Numerical approximation of discontinuous solutions of the semilinear wave equation

TL;DR

This paper introduces a high-frequency low-regularity integrator for the semilinear wave equation that accurately approximates discontinuous solutions without oscillations, achieving higher convergence rates than existing methods.

Contribution

A novel high-frequency recovery integrator is developed for rough and discontinuous solutions, with rigorous convergence analysis and demonstrated improved accuracy in numerical experiments.

Findings

Achieves almost first-order convergence for discontinuous solutions in 1D.

Effectively captures discontinuities without spurious oscillations.

Numerical results confirm theoretical convergence rates and efficiency.

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 $(u,\partial_{t}u)\in C([0,T];H^{\gamma}\times H^{\gamma-1})$ for $\gamma\in(0,1]$. 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 $\tau \sim N^{-1}$, where $\tau$ 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.