Using a (higher-order) Magnus method to solve the Sturm-Liouville problem

The main purpose of this paper is to describe techniques for the numerical solution of a Sturm-Liouville equation (in its Schrodinger form) by employing a Magnus expansion. With a suitable method to approximate the highly oscillatory integrals which appear in the Magnus series, high order schemes can be constructed. A method of order ten is presented. Even when the solution is highly-oscillatory, the scheme can accurately integrate the problem using stepsizes typically much larger than the solution "wavelength". This makes the method well suited to be applied in a shooting process to locate the eigenvalues of a boundary value problem.