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