High-order and Mass-conservative Regularized Implicit-explicit relaxation Runge-Kutta methods for the logarithmic Schr\"{o}dinger equation

TL;DR

This paper introduces high-order, mass-conservative, regularized implicit-explicit Runge-Kutta methods combined with Fourier spectral discretization to effectively solve the logarithmic Schrödinger equation with singular nonlinearity.

Contribution

It proposes an energy regularization technique and high-order relaxation Runge-Kutta methods that are linearly implicit and mass-conserving for the LogSE.

Findings

Methods effectively handle singular nonlinearity in LogSE.

Numerical results demonstrate high efficiency and accuracy.

Mass conservation is maintained throughout simulations.

Abstract

The non-differentiability of the singular nonlinearity (such as $f=\ln|u|^2$) at $u=0$ presents significant challenges in devising accurate and efficient numerical schemes for the logarithmic Schr\"{o}dinger equation (LogSE). To address this singularity, we propose an energy regularization technique for the LogSE. For the regularized model, we utilize Implicit-Explicit Relaxation Runge-Kutta methods, which are linearly implicit, high-order, and mass-conserving for temporal discretization, in conjunction with the Fourier pseudo-spectral method in space. Ultimately, numerical results are presented to validate the efficiency of the proposed methods.