A frequency-independent solver for systems of first order linear ordinary differential equations

TL;DR

This paper introduces a novel frequency-independent solver for systems of first order linear ordinary differential equations, enabling efficient solutions regardless of eigenvalue magnitudes by coupling scalar equation techniques with system transformation.

Contribution

The paper presents a new method that combines scalar equation solvers with system transformation to achieve eigenvalue magnitude independence in solving linear ODE systems.

Findings

Solver performance is unaffected by large eigenvalues.

Numerical experiments confirm efficiency and accuracy.

Method applies to a broad class of linear systems.

Abstract

When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing such solutions using standard techniques typically grows with the magnitudes of the eigenvalues. As a consequence, the running times of standard solvers for ordinary differential equations also grow with the size of these eigenvalues. The solutions of scalar equations with slowly-varying coefficients, however, can be efficiently represented via slowly-varying phase functions, regardless of the magnitudes of the eigenvalues of the corresponding coefficient matrix. Here, we couple an existing solver for scalar equations which exploits this observation with a well-known technique for transforming a system of linear ordinary differential equations into scalar form. The result is a method for solving a large class of systems of linear ordinary differential equations in time independent of the magnitudes of the eigenvalues of their coefficient matrices. We discuss the results of numerical experiments demonstrating the properties of our algorithm.