Nonlinear steepest descent and the numerical solution of Riemann-Hilbert problems

TL;DR

This paper develops a theoretical framework proving that numerical solutions to Riemann-Hilbert problems, based on nonlinear steepest descent, maintain accuracy in asymptotic regimes without needing local parametrices.

Contribution

It provides the first rigorous proof linking nonlinear steepest descent techniques to the accuracy of numerical solutions for Riemann-Hilbert problems.

Findings

Proves asymptotic accuracy of numerical methods in large parameter regimes.

Establishes sufficient conditions for numerical accuracy retention.

Demonstrates practical validity through solutions of Painlevé II and KdV equations.

Abstract

The effective and efficient numerical solution of Riemann-Hilbert problems has been demonstrated in recent work. With the aid of ideas from the method of nonlinear steepest descent for Riemann-Hilbert problems, the resulting numerical methods have been shown numerically to retain accuracy as values of certain parameters become arbitrarily large. Remarkably, this numerical approach does not require knowledge of local parametrices; rather, the deformed contour is scaled near stationary points at a specific rate. The primary aim of this paper is to prove that this observed asymptotic accuracy is indeed achieved. To do so, we first construct a general theoretical framework for the numerical solution of Riemann-Hilbert problems. Second, we demonstrate the precise link between nonlinear steepest descent and the success of numerics in asymptotic regimes. In particular, we prove sufficient conditions for numerical methods to retain accuracy. Finally, we compute solutions to the homogeneous Painlev\'e II equation and the modified Korteweg-de Vries equations to explicitly demonstrate the practical validity of the theory.