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

Jingye Yan, Hong Zhang, Yabing Wei, Xu Qian

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.

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 ) at 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.

Paper Structure

This paper contains 10 sections, 1 theorem, 22 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The energy regularized LogSE
  3. The construction of Regularized IMEX RRK methods
  4. High order Regularized IMEX RRK methods
  5. Numerical Experiments
  6. Accuracy test
  7. Convergence rate of the regularized model
  8. Convergence rate of the numerical scheme
  9. Dynamics
  10. Conclusion

Key Result

Lemma 3.1

Li2023Implicit Suppose that the given IMEX RK method is $p$-th order accurate with $p \geq 2$. For sufficiently small $\tau$, the relaxation coefficient $\gamma_n$ defined in gammarrk satisfies

Figures (4)

  • Figure 4.1: Convergence order ERLogSE to LogSE, i.e. $\|\hat{e}^{\varepsilon}(t=1)\|,\,\|\hat{e}^{\varepsilon}_{\rho}(t=1)\|,\,e^{\varepsilon}_{E}(t=1)$.
  • Figure 4.2: Convergence order of $max(|\gamma_n-1|)$ for different IMEX RRK schemes.
  • Figure 4.3: Convergence order in time of RRK with different $\varepsilon$.
  • Figure 4.4: Plots of $|u^{\varepsilon}(x,t)|$ (first column); $|u^{\varepsilon}(x,t)|$ at different time (second column) and evolution of mass error (third column) (Case for RRK(2,3)).

Theorems & Definitions (3)

  • proof
  • Lemma 3.1
  • proof