Structured Backward Error for the WKB method

TL;DR

This paper introduces a structured backward error analysis for the classical WKB method, revealing its exactness in certain cases and leading to new algorithms and hybrid approaches for solving singularly perturbed differential equations.

Contribution

It develops the first structured backward error analysis for the WKB method and proposes new iterative and hybrid algorithms to enhance solution accuracy.

Findings

WKB can exactly solve certain structured problems.

A new iterative algorithm improves WKB solution quality.

A hybrid Chebyshev polynomial approach simplifies implementation.

Abstract

The classical WKB method (also known as the WKBJ method, the LG method, or the phase integral method) for solving singularly perturbed linear differential equations has never, as far as we know, been looked at from the structured backward error (BEA) point of view. This is somewhat surprising, because a simple computation shows that for some important problems, the WKB method gives the exact solution of a problem of the same structure that can be expressed in finitely many terms. This kind of analysis can be extremely useful in assessing the validity of a solution provided by the WKB method. In this paper we show how to do this and explore some of the consequences, which include a new iterative algorithm to improve the quality of the WKB solution. We also explore a new hybrid method where the potential is approximated by Chebyshev polynomials, which can be implemented in a few lines of Chebfun.