Using the preconditioned Generalized Minimum RESidual (GMRES) method to solve the sea‐ice momentum equation
Using the preconditioned Generalized Minimum RESidual (GMRES) method to solve the sea‐ice momentum equation
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使用预处理广义最小残差（GMRES）方法求解海冰动量方程
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
L. Mysak
中科院分区：
文献类型：
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作者：
J. Lemieux;B. Tremblay;Stephen J. Thomas;J. Sedlacek;L. Mysak
[1] 我们引入了预处理广义最小残差（GMRES）方法以及外循环（OL）迭代来求解海冰动量方程。预处理 GMRES 方法是线性求解器。 GMRES 与 OL 一起用于求解非线性动量方程。 GMRES方法存储要求低，计算效率高且可并行。研究发现，预处理 GMRES 方法比独立连续过松弛 (SOR) 求解器快约 16 倍，比独立线 SOR (LSOR) 快约 3 倍。与独立SOR和独立LSOR不同，当松弛参数小于最优值时，预处理GMRES方法收敛所需的CPU时间弱依赖于松弛参数。结果还表明，在 6 小时的时间步长下，自由漂移速度场是比之前的时间步长解更好的初始猜测。对于 GMRES，系统矩阵的对称性不是先决条件。然后可以隐式处理科里奥利项和水阻力项的非对角线部分。隐式处理消除了以冰包总动能残余振荡为特征的不稳定性，当显式处理这些非对角项时，可能会出现这种不稳定性。明确处理这些项会阻止人们获得海冰动量方程的高精度解，除非应用校正器步骤。事实上，即使在大量 OL 迭代之后，当明确处理这些项时，也可能会出现与漂移本身相同量级的漂移误差。
[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.