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.

中文翻译：

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带有块迭代求解器的演化问题集成方法的数值研究

已经开发了集成方法来加速一系列演化问题的数值模拟。它的主要思想是，通过控制时间步长并对离散问题进行分组，以使同一组中的所有成员共享一个公共系数矩阵。因此，集成方法在每个时间步上都不会求解线性系统序列（每个线性系统仅包含一个右侧向量），而是针对每个组同时求解具有多个右侧向量的单个线性系统。当问题规模较小时，可以使用直接线性求解器来有效地解决这样的系统，因为相同的LU分解对于整个小组成员都适用。但是，对于大规模问题，通常必须使用迭代线性求解器，然后这种吸引人的优势就不明显了。在本文中，我们系统地研究了带有块迭代求解器的集成方法的数值性能，用于求解两个典型的演化问题：热方程和不可压缩的Navier-Stokes方程。特别地，前者考虑了块共轭梯度（CG）求解器，后者考虑了块广义最小残差（GMRES）求解器。我们的数值结果证明了与这些块迭代求解器一起使用时，集成方法的有效性和效率。前者考虑块共轭梯度（CG）求解器，后者考虑块广义最小残差（GMRES）求解器。我们的数值结果证明了与这些块迭代求解器一起使用时，集成方法的有效性和效率。前者考虑块共轭梯度（CG）求解器，后者考虑块广义最小残差（GMRES）求解器。我们的数值结果证明了与这些块迭代求解器一起使用时，集成方法的有效性和效率。