High-order and mass-conservative regularized implicit-explicit relaxation Runge-Kutta methods for the low regularity Schrödinger equations

The non-differentiability of the singular nonlinearities ($f=\ln |u{|}^2$ and $f=|u{|}^{2\alpha },\alpha <0$) at $u=0$ brings significant challenges in designing accurate and efficient numerical schemes for the low regularity Schrödinger equations (LorSE). In order to address the singularity, we propose an energy regularization for the LorSE. For the regularized models, we apply 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.