342 343 1. Start with an initial iterate u0 do k = 1, kmax 2. ‘‘Solve’’ J(uk 1 ) uk = F(uk 1 ) with FGMRES 3. uk = uk 1 + uk 4. If ||F(uk )|| < nl ||F(u0 )|| stop enddo 344 345 5. Calc hn = L(hn 1 , un ) and An = L(An 1 , un ) 346 347 348 349 350 where the initial iterate u0 is the previous time level solution and uk = un once it has converged. The matrix J and the vector F are functions of h and A at the previous time level, i.e. hn 1 and An 1 (note that SIT is technically an IMEX method, but it is not iterated). 351 352 353 354 355 The iterated IMEX approach (simply referred to as IMEX) now allows one to solve for un , vn , hn and An simultaneously. In order to do this, the explicit calculations of the thickness and concentration are moved inside the Newton loop. 13 356 357 358 359 360 361 362 363 1. Start with an initial iterate u0 do k = 1, kmax 2. Calc hk = L(hn 1 , uk 1 ) and Ak = L(An 1 , uk 1 ) 3. ‘‘Solve’’ J(uk 1 ) uk = F(uk 1 ) with FGMRES 4. uk = uk 1 + uk 5. If ||F(uk )|| < nl ||F(u0 )|| stop enddo 364 365 where in this case J and F are function of hk and Ak . 366 367 368 369 To obtain second-order accuracy in time, the latter algorithm can be modified by using the LRK2 advection operator and by using the BDF2 method. Hence, the BDF2-IMEX-RK2 algorithm is given by 370 371 372 373 374 375 376 377 1. Start with an initial iterate u0 do k = 1, kmax 2. Calc hk = LRK2 (hn 1 , un 1 , uk 1 ) and Ak = LRK2 (An 1 , un 1 , uk 1 ) 3. ‘‘Solve’’ J(uk 1 ) uk = F(uk 1 ) with FGMRES 4. uk = uk 1 + uk 5. If ||F(uk )|| < nl ||F(u0 )|| stop enddo 378 379 380 381 382 383 384 385 To ensure fast nonlinear convergence in the context of the IMEX or BDF2-IMEX-RK2 scheme, it is crucial to take into account the change in h and A associated with a change of velocity in the evaluation of J times a certain Krylov vector w (equation (28)). Hence, with the BDF2-IMEXRK2 scheme, F(uk 1 + ✏w) is a function of h+ = LRK2 (hn 1 , un 1 , u+ ) and A+ = LRK2 (An 1 , un 1 , u+ ) where u+ is uk 1 + ✏w (same idea for IMEX by using the simpler operator L). 386 387 388 389 For simplicity, the same notation is used for the three algorithms given above. However, as they do not solve the same nonlinear systems of equations, they lead to di↵erent Jacobian matrices, residual vectors and solutions. 390 391 392 A truncation error analysis, that demonstrates second-order accuracy in time for BDF2-IMEX-RK2, is given in Appendix B. 393 14 Symbol ⇢ ⇢a ⇢w Cda Cdw ✓da ✓dw f P* C e Definition sea ice density air density water density air drag coefficient water drag coefficient air turning angle water turning angle Coriolis parameter ice strength parameter ice concentration parameter ellipse ratio value 900 kg m 3 1.3 kg m 3 1026 kg m 3 1.2 ⇥ 10 3 5.5 ⇥ 10 3 25 25 1.46 ⇥ 10 4 s 1 27.5⇥103 N m 20 2 2 Table 1: Physical parameters for the numerical simulations 394 395 396 397 398 399 400 401 402 403 4. Information about the model Our pan-Arctic regional model can be run at four possible spatial resolutions: 10, 20, 40 and 80 km (square cartesian grids). The model uses two thickness categories and a zero-layer thermodynamics (described in [3]). The sea ice model is coupled thermodynamically to a slab ocean model. Climatological ocean currents are used to force the sea ice model and to advect heat in the ocean. The wind stress is calculated using the geostrophic winds derived from the National Centers for Environmental Prediction and National Center for Atmospheric Research (NCEP/NCAR) six hour reanalysis of sea level pressure [30]. 404 405 406 Tables (1) lists the values of the physical parameters used for the simulations in this paper. 407 408 409 410 411 412 413 For all the 2-D experiments, we use revision 317 of our model with small modifications to perform the experiments described below. The code is serial. All runs were performed on a machine with 2 Intel E5520 quad-core CPU at 2.26 GHz with 8 MB of cache and 72 GB of RAM. The compiler is GNU fortran (GCC) 4.1.2 20080704 (Red Hat 4.1.2-54), 64 bits. The optimization option O3-↵ast-math was used for all the runs. 414 15 415 416 417 418 To introduce and better illustrate the SIT instability, a few 1-D experiments are performed. Revision 89 of our 1-D model is used for all the 1-D experiments. A detailed description of the sea ice dynamic equations in 1-D can be found in [2]. 