Differential Equations, Difference Equations and Matrix Theory
Differential Equations, Difference Equations and Matrix Theory
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微分方程、差分方程和矩阵理论
DOI：
10.1002/cpa.3160110203
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
P. Lax
中科院分区：
文献类型：
--
作者：
P. Lax
Friedrichs 在他对对称双曲系统解的差分近似的研究中表明，具有正系数的差分格式是稳定的。在本文中，我们证明了弗里德里希斯的稳定性准则对于任意双曲系统的差分近似是有效的。该证明依赖于特征值的凸性和单调性，作为仅具有实特征值的矩阵线性空间上的矩阵函数。这些在对称情况下众所周知的定理借助有关双曲方程的解与其初始数据的依赖性的定理来证明。
Friedrichs, in his studies of difference approximations to solutions of symmetric hyperbolic systems, has shown that difference schemes with positive coefficients are stable. In this paper we show that this stability criterion of Friedrichs is valid for difference approximations to arbitrary hyperbolic systems. The proof relies on the convexity and monotonicity of eigenvalues as matrix functions over a linear space of matrices with only real eigenvalues. These theorems, well known in the symmetric case, are proved with the aid of theorems concerning the dependence of solutions of hyperbolic equations on their initial data.