0rr－Sommerfeld方程数值解法中的复广义矩阵特征值问题

THE COMPLEX GENERALIZED MATRIX EIGENVALUE PROBLEM IN NUMERICAL SOLVER FOR ORR-SOMMERFELD EQUATION

Ｏｒｒ－Ｓｏｍｍｅｒｆｅｌｄ方程的求解通常可以化为一个复广义矩阵特征值问题ＡＸ＝ωＢＸ。本文用酉变换分别约化Ａ，Ｂ为上Ｈｅｓｓｅｎｂｅｒｇ阵和上三角阵，然后利用Ｍｕｌｌｅｒ求根方法可以求出其全部特征值，其中特征多项式的值由Ｈｙｍａｎ方法给出。当仅需要判断有无不稳定模态时，利用一个简单的矩阵变换将其化为强特征值的求解问题，从而可使用最简单的幂迭代，Ｃｈｅｂｙｓｈｅｖ配置点法算例表明两种算法均快速有效。

The problem of solving Orr-Sommerfeld equation can usually be transformedinto a complex generalized matrix eigenvalue problem AX=ωBX。 This paper reduces A,Bto upper Hessenberg matrix and upper triangular matrix respectively by U-transformation,then uses Muller′s rootfinding method to obtain all eigenvalues,where the value of charac-teristic polynomial is given by Hyman′s indirect method.When only a judgement of whetherany mode is unstable is needee,the problem can be transformed into a dominant eigenvalueproblem by a simple matrix transformation, so the simplest power iterative method can beapplied,Example of Chebyshev collocation method shows that the two methods above areboth fast and effceint.