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 5. Results A series of one day numerical experiments in 1-D and 2-D are performed for the di↵erent time integration schemes at spatial resolutions of 40 and 20 km. The base set of numerical experiments use the SIT algorithm (referred to as SIT). The second set of numerical experiments use the iterated IMEX algorithm (referred to as IMEX). The final set of numerical experiments use the BDF2 scheme along with IMEX and the RK2 advection scheme (referred to as BDF2-IMEX-RK2). For each series, one day experiments are performed with di↵erent time steps ( t). To ensure that the CFL condition is respected, the maximum t at 40-km resolution is set to 360 min while it is 180 min for a resolution of 20 km (At these resolutions and maximum time steps, the CFL criterion is not violated for ice velocities 1 ms 1 ). It was observed that the solver had difficulties at the beginning of the time integration (with small wind and ice starting from rest). A value of ✏ = 10 7 , in the evaluation of the Jacobian times a vector (equation (28)), improves robustness compared to the value of 10 6 used in [4, 10]. Robustness was improved for the first few time levels by setting ✏ = 10 8 instead of 10 7 when the Newton iteration is larger than 50. This robustness issue is not a major problem as it has not been observed in realistic experiments. It is possible that a more sophisticated way of choosing ✏ (as described in [28]) or an exact Jacobian-times-vector operation by automatic di↵erentiation [11] could improve robustness for these idealized experiments, but this is not explored in this paper. As these few initial time levels are not representative of the usual behavior of the solver, only the last 12 hours of the one day integration are used to compute metrics to compare the di↵erent time integration approaches. 447 16 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 5.1. 1-D experiments For these 1-D experiments, the domain is 2000 km long with solid walls at both ends. There is a no inflow/outflow condition at the walls: i.e., the velocity is zero. The spatial resolution is 20 km. The initial thickness field is 1 m everywhere and the sea ice concentration is 0.95. The ice starts from rest. The westerly wind is zero at the beginning and is increased smoothy according to uga (t) = (1 e t/⌧ )ug⇤ a with ⌧ , a time constant set to 6 hours, 1 and |ug⇤ | =10 m s being the same everywhere. a To assess the quality of these approximate solutions, a 24-h reference solution is obtained by using a time step of 1 s (with BDF2-IMEX-RK2). We then compare the 24-h sea ice thickness field obtained with an integration scheme using a certain t with the reference solution. Thickness is used because it acts as an integrator of all the errors produced during the time integration. The Root Mean Square Error (RMSE) between a thickness field and the reference thickness field is calculated for all the experiments. The RMSE should decrease with t for all three series of experiments. BDF2IMEX-RK2 should be the most accurate and lead to second-order accuracy in time while the other two series (SIT and IMEX) are expected to be firstorder accurate in time. The termination criterion is nl = 10 6 for all the experiments. 469 470 471 472 473 474 475 476 477 478 479 Fig. 1a indeed confirms that SIT and IMEX are both first order accurate in time (the slope is ⇠1 on a log-log plot). This figure shows the RMSE between an approximate solution (thickness) and the reference solution as a function of the time step. Despite some wiggling, BDF2-IMEX-RK2 exhibits second-order accuracy in time. For any t, the BDF2-IMEX-RK2 solution is more than one order of magnitude more accurate than the IMEX and SIT ones. The improvement of IMEX over SIT is small except for large t. This implies that for smaller t, the splitting errors are smaller than the standard first-order discretization errors. The sudden increase in the RMSE for SIT for t larger than 60 min is due to noise in the thickness field near both walls. 480 481 482 483 484 485 The fact that the approximate solution for SIT is contaminated by noise makes it more difficult for JFNK to obtain the velocity field solution. This is illustrated in Fig. 1b. Whereas both IMEX and BDF2-IMEX-RK2 need less than 20 Newton iterations (on average), SIT behaves di↵erently than these two schemes for t larger than 15 min. Indeed, the mean number of Newton 17 486 487 iterations for SIT increases significantly for failure of JFNK for t = 120 min. t > 15 min. There was even a 488 489 490 491 492 493 494 These additional Newton iterations for SIT have an impact on the total CPU time as can be seen in Fig. 1c. While SIT is more efficient than IMEX and BDF2-IMEX-RK2 for small t, the additional Newton iterations for t > 15 min causes SIT to be more costly. Hence, BDF2-IMEX-RK2 is always significantly more accurate than SIT and it is also more computationally efficient than SIT for typical time steps (e.g. t = 60 min). 