The Levin approach to the numerical calculation of phase functions

TL;DR

This paper introduces two numerical algorithms inspired by Levin's method for efficiently constructing phase functions of scalar ODEs with slowly-varying coefficients, reducing computational complexity regardless of coefficient magnitude.

Contribution

The paper presents novel Levin-inspired algorithms for phase function calculation in scalar ODEs, improving efficiency for equations with large coefficients.

Findings

Algorithms have running times independent of coefficient magnitude.

Numerical experiments confirm the effectiveness of the methods.

Applicable to a broad class of scalar ODEs with slowly-varying coefficients.

Abstract

The solutions of scalar ordinary differential equations become more complex as their coefficients increase in magnitude. As a consequence, when a standard solver is applied to such an equation, its running time grows with the magnitudes of the equation's coefficients. It is well known, however, that scalar ordinary differential equations with slowly-varying coefficients admit slowly-varying phase functions whose cost to represent via standard techniques is largely independent of the magnitude of the equation's coefficients. This observation is the basis of most methods for the asymptotic approximation of the solutions of ordinary differential equations, including the WKB method. Here, we introduce two numerical algorithms for constructing phase functions for scalar ordinary differential equations inspired by the classical Levin method for the calculation of oscillatory integrals. In the case of a large class of scalar ordinary differential equations with slowly-varying coefficients, their running times are independent of the magnitude of the equation's coefficients. The results of extensive numerical experiments demonstrating the properties of our algorithms are presented.