发展方程差分格式的非线性计算稳定性问题

Problems on Nonlinear Computational stability of the Difference Schemes of Evolution Equations

非线性计算稳定性是计算数学、计算物理、计算气象等学科中的一个重要问题。讨论非定常运动的两类典型非线性发展方程（双曲型守恒系统和耗散系统）数值格式的计算稳定性，给出一种判定差分格式计算稳定性的新方法。数值试验证明该方法是实用的和有效的，得到的判据的确是保证差分格式计算稳定的必要条件；指出非线性计算不稳定与原始微分方程解的性质密切有关，同时还和具体格式结构及初值形式有关。

The stability of nonlinear computation is important in numerical mathematics, computing physics and numerical weather forecasting. Taken two kinds of evolution equations as classical models (The hyperbolic conservation system and dissipation system), a new method for evaluating the computational stability is proposed. It is proved to be practical and effective by several numerical examples. The stability criteria obtained by this method are the necessary conditions of computational stability. It is emphasized the strong dependence of nonlinear computational stability on the properties of original differential equation solution. It depends on structure of the schemes and the initial condition too.