Uniformly accurate time-splitting methods for the semiclassical Schr\"odinger equation Part 1 : Construction of the schemes and simulations

TL;DR

This paper introduces new time-splitting numerical methods for the semiclassical Schrödinger equation that are accurate, spectral in space, and preserve key properties, with the first part focusing on scheme construction and simulations.

Contribution

It presents a phase-amplitude reformulation enabling the development of high-order, epsilon-independent splitting schemes for the semiclassical Schrödinger equation, including a norm-preserving second-order method.

Findings

Spectral accuracy in space achieved

Fourth-order accuracy in time demonstrated

Second-order method preserves L^2-norm

Abstract

This article is devoted to the construction of new numerical methods for the semiclassical Schr\"odinger equation. A phase-amplitude reformulation of the equation is described where the Planck constant epsilon is not a singular parameter. This allows to build splitting schemes whose accuracy is spectral in space, of up to fourth order in time, and independent of epsilon before the caustics. The second-order method additionally preserves the L^2-norm of the solution just as the exact flow does. In this first part of the paper, we introduce the basic splitting scheme in the nonlinear case, reveal our strategy for constructing higher-order methods, and illustrate their properties with simulations. In the second part, we shall prove a uniform convergence result for the first-order splitting scheme applied to the linear Schr\"odinger equation with a potential.