495 496 497 498 499 Fig. 2 displays how the errors are spatially distributed. The reference thickness and velocity solutions are respectively shown on Fig. 2a and 2b. The ice has piled up and the velocity exhibits strong convergence at the wall. The ice concentration has reached 1.0 close to the wall (not shown). 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 The di↵erence between the thickness obtained with SIT when using a time step of 120 min or 180 min and the reference solution are respectively shown on Fig. 2c and Fig. 2d in black. Similar to the results of Lipscomb et al. [2], there is noise in the approximate solution in the region of convergence. It is also observed that errors are also present on the western side of the domain where the ice is diverging. The error is, however, more localized than close to the eastern wall. The maximum errors are respectively 2.5 cm and 8.1 cm for t of 120 and 180 min. These figures also demonstrate that the noise is notably smaller everywhere on the domain with BDF2-IMEX-RK2 (in blue). In this case, the maximum errors are 0.1 cm ( t = 120 min) and 0.32 cm ( t = 180 min). As opposed to the SIT scheme, the IMEX approach decreases the errors close to the eastern wall but does not significantly a↵ect the noise on the other side of the domain where the ice diverges (not shown). 5.2. 2-D experiments Experiments in 2-D are performed at 40 and 20-km resolutions. The initial conditions for these one day are the same than in [10]. These experiments are performed starting on 17 January 2002 00Z. As in Lemieux et al. [10], this 24-hour period was chosen because it is characterized by typical conditions in the Arctic: a high pressure system close to the Beaufort Sea, convergence north of Greenland and ice flowing south through Fram Strait. The thermodynamics and the ocean currents are set to zero for these idealized experiments. The ice starts from rest. It is then accelerated by a smoothly 18 (a) 100 −3 10 −4 RMSE (m) 10 (b) 90 Mean nb of Newton iterations −2 10 SIT IMEX BDF2−IMEX−RK2 80 70 60 50 40 30 20 10 0 −5 10 0 20 40 60 80 100 120 140 160 180 Time step (min) 1 10 (c) −6 SIT IMEX BDF2−IMEX−RK2 Total CPU time (s) 10 −7 10 −8 10 SIT IMEX BDF2−IMEX−RK2 0 10 1 2 10 10 Time step (min) 0 10 −1 10 0 3 10 20 40 60 80 100 120 140 160 180 Time step (min) Figure 1: RMSE (a), mean number of Newton iterations per time level (b) and total CPU time (c) as a function of the time step. The mean number of Newton iterations and total CPU time were calculated for the last 12 h of the integration. Black curve with triangles is for the SIT scheme, red curve with diamonds is for IMEX while the blue curve with circles is BDF2-IMEX-RK2. This is a 1-D experiment with a spatial resolution of 20 km. 523 524 increased wind stress field. The geostrophic wind field on 18 January 2002 00Z is used but it is ramped up according to uga (t) = (1 525 526 527 e t/⌧ )ug⇤ a , (30) where ug⇤ a is the geostrophic wind field on 18 January 2002 00Z, t is the time (starting on 17 January 2002 00Z) and ⌧ is set to 6 hours as in the 1-D experiments. 528 529 530 531 532 A reference solution is again obtained by using a time step of 1 s (with BDF2-IMEX-RK2). We then compare the sea ice thickness field obtained on 18 January 2002 00Z with the reference solution valid at the same time. As in the 1-D experiments, the termination criterion is set to nl = 10 6 . 533 534 535 Fig. 3a shows the 20-km reference solution concentration field on 18 January 2002 00Z while Fig. 3b displays the reference solution velocity field at 19 1.2 0.2 (a) (b) 1 0.15 u (m/s) h (m) 0.8 0.6 0.1 0.4 0.05 0.2 0 0.1 t=1 s (Ref) 0 500 1000 1500 0 2000 0.1 (c) SIT − Ref BDF2−IMEX−RK2 − Ref 0 −0.05 −0.1 500 (d) 1000 1500 2000 SIT − Ref BDF2−IMEX−RK2 − Ref 0.05 h (m) h (m) 0.05 t=1 s (Ref) 0 0 −0.05 t=2 h 0 500 1000 1500 2000 Distance (km) −0.1 t=3 h 0 500 1000 1500 2000 Distance (km) Figure 2: 1-D reference solution ice thickness (a) and velocity (b) fields. Di↵erence between the thickness field obtained with the SIT approach (in black) or with BDF2-IMEX-RK2 (in blue) and the reference solution for t = 120 min (c) and t = 180 min (d). The spatial resolution