Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics

TL;DR

This paper presents advanced numerical techniques for efficiently computing high-index eigenvalues of Sturm-Liouville problems, overcoming oscillatory challenges with coefficient approximation and high-order integrators.

Contribution

It introduces methods based on coefficient approximation combined with Magnus and Neumann integrators to uniformly approximate and compute large eigenvalues efficiently.

Findings

Methods enable large step sizes for high eigenvalues

Coefficient approximation simplifies complex problem coefficients

High-order integrators improve accuracy and efficiency

Abstract

Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behaviour of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss some techniques that yield uniform approximation over the whole eigenvalue spectrum and can take large steps even for high eigenvalues. In particular, we will focus on methods based on coefficient approximation which replace the coefficient functions of the Sturm-Liouville problem by simpler approximations and then solve the approximating problem. The use of (modified) Magnus or Neumann integrators allows to extend the coefficient approximation idea to higher order methods.