High-order and Mass-conservative Regularized Implicit-explicit relaxation Runge-Kutta methods for the logarithmic Schrödinger equation

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 Schrödinger 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.