is 20 km. The x-axis for these graphs is the distance in km from the western wall. 536 the same valid time. The reference thickness solution is shown in Fig. 7a. 537 538 539 540 541 542 543 544 545 546 Fig. 4 shows, for the di↵erent schemes, the RMSE as a function of the time step on a log-log plot for spatial resolutions of 40 km (a) and 20 km (b). The RMSE is calculated only where the concentration of the reference solution is above 50%. The behavior of the time integration scheme is qualitatively the same at both resolutions. We therefore concentrate on the 20-km resolution results. The SIT and IMEX schemes lead to first-order accuracy in time while BDF2-IMEX-RK2 clearly demonstrates that it is second-order accurate in time over a wide range of t. There seems to be error saturation for large t as a flattening of the curve is observed. 547 548 As the continuity and momentum equations are solved simultaneously 20 (b) (a) Figure 3: Ice concentration (a) and velocity field (b) at 20-km resolution on 18 January 2002 00Z obtained with BDF2-IMEX-RK2 with a time step of 1 s. These 2-D fields form the reference solution. For clarity, only one velocity vector out of 16 is shown. The continents are in gray. 549 550 551 552 553 554 with BDF2-IMEX-RK2, we verify that the scheme also leads to second-order accuracy in time for the velocity field. Fig. 5 shows the RMS of the magnitude of the velocity error (referred to as RMSEv) between an approximate solution and the reference solution as a function of t. This result demonstrates second-order accuracy in time for the velocity field when using the BDF2-IMEX-RK2 scheme. 555 556 557 558 559 560 561 562 563 564 Consistent with the findings of Lipscomb et al. [2], we observe that SIT is less sensitive in 2-D than in 1-D. Shear stress tends to help the numerical scheme. A test with an elliptical yield curve with a very large aspect ratio of 1000 (i.e., with very small resistance to shear deformations) shows that results in 2-D exhibit a similar behavior to results in 1-D (the mean number of Newton iterations and RMSE for SIT increases significantly for large t, not shown). Our results also suggest that our model is less sensitive to the SIT instability than the one of Lipscomb et al. [2]. This is likely because we use a two-thickness category model as opposed to their multi-category model. 565 566 567 568 569 570 Fig. 6a and Fig. 6b respectively show the mean number of Newton iterations per time level (last 12 h) and the total CPU time required for the last 12 h of the one day integration, as a function of t, for the di↵erent time integration schemes. As opposed to the 1-D experiments, the number of Newton iterations for SIT is about the same as for IMEX and BDF221 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 IMEX-RK2 even for large t. BDF2-IMEX-RK2 requires roughly 10-25% more total CPU time than SIT for the same t. As this is not due to an increase in the number of Newton iterations (the number is even slightly lower for BDF2-IMEX-RK2), the extra CPU time for BDF2-IMEX-RK2 is rather a consequence of the additional operations inside the Newton loop (the twostep advection operator). However, comparing the computational efficiency of SIT and BDF2-IMEX-RK2 for the same t is not a fair comparison as the integration schemes do not lead to the same accuracy. As an example, BDF2-IMEX-RK2 with a t of 90 min leads to an approximate solution that is more accurate (RMSE of 1.77 ⇥ 10 4 m) than the one obtained with SIT with t= 10 min (RMSE of 2.86 ⇥ 10 4 m, Fig. 4b). As the total CPU time required by BDF2-IMEX-RK2 with t= 90 min is 146 s and the one for SIT with t= 10 min is 775 s, this means that the second-order scheme is more than five times faster than the SIT integration scheme to obtain the same accuracy. 586 587 588 589 590 591 592 593 594 595 596 597 598 599 Fig. 7c shows how the thickness errors are spatially distributed on the pan-Arctic domain when using BDF2-IMEX-RK2 with t=90 min. This can be compared to the errors obtained with SIT for the same t of 90 min (Fig. 7b). Fig. 7b shows that notable errors are found at many places in the domain, with the largest errors close to the coast lines. The largest errors in SIT with t=90 min is -7.6 cm while the maximum error is reduced to 0.34 cm with BDF2-IMEX-RK2 when using the same time step. As mentioned earlier, SIT needs a t=10 min to obtain a comparable RMSE than the one obtained with BDF2-IMEX-RK2 with t=90 min. The spatial errors for SIT for a t of 