Phase function methods for second order inhomogeneous linear ordinary differential equations

TL;DR

This paper introduces an efficient numerical approach combining phase function methods with adaptive Levin techniques to solve a broad class of second order inhomogeneous linear ordinary differential equations accurately and efficiently.

Contribution

It extends phase function methods to inhomogeneous equations by integrating them with adaptive Levin methods for oscillatory integral evaluation.

Findings

Method achieves accuracy comparable to condition number predictions.

Algorithm runs in time independent of coefficient magnitude.

Applicable to a large class of second order inhomogeneous equations.

Abstract

It is well known that second order homogeneous linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation underlies the Liouville-Green method and many other techniques for the asymptotic approximation of the solutions of such equations. It is also the basis of a recently developed numerical algorithm that, in many cases of interest, runs in time independent of the magnitude of the equation's coefficients and achieves accuracy on par with that predicted by its condition number. Here we point out that a large class of second order inhomogeneous linear ordinary differential equations can be efficiently and accurately solved by combining phase function methods for second order homogeneous linear ordinary differential equations with a variant of the adaptive Levin method for evaluating oscillatory integrals.