Asymptotic solvers for ordinary differential equations with multiple frequencies

TL;DR

This paper develops asymptotic expansion methods for efficiently solving ordinary differential equations with multiple high-frequency oscillatory forcing terms, enabling accurate discretization and numerical solutions.

Contribution

It introduces a novel asymptotic expansion approach for ODEs with multiple non-commensurate frequencies, improving discretization accuracy.

Findings

Asymptotic expansions effectively approximate solutions with highly oscillatory forcing.

Truncated expansions provide accurate discretization methods.

Numerical examples demonstrate the method's effectiveness.

Abstract

We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focussing on the case of multiple, non-commensurate frequencies. We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question. Numerical examples illustrate the effectiveness of the method.