波形松弛方法的绝对稳定与压缩

ABSOLUTE STABILITY AND CONTRACTION OF WAVEFORM RELAXATION METHODS

波形松弛(WR)方法是求常微分方程近似解的数值方法,对它的研究多集中于收敛性,极少见到稳定性研究报告,而不稳定的数值方法是没有意义的.借鉴常微分方程数值方法绝对稳定的思想,提出了WR方法的绝对稳定定义.分析连续基本WR方法和基于Θ方法的离散基本WR方法的稳定性,给出了连续和离散WR方法的绝对稳定条件,以及离散WR方法的压缩条件.对于WR方法,分裂函数和数值方法(用于离散连续WR方法)的选择是两个基础问题.论文结论部分地揭示了WR方法的稳定性与分裂函数和数值方法的关系.

The waveform relaxation method is the numerical method of solving approximately the ordinary differential equation. Most of research works on WR mehtods focus on convergence, few of them concern the stability although an unstable numerical method is unmeaning. The definition of absolute stability of WR methods is presented by generalizing absolute stability of numerical methods of ordinary differential equations. The analysis for stability of con-tinuous basic WR methods and discrete basic WR methods based on Θ-methods leads to the absolutely stable conditions of the continuous and discrete WR methods, and the con-tracting condition of discrete WR methods. The choice of splitting functions and numerical methods (used to discrete continuous WR methods) is two basic problems of WR methods. The results of this paper shows partly the relationship between the stability of WR methods and the splitting functions and numerical methods used.