High-order structure-preserving schemes for the regularized logarithmic Schr\"{o}dinger equation

TL;DR

This paper introduces a high-order numerical scheme for the regularized logarithmic Schrödinger equation that conserves mass and energy, utilizing a supplementary variable method combined with spectral discretization for improved efficiency and accuracy.

Contribution

It presents a novel high-order, structure-preserving scheme based on the supplementary variable method for the RLogSE, ensuring conservation and computational efficiency.

Findings

The scheme accurately conserves mass and energy.

Numerical experiments confirm high accuracy and efficiency.

The method outperforms existing schemes in preserving physical invariants.

Abstract

In this paper, a novel high-order, mass and energy-conserving scheme is proposed for the regularized logarithmic Schr\"{o}dinger equation(RLogSE). Based on the idea of the supplementary variable method (SVM), we firstly reformulate the original system into an equivalent form by introducing two supplementary variables, and the resulting SVM reformulation is then discretized by applying a high-order prediction-correction scheme in time and a Fourier pseudo-spectral method in space, respectively. The newly developed scheme can produce numerical solutions along which the mass and original energy are precisely conserved, as is the case with the analytical solution. Additionally, it is extremely efficient in the sense that only requires solving a constant-coefficient linear systems plus two algebraic equations, which can be efficiently solved by the Newton iteration at every time step. Numerical experiments are presented to confirm the accuracy and structure-preserving properties of the new scheme.