A new class of uniformly accurate numerical schemes for highly oscillatory evolution equations

TL;DR

This paper presents a novel methodology for designing uniformly accurate numerical schemes that effectively handle highly oscillatory evolution equations across various regimes, maintaining accuracy and efficiency regardless of oscillation intensity.

Contribution

The authors introduce an averaging transformation-based approach that enables high-order numerical schemes to be uniformly accurate for oscillatory equations, regardless of stiffness or oscillation frequency.

Findings

Achieves high-order accuracy with errors independent of oscillation regime

Maintains computational cost regardless of oscillation frequency

Applicable across a wide spectrum of oscillatory regimes

Abstract

We introduce a new methodology to design uniformly accurate methods for oscillatory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly-oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical approximations are obtained with errors and at a cost independent of the regime.