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Numerical approximation of discontinuous solutions of the semilinear wave equation

Jiachuan Cao, Buyang Li, Yanping Lin, Fangyan Yao

Abstract

A high-frequency recovered fully discrete low-regularity integrator is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method, with high-frequency recovery techniques, can capture the discontinuities of the solutions correctly without spurious oscillations and approximate rough and discontinuous solutions with a higher convergence rate than pre-existing methods. Rigorous analysis is presented for the convergence rates of the proposed method in approximating solutions such that $(u,\partial_{t}u)\in C([0,T];H^γ\times H^{γ-1})$ for $γ\in(0,1]$. For discontinuous solutions of bounded variation in one dimension (which allow jump discontinuities), the proposed method is proved to have almost first-order convergence under the step size condition $τ\sim N^{-1}$, where $τ$ and $N$ denote the time step size and the number of Fourier terms in the space discretization, respectively. Numerical examples are presented in both one and two dimensions to illustrate the advantages of the proposed method in improving the accuracy in approximating rough and discontinuous solutions of the semilinear wave equation. The numerical results are consistent with the theoretical results and show the efficiency of the proposed method.

Numerical approximation of discontinuous solutions of the semilinear wave equation

Abstract

A high-frequency recovered fully discrete low-regularity integrator is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method, with high-frequency recovery techniques, can capture the discontinuities of the solutions correctly without spurious oscillations and approximate rough and discontinuous solutions with a higher convergence rate than pre-existing methods. Rigorous analysis is presented for the convergence rates of the proposed method in approximating solutions such that for . For discontinuous solutions of bounded variation in one dimension (which allow jump discontinuities), the proposed method is proved to have almost first-order convergence under the step size condition , where and denote the time step size and the number of Fourier terms in the space discretization, respectively. Numerical examples are presented in both one and two dimensions to illustrate the advantages of the proposed method in improving the accuracy in approximating rough and discontinuous solutions of the semilinear wave equation. The numerical results are consistent with the theoretical results and show the efficiency of the proposed method.
Paper Structure (12 sections, 15 theorems, 171 equations, 8 figures)

This paper contains 12 sections, 15 theorems, 171 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Notations and numerical scheme
  4. Main theoretical results
  5. Preliminary results
  6. Construction of the numerical scheme
  7. Proof of Theorem \ref{['thm:convergence']}
  8. Proof of Theorem \ref{['u11']}
  9. Numerical examples
  10. The Sine--Gordon equation in one dimension
  11. The Sine--Gordon equation in two dimensions
  12. The Klein--Gordon equation in one dimension

Key Result

Theorem 2.1

Let $d=1,2$ and $\gamma\in (0,1]$, and assume that the nonlinear function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the following condition: Then, under the regularity condition $U \in C([0,T]; H^\gamma(\Omega) \times H^{\gamma-1}(\Omega))$ and the step size condition $N\lesssim \tau^{-\frac{1}{1- \gamma}+\epsilon_{0}}$ (for an abitrary $\epsilon_{0} \in(0,1]$), the numerical solutions given

Figures (8)

  • Figure 1: Numerical solution of the 1D problem with discontinuous initial value in \ref{['1d-initial-value']}.
  • Figure 2: Comparison of numerical solutions given by several different methods.
  • Figure 3: Numerical results of the 1D problem with the initial value in \ref{['1d-initial-value']}.
  • Figure 4: Comparison of the numerical solutions at $t=0.25$ computed by two different methods
  • Figure 5: Comparison of the numerical solutions at $t=0.25$ computed by two different methods
  • ...and 3 more figures

Theorems & Definitions (32)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.2
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 3.1: Bernstein's inequality; cf. Guo1998
  • ...and 22 more