10 min are shown on Fig. 7d. Qualitatively speaking, it can be observed that the errors in Fig. 7c and Fig. 7d are of similar magnitude, although the spatial patterns are di↵erent. The largest error for SIT with t=10 min is -0.78 cm. 600 601 602 603 604 605 606 607 5.3. Robustness We have first assessed the robustness of the BDF2-IMEX-RK2 scheme when using winds that change more abruptly. We repeated the 40 km resolution experiments of Section 5.2 but with winds that change a lot more quickly. The time constant in equation (30), that determines how quickly the winds are ramped up, was set to 1 hour (instead of 6 hours). Results demonstrate that the BDF2-IMEX-RK2 scheme still leads to second-order 22 608 accuracy in time (not shown). 609 610 611 612 613 614 615 616 617 We have also investigated how robust is our JFNK solver when used in the context of the BDF2-IMEX-RK2 scheme or in the context of the SIT first-order approach. We ran the 2-D model for five years (2002-2007) at 40 and 20-km resolutions with either BDF2-IMEX-RK2 or SIT and counted the number of failures of JFNK. For all these experiments, t is 30 min and 4 nl = 10 . Note that realistic wind forcing was used and thermodynamic source terms were included (through operator splitting) for these long simulations. 618 619 620 621 622 623 624 625 626 627 628 The introduction of the linesearch globalization and to a lesser extent of the Bouillon et al. [9] approach for the calculation of the viscous coefficients clearly improved the robustness of our JFNK solver when compared to the first version described in [4]. For these five-year integrations, JFNK within both the SIT and BDF2-IMEX-RK2 schemes did not fail at 40-km resolution. However, at 20-km resolution, JFNK failed a few times for both integration schemes. In terms of percentage, the failure rate is 0.027 % for SIT while it is 0.025 % for BDF2-IMEX-RK2. Losch et al. [11] report a failure rate of 0.006% with a SIT approach over a 50 year simulations for a spatial resolution of 27 km. 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 6. Discussion and concluding remarks To our knowledge, we have demonstrated for the first time second-order temporal accuracy in a sea ice dynamic model. This second-order scheme was implemented relatively easily from a Splitting In Time (SIT) scheme using a Jacobian-free Newton-Krylov (JFNK) nonlinear solver. Basically, three minor modifications were made to this configuration to get second-order accuracy in time. First, the advection operation was moved inside the Newton loop such that the ice thickness and concentration fields are updated along with the velocity field during the Newton iteration. Secondly, the first-order explicit advection operation was upgraded to a second-order Runge-Kutta (RK2) predictor-corrector approach. Finally, in order to get second-order accuracy, the backward Euler time discretization in the momentum equation was replaced by a second-order backward di↵erence formula (BDF2) integration scheme. We refer to this new iterated IMplicit-EXplicit (IMEX) scheme 23 644 645 as BDF2-IMEX-RK2. This implementation is a lot more straightforward than the development of a fully implicit scheme would have been. 646 647 648 649 650 651 652 653 654 655 656 657 658 The Root Mean Square Error (RMSE) between thickness fields obtained with di↵erent time steps ( t) and a reference solution thickness field demonstrates that BDF2-IMEX-RK2 is second-order accurate in time. The supporting analysis can be found in Appendix B. Results at 40 and 20-km resolutions lead qualitatively to the same conclusions. For the same t, BDF2IMEX-RK2 is always more than one order of magnitude more accurate than the SIT approach. As an example, the approximate solution obtained with BDF2-IMEX-RK2 with t= 90 min is more accurate than the one obtained with SIT with t=10 min. Hence, to get the same level of accuracy than SIT, significantly larger time steps can be used with BDF2-IMEX-RK2 which leads to a decrease in the computational time. This efficiency gain is greater than a factor of 5 at 20-km resolution. 659 660 661 662 663 664 665 666 The implementation of this efficient second-order accurate in time scheme was possible because our nonlinear solver for the momentum equation is a Newton-Krylov scheme. As the EVP solver [8] is an explicit scheme, the IMEX approach would not be possible with this method. On the other hand, IMEX could be implemented in the framework of a Picard iteration (e.g. [5, 6, 7]) although the Picard solver is known to exhibit a very inefficient nonlinear convergence [7, 11]. 