Smoothed Wigner transforms in the numerical simulation of semiclassical (high-frequency) wave propagation

TL;DR

This paper introduces a new phase-space numerical method using the smoothed Wigner Transform for efficient simulation of high-frequency wave propagation in semiclassical regimes, outperforming traditional Wigner Transform approaches.

Contribution

The paper proposes a novel SWT-based numerical scheme that simplifies and accelerates the simulation of semiclassical wave propagation compared to existing WT methods.

Findings

SWT approach is significantly faster than WT and full numerical solutions.

The method maintains accuracy through comparison with exact and numerical solutions.

Applicable to linear Schrödinger equations in semiclassical regimes.

Abstract

The numerical simulation of wave propagation in semiclassical (high-frequency) problems is well known to pose a formidable challenge. In this work, a new phase-space approach for the numerical simulation of semiclassical wave propagation, making use of the smoothed Wigner Transform (SWT), is proposed. There are numerous works which use the Wigner Transform (WT) in the study of a variety of wave propagation problems including high-frequency limits for linear, nonlinear and/or random waves. The WT however is well known to present significant difficulties in the formulation of numerical schemes. Working with concrete examples for the semiclassical linear Schrodinger equation it is seen that the SWT approach is indeed significantly faster (in a well-defined sense) to work with than the WT and than full numerical solutions of the original equation in the semiclassical regime. Comparisons with exact and numerical solutions are used to keep track of numerical errors.