Robin型离散Schwarz波形松弛算法的收敛性分析

Convergence analysis for discrete Schwarz waveform relaxation algorithm of Robin type

Schwarz波形松弛(Schwarz waveform relaxation,SWR)是一种新型区域分解算法,是当今并行计算研究领域的焦点之一,但针对该算法的收敛性分析基本上都停留在时空连续层面.从实际计算角度看,分析离散SWR算法的收敛性更重要.本文考虑SWR研究领域中非常流行的Robin型人工边界条件,分析时空离散参数t和x、模型参数等因素对算法收敛速度的影响.Robin型人工边界条件中含有一个自由参数p,可以用来优化算法的收敛速度,但最优参数的选取却需要求解一个非常复杂的极小-极大问题.本文对该极小-极大问题进行深入分析,给出最优参数的计算方法.本文给出的数值实验结果表明所获最优参数具有以下优点:(1)相比连续情形下所获最优参数,利用离散情形下获得的参数可以进一步提高Robin型SWR算法在实际计算中的收敛速度,当固定t或x而令另一个趋于零时,利用离散情形下所获参数可以使算法的收敛速度具有鲁棒性(即收敛速度不随离散参数的减小而持续变慢).(2)相比连续情形下所获收敛速度估计,离散情形下获得的收敛速度估计可以更加准确地预测算法的实际收敛速度.

t Schwarz waveform relaxation （SWR） algorithm is a new kind of domain decomposition methods and receives lots of attention in the field of parallel computation. However, up to now the algorithm is only analyzed at continuous level. From a computational point of view, it is more important to investigate the SWR algorithm at discrete level, since the continuous SWR algorithm can not be straightforwardly used to calculate the solution of a PDE problem. In this paper, we analyze the SWR algorithm at discrete level and investigate how the discrezation parameters △x and △t affect the convergence rate of the algorithm. We consider Robin condition as the artificial boundary condition in our analysis and a free parameter, namely p, is involved in this artificial boundary condition which can be optimized to accelerate the convergence rate of the SWR algorithm. However, the best choice of p is concerned with a min-max problem which is much more complex than the one arising at continuous level. We investigate this min-max problem deeply and propose a computational formula for the best parameter p. The numerical results given in this paper show that the obtained optimized parameter has the following advantages： （1） Compared with the parameter obtained at continuous level, the convergence rate of the SWR algorithm in practical computation can be further improved by using the parameter obtained in this paper. Moreover, if we fix △x or △t and let the other one approach to zero, the SWR algorithm behaves robustly, i.e., refining △x or △t does not deteriorate the convergence rate. （2） Compared with the bound obtained at continuous level, the bound presented in this paper can predict the convergence rate of the SWR algorithm more accurately in practical computation.