667 668 669 670 671 672 673 674 675 676 677 678 679 To maintain the fast nonlinear convergence of JFNK with the IMEX approach, it is crucial to take into account the changes in thickness and concentration associated with a change of velocity when performing the calculation of the Jacobian times a vector. This operation is performed correctly in our BDF2-IMEX-RK2 as can be seen in Fig. 6a. This figure shows that the mean number of Newton iterations is about the same with BDF2-IMEX-RK2 than it is with the SIT scheme (it is even a little lower). To reinforce this conclusion, we show in Fig. 8 a typical nonlinear evolution of the L2-norm of the residual for BDF2-IMEX-RK2 and for the SIT schemes. The time step is 30 min and the resolution is 20 km. Both schemes exhibit a very similar nonlinear convergence. They both need 12 Newton iterations to reach the nonlinear convergence criterion ( nl = 10 6 ). 680 681 As in Lipscomb et al. [2], we found that the 2-D model is less sensitive 24 682 683 684 685 686 687 688 than the 1-D model to the SIT instability. The BDF2-IMEX-RK2 scheme is nevertheless useful as the SIT instability is more severe as the grid is refined and when using a multi-category sea ice model [2]. Note that our method could easily be applied to a multi-category model. Furthermore, a sea ice model using a yield curve having less shear strength than the standard elliptical yield curve would also be more exposed to this instability and would therefore benefit from the more stable BDF2-IMEX-RK2 scheme. 689 690 691 692 693 An obvious extension to this work would be to develop a second-order scheme that would also include thermodynamic processes. To do so, the predictor-corrector approach would include the source terms and would become (h⇤ hn 1 ) = t/2 r · (un 1 hn 1 ) + Sh (hn 1 , An 1 ), (hn 694 1 hn 1 ) = r · (un 2 h⇤ ) + Sh (h⇤ , A⇤ ), t ⇤ n where A and A would be obtained in a similar way. (31) (32) 695 696 697 698 699 700 701 Another improvement would be to replace our di↵usive first-order in space upstream scheme by a more sophisticated advection operator. For example, second-order accuracy in space could also be achieved by using the remapping scheme of Lipscomb and Hunke [31]. Note that a stabilization method (di↵erent time-stepping approach) may be required as higher order advection schemes are less di↵usive than a first-order upstream operator. 702 703 704 705 706 707 708 709 The JFNK solver is remarkably robust in longer simulations (five years). At 40-km resolution, JFNK did not fail for either the SIT or the BDF2IMEX-RK2 integration scheme. At 20-km resolution, convergence was not reached on rare occasions for both integration schemes. With SIT, JFNK had a failure rate as low as 0.027 % while JFNK with the BDF2-IMEX-RK2 scheme failed for only 0.025 % of the time levels (this is slightly smaller than for SIT but probably not statistically significant). 710 711 712 Even though these failure rates are very small and when a failure occurs it usually a↵ects only a few grid cells (not shown), the increase in the failure 25 713 714 715 716 rates with resolution indicates that further work is needed to improve the robustness. A more sophisticated approach than the linesearch method might help (e.g. [32]) but we also suspect that our preconditioning approach might need to be revisited as we refine the grid. 717 718 719 720 721 722 723 724 725 726 Indeed, as the spatial resolution increases, the rheology term makes the problem more and more nonlinear. We have observed occasional failures of the preconditioned FGMRES at 10-km resolution for a linear tolerance of 0.1. To improve our preconditioning operator, we are currently working on using the MultiLevel (ML) preconditioner from the Trilinos library [33]. It is possible, however, that this might not be sufficient and that we might have to reconsider the use of the Picard matrix for the preconditioning step. In other words, our preconditioning matrix might have to be closer to the Jacobian matrix than what the Picard matrix is. 727 728 729 730 731 732 733 734 735 736 737 738 This study was done using a serial code. Losch et al. [11] have recently implemented a parallel JFNK solver for sea ice dynamics. They have demonstrated that the scaling of JFNK with a similar line relaxation approach for the preconditioner is almost as good as for other solvers (Picard and EVP); in their case for domain decompositions