Uniformly accurate time-splitting methods for the semiclassical linear Schr{\"o}dinger equation

TL;DR

This paper develops and analyzes time-splitting numerical methods for the semiclassical linear Schrödinger equation that maintain accuracy regardless of the small semiclassical parameter, supported by theoretical proofs and simulations.

Contribution

It introduces uniformly accurate time-splitting methods for the semiclassical Schrödinger equation, ensuring consistent accuracy as the semiclassical parameter varies.

Findings

Proved uniform accuracy of the proposed methods.

Validated methods through numerical simulations.

Demonstrated robustness in the semiclassical limit.

Abstract

This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr{\"o}dinger equation in the semiclassical limit. We specifically analyse the convergence behavior of the first-order splitting. Our main result is a proof of uniform accuracy. We illustrate the properties of our methods with simulations.