Implicit-explicit time discretization schemes for a class of semilinear wave equations with nonautonomous dampings

TL;DR

This paper develops and analyzes implicit-explicit schemes for dissipative wave equations with time-varying damping, addressing order reduction issues and proposing a revised method that restores second-order accuracy and enhances efficiency.

Contribution

The paper introduces a revised IMEX scheme for nonautonomous wave equations that maintains second-order accuracy, overcoming order reduction caused by time-dependent damping.

Findings

Revised scheme achieves second-order accuracy.

Original scheme's error rate drops to first order with nonautonomous damping.

Numerical experiments confirm improved efficiency and accuracy.

Abstract

This paper is concerned about the implicit-explicit (IMEX) methods for a class of dissipative wave systems with time-varying velocity feedbacks and nonlinear potential energies, equipped with different boundary conditions. Firstly, we approximate the problems by using a vanilla IMEX method, which is a second-order scheme for the problems when the damping terms are time-independent. However, rigors analysis shows that the error rate declines from second to first order due to the nonautonomous dampings. To recover the convergence order, we propose a revised IMEX scheme and apply it to the nonautonomous wave equations with a kinetic boundary condition. Our numerical experiments demonstrate that the revised scheme can not only achieve second-order accuracy but also improve the computational efficiency.