of up to 1000 CPUs. There is no reason to believe that our BDF2-IMEX-RK2 approach would not exhibit similar performances as the additional thickness and concentration calculations performed in the Newton loop are explicit and do not require extra communication overheads. Using a di↵erent preconditioner (such as ML) might lead to an improved scalability of JFNK. This is the subject of future work. 739 740 741 742 743 744 Appendix A: Undamped oscillation with a Crank-Nicolson approach By centering in time (at n 12 ) the terms in the momentum equation, a Crank-Nicolson approach also leads to second-order accuracy (not shown). However, as explained here, it can lead to an undamped oscillation in zones with little ice. With this approach, the u and v equations are written as ⇢h n 1 2 (un un 1 ) = ⇢hn t 1 2 fv n 1 2 n +⌧au 1 2 n ⌧wu 1 2 + @ n 11 1 2 n (Pp @x 1 2) + @ n 12 1 2 n (Pp @y 1 2) , (33) 26 ⇢h n 1 2 (v n 746 where hn that the 747 and that 745 vn 1) = t 1 2 ⇢h n 1 2 fu n 1 2 n +⌧av 1 2 n ⌧wv 1 2 + @ n 22 1 2 n (Pp + @ n 12 1 2 n (Pp 1 2) @x (34) hn +hn 1 n 12 n 12 n 12 = and A ,u and v are similarly defined. Note 2 1 1 and the water stress components are functions of un 2 and v n 2 ij n Pp 1 2 = P ⇤ hn 1 2 exp[ C(1 An 1 2 @y 1 2) )]. 748 749 750 751 752 753 754 Assuming a region with very thin ice, the balance of force is then between the water stress and the wind stress. To explain the oscillation, we further simplify the problem by setting the water turning angle to zero and by assuming that the ocean is at rest and that the wind is blowing from the west (such that the ice velocity is positive). The momentum balance then becomes u n + un 1 2 ), (35) 2 Assume that the wind stress was zero before such that un 1 = 0 and that after that it is constant and equal to ⌧au . The velocity at time level n is then r ⌧au n u =2 , (36) ⇢w Cdw n ⌧au 755 756 757 759 760 761 762 763 = ⇢w Cdw ( while at n + 1 it is equal to u 758 1 2 n+1 =2 r ⌧au ⇢w Cdw un = 0, (37) and we q find that un+2 = un , i.e., the solution oscillates between two values: 0 and 2 ⇢w⌧Caudw . This undamped oscillation is more severe when using large time steps as a significant time di↵erence between two time levels is more likely to lead to a large change in the wind stress. This oscillation is not observed when using the second-order backward di↵erence time integration approach. 764 27 , 765 766 767 768 769 770 771 772 Appendix B: Truncation error analysis We perform a truncation error analysis similar to the one described in Kadioglu and Knoll [14]. We assume a 1-D problem, that the velocity is positive, that the concentration is 1 everywhere and that the viscous coefficients are constant in space and in time. The replacement closure (equation (9)) is not used such that P = Pp . We also assume that the Newton iteration has already converged such that uk = un and hk = hn . The momentum equation is then given by @u @ 2 u 1 @P = R = ⌧a Cu2 + ⇣ 2 , (38) @t @x 2 @x where C = ⇢w Cdw , P = P ⇤ h and R is just the sum of all the terms on the 2 RHS. To simplify the notation, we introduce Lu (u) = @@xu2 and Lp (P ) = @P . @x The continuity equation for h is ⇢h 773 774 775 @h = @t 776 @(uh) , @x for which we introduce the operator Luh (uh) = (39) @(uh) . @x 777 778 779 At time level n we solve with our BDF2-IMEX-RK2 method the following equations ⇢hn ( 3un 2 2un hn = hn 780 with un 1 2 1 + un 2 )= 2 tLuh (un 1 2 tRn , h⇤ ), (40) (41) and h⇤ given by u n 1 2 (un + un 1 ) = , 2 (42) t Luh (un 1 hn 1 ). (43) 2 We use the following Taylor series to express un as a function of un 1 h⇤ = hn 781 1 un = un 1 + t 1 @un @t 1 + 28 t 2 @ 2 un 2 @t2 1 + O( t3 ). (44) 782 783 784 785 786 787 788 789 We now prove that our BDF2-IMEX-RK2 method leads to second-order accuracy in time for the calculation of the velocity and the thickness. If h and u are both second-order accurate in time, their product is also secondorder accurate in time. We can demonstrate this by starting from equation (40) and then by using the other equations we introduced above (the LHS of equation (40) is expressed in terms of products of h and u). Using equation (44) and also a Taylor expansion around un 1 for un 2 , the LHS of equation (40) can be written as 3 ⇢hn (un 2 1 ⇢hn (un 2 790 791 n 1 ⌧✏ = ⇢ h 795 797 1 1 1 + O( t3 ) . (45) (46) tLuh (u n 1 2 ⇤ h) i t @un @t 1 + t2 @ 2 un @t2 1 + O( t3 ) . (47) n 1 tLuh (u n 1 2 ⇤ h) i t @un @t 1 + t2 @ 2 un @t2 1 + O( t3 ) tRn , (48) where Rn is expanded below. The terms can be rearranged such