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

Zhe Jiao, Yaxu Li, Lijing Zhao

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.

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

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.

Paper Structure

This paper contains 12 sections, 2 theorems, 100 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Abstract full discretization
  4. Abstract space discretization
  5. Full discretization with vanilla IMEX scheme
  6. Full discretization with revised IMEX scheme
  7. Numerical simulation
  8. Proof of Theorem \ref{['main_result']}
  9. Conclusions
  10. Crank-Nicolson scheme for Equation \ref{['eq:evolution']}
  11. vanilla IMEX scheme for Equation \ref{['eq:evolution']}
  12. Numerical tests for different cases of $\gamma(t)$

Key Result

Lemma 2.1

Let Assumptions $\mathrm{(H1)}$-$\mathrm{(H4)}$ be satisfied. Then for all $u^0\in V$, $v^0\in H$, eq:wave has a unique weak solution with $T = T(u^0, v^0)$. Moreover, if $u^0\in D(A)$, $v^0\in V$, there exists a strong solution to eq:wave

Figures (7)

  • Figure 1: Error $\mathcal{E}(0.8)$ of the full discretization \ref{['full_discretization']} plots for the problem with the damping coefficient $\gamma(t) = 0$. Left: coarse space discretization ($328193$ degrees of freedom, $h = 0.0139201$); Right: fine space discretization ($1311745$ degrees of freedom, $h = 0.00697294$).
  • Figure 2: Error $\mathcal{E}(0.8)$ of the full discretizations plots for the problem with the damping coefficient satisfying $r_1 =r_2 =1$ and $\eta = -2$.
  • Figure 3: Error $\mathcal{E}(0.8)$ of the full discretization plots for the problem with the damping coefficient satisfying $r_1 =r_2 = 1$ and $\eta = -1$.
  • Figure 4: Error $\mathcal{E}(0.8)$ of different time schemes with the same space discretization ($h = 0.0139201$) plotted against the runtime. Here, vIMEX: the vanilla IMEX scheme, rIMEX: the revised IMEX scheme, CN: the Crank--Nicolson scheme, and RK: the Runge--Kutta scheme.
  • Figure 5: The profiles of numerical solutions to the problem with the damping coefficient satisfying $r_1 =r_2 = 1$ and $\eta = -1$.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Lemma 2.1
  • Theorem 3.1
  • proof