Eigensolutions of boundary value problems using inverse iteration

摘要

Matrix eigenvalue problems arise when the differential operators in a system of ordinary or partial differential equations are replaced by finite-difference operators. We describe the use of the method of inverse iteration to solve such eigenvalue problems. The key to the success of this method is that it can take full advantage of the band structure of the matrix, resulting in a very considerable savings in storage and CPU-time compared with other matrix methods. For ordinary differential equations, the time taken is proportional to the number of grid points chosen. To illustrate the method, we solve the Orr—Sommerfeld problem, using both second- and fourth-order difference schemes. For a given accuracy of solution, the latter requires a similar CPU-time to shooting with orthonormalization. We show that the inverse iteration method has no trouble coping with very stiff problems.