分解算法中的两个问题

Two Problems Existing in the Split Algorithm

文中指出了分解算法中的两个问题:(1)为了使算法获得最佳效果,除按快、慢过程分解方程外,进一步的分解应按什么原则?(2)时间离散是否可以任意,如果不然,应采取什么形式?针对这两个问题得到的研究结果是:和一般设计简化模式一样,分解后的方程应保持原方程的重要物理性质,特别是其整体性质(如能量守恒和质量守恒等),而且,其分解后方程的解还应和原方程的解一致,至少对其中重要的解应一致.至于时间离散,在差分的情况则应采用二时间层格式;如采用三时间层格式,则会导致差分方程(或差分微分方程)和偏微分方程的不相容.

It is pointed out in this paper that there exist two problems to be settled in the split algorithm. They are: a) In addition to splitting the original equations according to fast and slow processes, which principle should be based on to split the equations again in order to get optimal efficiency? b) Whether or not the form of time discretization is allowed to be arbitrary, or in which form time discretization should be taken? The research to solve the first problem shows that the secondly split equation should preserve the important physical properties of the original equations, especially their overall properties (such as conservation of energy and mass) and the solutions of the secondly split equations should be consistent with those of the original equations, at least be consistent with some important ones of them. The result for settling the second problem indicates that in the case of finite difference scheme, two-time-level scheme is desirable; if three-time-level scheme is adopted, inconsistency would happen between the scheme and the corresponding set of partial differential equations.