An Isaac Newton Institute Workshop

Effective Computational Methods for Highly Oscillatory Problems: The Interplay between Mathematical Theory and Applications

High order coefficient approximations methods for the Sturm-Liouville boundary value problem

Authors: Veerle Ledoux (Ghent University), Marnix Van Daele (Ghent University), Guido Vanden Berghe (Ghent University)

Abstract

The determination of the eigenvalues of Sturm-Liouville problems is of great interest in mathematics and its applications. However most eigenvalue problems cannot be solved analytically, and computationally efficient approximation techniques are of great applicability. An important class of methods obtain eigenvalue approximations by applying an integrator based on coefficient approximation in a shooting process. These coefficient approximation methods replace the coefficient functions of the Sturm-Liouville equation by simpler approximations and then solve the approximating problem. The standard reference in the piecewise constant approximation case is due to S. Pruess [1], and therefore the methods are often referred to as Pruess methods. The Pruess method has some significant advantages. While a naive integrator is forced to make increasingly smaller steps in the search for large eigenvalues (due to the increasingly oscillatory nature of the solution), the stepsize is not restricted by the oscillations in the solution for a Pruess method. A drawback of the Pruess methods is the difficulty in obtaining higher order methods; unless Richardson extrapolation is used the method is only second order.

Higher order methods based on coefficient approximation can be realized using a perturbation technique. This approach leads to the so-called Piecewise Perturbation Methods (PPM) [2]. The PPM add some perturbation corrections to the solution of the approximating problem in order to obtain a more accurate approximation to the solution of the original problem. High order PPM were found to be well suited to be used in a shooting procedure to compute the eigenvalues efficiently and accurately. This resulted in a Matlab software package which can be used to compute the eigenvalues of a Sturm-Liouville or Schr\"odinger problem up to high accuracy ({\sc MATSLISE} [3]).

Recently it was shown that the piecewise perturbation approach may be viewed as the application of a modified Neumann expansion [4]. The excellent performance of piecewise perturbation methods for the Sturm-Liouville problem can thus be seen as a convincing illustration of the power and potential of the Neumann series integrators. Another integral series which has been recognized as a very effective computational tool for problems with highly oscillatory solution, is the Magnus expansion. Also integrators based on this Magnus expansion can be combined with coefficient approximation and form another extension of the Pruess ideas to high order approximations. \\

[1] Pruess, S. Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation. SIAM J. Numer. Anal. 10 (1973).

[2] Ixaru, L. Gr., De Meyer, H. and Vanden Berghe, G. CP methods for the Schr\"odinger equation revisited, J. Comput. Appl. Math. 88 (1997).

[3] Ledoux, V., Van Daele, M., and Vanden Berghe, G. Matslise: A matlab package for the numerical solution of Sturm-Liouville and Schrodinger equations. ACM Trans. Math. Software 31 (2005).

[4] Degani, I., AND Schiff, J. RCMS: Right Correction Magnus Series approach for oscillatory ODEs. J. Comput. Appl. Math. 193 (2006).