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Error analysis of an implicit-explicit time discretization scheme for semilinear wave equations with application to multiscale problems

Daniel Eckhardt, Marlis Hochbruck, Barbara Verfürth

TL;DR

This work develops and analyzes an implicit–explicit (IMEX) time discretization for semilinear damped wave equations, achieving unconditional stability by implicitly treating the stiff linear part and explicitly handling the nonlinear nonstiff part. A rigorous error framework is established for full discretization, combining the IMEX time stepping with abstract space discretizations, and a second-order temporal convergence result is proved under suitable regularity. The theory is then extended to a multiscale setting by applying it to the FE-Heterogeneous Multiscale Method (FE-HMM) for wave equations with highly oscillatory coefficients, deriving an $\varepsilon$-independent homogenized model and quantifying the spatial and temporal discretization errors. The paper also provides a detailed multiscale implementation and numerical evidence confirming the predicted convergence rates, illustrating the practical relevance of IMEX schemes for semilinear wave equations and HMM-based multiscale simulations. Overall, the contribution offers a robust, efficient framework for accurate time integration of semilinear damped waves in multiscale environments and delivers concrete error guarantees for fully discrete FE-HMM discretizations.

Abstract

We present an implicit-explicit (IMEX) scheme for semilinear wave equations with strong damping. By treating the nonlinear, nonstiff term explicitly and the linear, stiff part implicitly, we obtain a method which is not only unconditionally stable but also highly efficient. Our main results are error bounds of the full discretization in space and time for the IMEX scheme combined with a general abstract space discretization. As an application, we consider the heterogeneous multiscale method for wave equations with highly oscillating coefficients in space for which we show spatial and temporal convergence rates by using the abstract result.

Error analysis of an implicit-explicit time discretization scheme for semilinear wave equations with application to multiscale problems

TL;DR

This work develops and analyzes an implicit–explicit (IMEX) time discretization for semilinear damped wave equations, achieving unconditional stability by implicitly treating the stiff linear part and explicitly handling the nonlinear nonstiff part. A rigorous error framework is established for full discretization, combining the IMEX time stepping with abstract space discretizations, and a second-order temporal convergence result is proved under suitable regularity. The theory is then extended to a multiscale setting by applying it to the FE-Heterogeneous Multiscale Method (FE-HMM) for wave equations with highly oscillatory coefficients, deriving an -independent homogenized model and quantifying the spatial and temporal discretization errors. The paper also provides a detailed multiscale implementation and numerical evidence confirming the predicted convergence rates, illustrating the practical relevance of IMEX schemes for semilinear wave equations and HMM-based multiscale simulations. Overall, the contribution offers a robust, efficient framework for accurate time integration of semilinear damped waves in multiscale environments and delivers concrete error guarantees for fully discrete FE-HMM discretizations.

Abstract

We present an implicit-explicit (IMEX) scheme for semilinear wave equations with strong damping. By treating the nonlinear, nonstiff term explicitly and the linear, stiff part implicitly, we obtain a method which is not only unconditionally stable but also highly efficient. Our main results are error bounds of the full discretization in space and time for the IMEX scheme combined with a general abstract space discretization. As an application, we consider the heterogeneous multiscale method for wave equations with highly oscillating coefficients in space for which we show spatial and temporal convergence rates by using the abstract result.
Paper Structure (17 sections, 13 theorems, 120 equations, 1 figure)

This paper contains 17 sections, 13 theorems, 120 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Time Integration
  3. Wellposedness of the continuous problem
  4. First-order formulation
  5. The IMEX scheme
  6. Full discretization
  7. Space discretization
  8. Error bounds for the full discretization
  9. Application: Semilinear damped wave equations with highly oscillatory coefficients
  10. Notations
  11. Model problem and G-convergence
  12. Heterogeneous Multiscale Method
  13. Error bounds for the fully discrete FE-HMM
  14. Numerical Examples
  15. Implementation
  16. ...and 2 more sections

Key Result

Lemma 2.2

The operator $S: D(S) \to \mathbf{X}$ is the generator of $C_0$-semigroup, which satisfies with $c^{S}_{qm} :=\frac{1}{2}c_{d} C_{\mathbf{H},\mathbf{V}} + c_{qm}$.

Figures (1)

  • Figure 1: The error $\mathbf{E}_\tau(1)$ of finite element heterogeneous multiscale method plotted against the macroscopic mesh size $H$ on the left and the time step-size $\tau$ on the right.

Theorems & Definitions (30)

  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Theorem 2.4
  • Remark 3.3
  • Definition 3.4
  • Definition 3.5
  • Theorem 3.6
  • Definition 3.7
  • Lemma 3.8
  • ...and 20 more