Numerical integration of ordinary differential equations with rapidly oscillatory factors

TL;DR

This paper introduces a novel numerical method for integrating ordinary differential equations with rapidly oscillatory terms, generalizing Filon quadrature and adaptable to different accuracy levels based on user-defined truncation.

Contribution

The paper develops a generalized numerical integration technique for ODEs with oscillatory factors, extending Filon quadrature and allowing adjustable accuracy.

Findings

Method effectively handles oscillatory terms in ODEs

Flexible accuracy achieved through truncation levels

Applicable to a wide range of oscillatory ODEs

Abstract

We present a methodology for numerically integrating ordinary differential equations containing rapidly oscillatory terms. This challenge is distinct from that for differential equations which have rapidly oscillatory solutions: here the differential equation itself has the oscillatory terms. Our method generalises Filon quadrature for integrals, and is analogous to integral techniques designed to solve stochastic differential equations and, as such, is applicable to a wide variety of ordinary differential equations with rapidly oscillating factors. The proposed method flexibly achieves varying levels of accuracy depending upon the truncation of the expansion of certain integrals. Users will choose the level of truncation to suit the parameter regime of interest in their numerical integration.