that one obtains @un 1 +O( t3 ) tRn . @t @t (49) Using equations (41) and (44) and introducing a Taylor series for the wind stress, Rn can be written as ⌧✏ = 796 t 2 @ 2 un 1 ) ⇢hn 2un 1 + 2 @t2 t 2 @ 2 un 1 + ) + O( t3 ), 2 2 @t + From the latest equation, the truncation error (⌧✏ ) can be obtained by subtracting the RHS of equation (40) from expression (47) h 794 1 t Substituting hn from equation (41) in (46) we get ⇢ h 793 @un @t @un t @t + which after regrouping the terms becomes @un 1 @ 2 un n ⇢h t + t2 @t @t2 h 792 1 t⇢hn 1 @u n 1 + t2 ⇢hn [email protected] 2 n 1 u @t2 29 t2 ⇢Luh (un 1 2 h⇤ ) @⌧an 1 @un R + t C (un 1 )2 + 2 tun 1 @t @t @un 1 P⇤ + t⇣Lu ( ) Lp (hn ) + O( t2 ). @t 2 Using again equation (41) for hn , we get n 798 R n =⌧an 1 =⌧an 1 + 799 800 Simplifying and using Luh (un (51) we get 1 + t @⌧an @t 1 2 1 1 @u 2 tCun n 1 @tn tP ⇤ @h Lp 2 @t where we have used the fact that Luh (un 1 hn 1 ) = can write the previous equation as n R =R (50) h⇤ ) = Luh (un 1 hn 1 ) + O( t) in equation @ t ⇣Lu (un 1 ) @t + 802 + ⇣Lu (un 1 ) @⌧an 1 @un 1 + t C (un 1 )2 + 2 tun 1 + ⇣Lu (un 1 ) @t @t i 1 @un 1 P⇤ tP ⇤ h t⇣Lu ( ) Lp (hn 1 ) + Lp Luh (un 2 h⇤ ) + O( t2 ), @t 2 2 (51) Rn =Rn 801 1 n 1 @ + t ⌧an @t 1 1 (52) + O( t2 ), @hn @t 1 . Rearranging, we P⇤ Lp (hn 1 ) + O( t2 ). 2 (53) so we can write C(un 1 )2 + ⇣Lu (un 1 ) 803 The term inside the brackets is just Rn 804 @Rn 1 + O( t2 ). @t We replace Rn in equation (49) using equation (54) and obtain Rn = Rn ⌧✏ = t⇢h n 1 @u tR n 1 @t n 1 + 2 1 2 t ⇢h @Rn t @t + n [email protected] 1 t 2 n 1 u @t2 1 t ⇢Luh (u 3 + O( t ). 30 2 n 1 2 @un h) @t ⇤ (54) 1 (55) 805 1 2 Using again Luh (un ⌧✏ = t ⇢h n 1 @u n 1 R @t h⇤ ) = Luh (un 1 hn 1 ) + O( t), we can write n 1 + 3 + O( t ). t 2 ⇢hn [email protected] 2 n 1 u @t2 +⇢ @hn 1 @un @t @t 1 @Rn @t 1 (56) 806 807 Using equation (38), we can eliminate the O( t) terms. We now use 2 @h = h @@t2u + @u to get @t @t @ [email protected] @t @t ⌧✏ = t 2 ⇢ @ n h @t 1 @u n 1 @t 809 810 @un 1 @hn @t @t 1 +⇢ @hn 1 @un @t @t 1 @Rn @t 1 + O( t3 ). (57) ⌧✏ = 808 ⇢ @ t⇢ hn @t 2 1 @u n 1 @t Rn 1 + O( t3 ). 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Soft. 31 (2005) 397–423. 34 −2 −2 10 10 (a) −3 −3 10 10 −4 −4 10 RMSE (m) RMSE (m) 10 −5 10 −6 −5 10 −6 10 10 x=40 km −7 x=20 km −7 10 10 SIT IMEX BDF2−IMEX−RK2 −8 10 (b) 0 10 1 2 10 10 Time step (min) SIT IMEX BDF2−IMEX−RK2 −8 3 10 10 0 10 1 2 10 10 Time step (min) 3 10 Figure 4: RMSE between an approximate solution and the reference thickness as a function of t for spatial resolutions of 40 km (a) and 20 km (b). The black curve with triangles is the SIT method, the red curve with diamonds is the IMEX method while the blue curves with circles is for BDF2-IMEX-RK2. 35 −1 10 −2 10 −3 RMSEv (cm/s) 10 −4 10 −5 10 −6 10 36 −7 10 0 10 1 SIT IMEX BDF2−IMEX−RK2 2 10 10 Time step (min) 3 10 4 10 (a) (b) SIT IMEX BDF2−IMEX−RK2 20 Total CPU time (s) Mean nb of Newton iterations 25 15 10 3 10 5 SIT IMEX BDF2−IMEX−RK2 0 0 20 40 60 80 100 120 140 160 2 10 180 0 20 40 60 80 100 120 140 160 180 Time step (min) Time step (min) Figure 6: (a) Mean number of Newton iterations per time level as a function of t. (b) Total CPU time as a function of t. These two quantities were calculated for the last 12 h of the integration. The black curve with triangles is the SIT method, the red curve with diamonds is the IMEX method while the blue curves with circle is for BDF2-IMEX-RK2. The spatial resolution is 20 km. 37 (a) (b) (c) (d) Figure 7: (a) Reference solution thickness field (in m) on 18 January 2002 00Z. This field is capped to 4 m on the figure to see more details. (b) Di↵erence (in m) between the approximate solution obtained with SIT with t = 90 min and the reference solution. (c) Di↵erence (in m) between the approximate solution obtained with BDF2-IMEX-RK2 with t = 90 min and the reference solution. (d) Di↵erence (in m) between the approximate solution obtained with SIT with t = 10 min and the reference solution. The di↵erence fields are capped to ±0.01 m. Note that the scale is di↵erent in (a). The spatial resolution is 20 km. 38 0 10 BDF2−IMEX−RK2 SIT −1 10 −2 L2−norm 10 −3 10 −4 10 −5 10 −6 10 −7 10 0 5 10 15 Newton iteration Figure 8: L2-norm on 18 January 2002 00Z as a function of the number of Newton iterations when using the SIT scheme (black curve with triangles) and the BDF2-IMEX-RK2 scheme (blue curve with circles). The time step is 30 min and the spatial resolution is 20 km. 39

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