Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

High-order structure-preserving schemes for the regularized logarithmic Schrödinger equation

Fan Yang, Zhida Zhou, Chaolong Jiang

Abstract

In this paper, a novel high-order, mass and energy-conserving scheme is proposed for the regularized logarithmic Schrödinger equation(RLogSE). Based on the idea of the supplementary variable method (SVM), we firstly reformulate the original system into an equivalent form by introducing two supplementary variables, and the resulting SVM reformulation is then discretized by applying a high-order prediction-correction scheme in time and a Fourier pseudo-spectral method in space, respectively. The newly developed scheme can produce numerical solutions along which the mass and original energy are precisely conserved, as is the case with the analytical solution. Additionally, it is extremely efficient in the sense that only requires solving a constant-coefficient linear systems plus two algebraic equations, which can be efficiently solved by the Newton iteration at every time step. Numerical experiments are presented to confirm the accuracy and structure-preserving properties of the new scheme.

High-order structure-preserving schemes for the regularized logarithmic Schrödinger equation

Abstract

In this paper, a novel high-order, mass and energy-conserving scheme is proposed for the regularized logarithmic Schrödinger equation(RLogSE). Based on the idea of the supplementary variable method (SVM), we firstly reformulate the original system into an equivalent form by introducing two supplementary variables, and the resulting SVM reformulation is then discretized by applying a high-order prediction-correction scheme in time and a Fourier pseudo-spectral method in space, respectively. The newly developed scheme can produce numerical solutions along which the mass and original energy are precisely conserved, as is the case with the analytical solution. Additionally, it is extremely efficient in the sense that only requires solving a constant-coefficient linear systems plus two algebraic equations, which can be efficiently solved by the Newton iteration at every time step. Numerical experiments are presented to confirm the accuracy and structure-preserving properties of the new scheme.

Paper Structure

This paper contains 9 sections, 5 theorems, 51 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model reformulation
  3. High-order mass- and energy-conserving scheme
  4. Temporal semi-discretization
  5. Full discretization
  6. Numerical experiments
  7. RlogSE in 1D
  8. RlogSE in 2D
  9. Concluding remarks

Key Result

Theorem 3.1

It is evident that Scheme svm-scheme1 preserves the following semi-discrete mass and Hamiltonian energy where

Figures (13)

  • Figure 1: Case I : Evolution of $\sqrt{\left| u^\epsilon \right|}$ from $t = 0$ to $t = 500$ (left), and plots of $\left| u^\epsilon \right|$ at different times (right) produced by SVM4 in example \ref{['EX4-2']}.
  • Figure 2: Case II : Evolution of $\sqrt{\left| u^\epsilon \right|}$ from $t = 0$ to $t = 100$ (left), and plots of $\left| u^\epsilon \right|$ at different times (right) produced by SVM4 in example \ref{['EX4-2']}.
  • Figure 3: Case III : Evolution of $\sqrt{\left| u^\epsilon \right|}$ from $t = 0$ to $t = 16$ (left), and plots of $\left| u^\epsilon \right|$ at different times (right) produced by SVM4 in example \ref{['EX4-2']}.
  • Figure 4: Case IV : Evolution of $\sqrt{\left| u^\epsilon \right|}$ from $t = 0$ to $t = 3$ (left), and plots of $\left| u^\epsilon \right|$ at different times (right) produced by SVM4 in example \ref{['EX4-2']}.
  • Figure 5: Case I: The long-time evolution of $e_{\mathcal{M}}$ & $e_{\mathcal{E}}$ for the two schemes in example \ref{['EX4-2']}.
  • ...and 8 more figures

Theorems & Definitions (16)

  • Remark 2.1
  • Theorem 3.1
  • Proof
  • Theorem 3.2
  • Theorem 3.3
  • Proof
  • Theorem 3.4
  • Proof
  • Remark 3.1
  • Remark 3.2
  • ...and 6 more