Numerical investigation of ensemble methods with block iterative solvers for evolution problems
Numerical investigation of ensemble methods with block iterative solvers for evolution problems
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DOI：
10.3934/dcdsb.2020132
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
L. Ju;W. Leng;Zhu Wang;Shuai Yuan
中科院分区：
文献类型：
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作者：
L. Ju;W. Leng;Zhu Wang;Shuai Yuan
系综方法已被开发用于加速演化问题的一系列数值模拟。其主要思想是通过对离散问题的时间步长和分组处理，使同一组中的所有成员共享一个共同的系数矩阵。因此，在每个时间步，而不是解决一系列的线性系统，其中每个只包含一个右手侧向量，集成方法同时解决了一个单一的线性系统与多个右手侧向量为每组。这样的系统可以有效地解决使用直接线性求解器时，问题是小规模的，因为相同的LU分解将工作的整个组成员。然而，对于大规模的问题，迭代线性求解器往往必须使用，然后这种吸引人的优势变得不明显。本文系统地研究了两类典型发展问题：热传导方程和不可压Navier-Stokes方程的块迭代系综方法的数值性能。特别是，块共轭梯度（CG）求解器被认为是前者和块广义最小残差（GMRES）求解器为后者。我们的数值结果表明，这些块迭代求解器一起工作时，合奏方法的有效性和效率。
The ensemble method has been developed for accelerating a sequence of numerical simulations of evolution problems. Its main idea is, by manipulating the time stepping and grouping discrete problems, to make all members in the same group share a common coefficient matrix. Thus, at each time step, instead of solving a sequence of linear systems each of which contains only one right-hand-side vector, the ensemble method simultaneously solves a single linear system with multiple right-hand-side vectors for each group. Such a system could be solved efficiently by using direct linear solvers when the problems are of small scale, as the same LU factorization would work for the entire group members. However, for large-scale problems, iterative linear solvers often have to be used and then this appealing advantage becomes not obvious. In this paper we systematically investigate numerical performance of the ensemble method with block iterative solvers for two typical evolution problems: the heat equation and the incompressible Navier-Stokes equations. In particular, the block conjugate gradient (CG) solver is considered for the former and the block generalized minimal residual (GMRES) solver for the latter. Our numerical results demonstrate the effectiveness and efficiency of the ensemble method when working together with these block iterative solvers.