Numerical solution of eigenvalue problems for linear boundary value ODEs
The authors consider the numerical solution of eigenvalue problems for linear ordinary differential equations, although some of the techniques discussed apply naturally to nonlinear differential equations and/or to problems where the eigenvalue dependence may be nonlinear. Actually all these problems, including the one considered here are nonlinear (i.e. at least quadratic in the eigenfunction/eigenvalue unknowns), although special techniques can be used when a linear differential operator is involved. Several different methods are considered for the case in which only a few eigenpairs are desired and interrelationships among these methods are shown. Both shooting, i.e., initial value problem based, and global, i.e., finite difference or collocation methods are discussed. A Riccati type, a continuous orthonormalization, and a collocation method are used to solve numerically the complex Orr-Sommerfeld eigenvalue problem for large Reynolds numbers. Unfortunately, none of the approaches mentioned here can guarantee that the sought after eigenpairs will be obtained. This, as any other nonlinear problem has a mind of itself, and the iterative methods used for their solutions will, in the best of cases, converge to a solution, and as the authors point out, sometimes to a spurious one generated by the discretization. The fact that very accurate initial guesses are required to guarantee convergence should be a warning flag. In recognizing this inherent difficulty the authors resort to continuation on Reynolds number, a time honored solution, which amounts to realizing that a reasonable way to try to reach a particular solution of a non- monotone problem (i.e. one with multiple solutions), is by considering the more complex problem of solution branch following and bifurcation. Of course, having taken that line of thought, it would have been reasonable to consider other software available for solving these much harder problems, like that of E. Doedel and R. Seydel.
complex Orr-Sommerfeld eigenvalue problem
scientific article; zbMATH DE number 4199533
scientific article