非线性平流方程差分格式的稳定性分析

Stability Analysis for Difference Schemes of Nonlinear Advection Equations

非线性计算稳定性是计算数学、计算物理中的一个重要问题。本文以一维非线性平流方程为模型，给出一个较普遍的差分格式（以往若于常见平流格式均为其特例），对格式进行定性和定量的稳定性分析与比较，总结出若干容易出现非线性计算不稳定的情况，特别强调了非线性计算稳定性与非线性发展方程解的性质和格式构造及初值形式的依赖关系，从稳定性的启发性分析侧面，揭示了非线性计算不稳定的特性和机理。

The stability of nonlinear computation is an important problem in numerical mathematics and computing physics. In this paper, with the model of one-dimensional nonlinear advection equation, making auniversal difference scheme (Some common advection schemes before are its special examples). The comparative analysis for stability is carried out qualitatively and quantitatively for these schemes, and sum up thateasy to produce some things of nonlinear computational instability. It is emphasized that nonlinear computationalstability closely depends on the properties of solution to original equation, the structure of the scheme and theinitial condition adopted. From the method of elicitation analysis on stability, the nature and mechanism ofnonlinear computational instability is disscussed.