Using the preconditioned Generalized Minimum RESidual (GMRES) method to solve the sea‐ice momentum equation

Abstract

[1] We introduce the preconditioned generalized minimum residual (GMRES) method, along with an outer loop (OL) iteration to solve the sea-ice momentum equation. The preconditioned GMRES method is the linear solver. GMRES together with the OL is used to solve the nonlinear momentum equation. The GMRES method has low storage requirements, and it is computationally efficient and parallelizable. It was found that the preconditioned GMRES method is about 16 times faster than a stand-alone successive overrelaxation (SOR) solver and three times faster than a stand-alone line SOR (LSOR). Unlike stand-alone SOR and stand-alone LSOR, the cpu time needed by the preconditioned GMRES method for convergence weakly depends on the relaxation parameter when it is smaller than the optimal value. Results also show that with a 6-hour time step, the free drift velocity field is a better initial guess than the previous time step solution. For GMRES, the symmetry of the system matrix is not a prerequisite. The Coriolis term and the off-diagonal part of the water drag term can then be treated implicitly. The implicit treatment eliminates an instability characterized by a residual oscillation in the total kinetic energy of the ice pack that can be present when these off-diagonal terms are handled explicitly. Treating these terms explicitly prevents one from obtaining a high-accuracy solution of the sea-ice momentum equation unless a corrector step is applied. In fact, even after a large number of OL iterations, errors in the drift of the same magnitude as the drift itself can be present when these terms are treated explicitly.

Improving the numerical convergence of viscous-plastic sea ice models with the Jacobian-free Newton–Krylov method

“…The advection is performed after the last OL (or Newton) iteration with a simple upstream scheme. More details on the model can be found in Lemieux et al [5].…”

“…This is not a problem because the GMRES method, as opposed to the conjugate gradient method for example, does not require a symmetric system matrix [20] and therefore permits an implicit treatment of the off-diagonal terms. Our implementation of the preconditioned GMRES method is described in Lemieux et al [5].…”

“…The advantage of using this matrix M is that we use the same preconditioner as used in the standard method. During preconditioning, a fixed number of iterations (8 iterations) of a LSOR solver are performed on Mz j ¼ v j where z j and v j are vectors formed during the jth Krylov iteration [5].…”

“…The linearized system of equations formed at each OL iteration is solved by a linear solver such as a successive overrelaxation (SOR) method [3], a line SOR (LSOR, [4]) or the preconditioned generalized minimum residual (GMRES) method [5].…”

Is ice-rafted sediment in a North Pole marine record evidence for perennial sea-ice cover?

“…Furthermore, as we are specifically interested in the dynamics of the seasonal ice pack, the details of the growth/melt of the seasonal ice cover do not impact our conclusions. More details about the model and numerical method used to solve the sea-ice momentum equation can be found in [38,39].…”