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Explicit Local Time-Stepping for the Inhomogeneous Wave Equation with Optimal Convergence

Marcus J. Grote, Simon R. J. Michel, Stefan A. Sauter

Abstract

Adaptivity and local mesh refinement are crucial for the efficient numerical simulation of wave phenomena in complex geometry. Local mesh refinement, however, can impose a tiny time-step across the entire computational domain when using explicit time integration. By taking smaller time-steps yet only inside locally refined regions, local time-stepping methods overcome the stringent CFL stability restriction imposed on the global time-step by a small fraction of the elements without sacrificing explicitness. In [21], a leapfrog based local time-stepping method was proposed for the inhomogeneous wave equation, which applies standard leapfrog time-marching with a smaller time-step inside the refined region. Here, to remove potential instability at certain time-steps, a stabilized version is proposed which leads to optimal L2-error estimates under a CFL condition independent of the coarse-to-fine mesh ratio. Moreover, a weighted transition is introduced to restore optimal H1-convergence when the source is nonzero across the coarse-to-fine mesh interface. Numerical experiments corroborate the theoretical error estimates and illustrate the usefulness of these improvements.

Explicit Local Time-Stepping for the Inhomogeneous Wave Equation with Optimal Convergence

Abstract

Adaptivity and local mesh refinement are crucial for the efficient numerical simulation of wave phenomena in complex geometry. Local mesh refinement, however, can impose a tiny time-step across the entire computational domain when using explicit time integration. By taking smaller time-steps yet only inside locally refined regions, local time-stepping methods overcome the stringent CFL stability restriction imposed on the global time-step by a small fraction of the elements without sacrificing explicitness. In [21], a leapfrog based local time-stepping method was proposed for the inhomogeneous wave equation, which applies standard leapfrog time-marching with a smaller time-step inside the refined region. Here, to remove potential instability at certain time-steps, a stabilized version is proposed which leads to optimal L2-error estimates under a CFL condition independent of the coarse-to-fine mesh ratio. Moreover, a weighted transition is introduced to restore optimal H1-convergence when the source is nonzero across the coarse-to-fine mesh interface. Numerical experiments corroborate the theoretical error estimates and illustrate the usefulness of these improvements.
Paper Structure (15 sections, 8 theorems, 164 equations, 7 figures, 1 algorithm)

This paper contains 15 sections, 8 theorems, 164 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Stabilized LF-LTS Method
  3. The Wave Equation
  4. Galerkin Finite Element Spatial Discretization
  5. Stabilized LF-LTS($\nu$) Algorithm
  6. Convergence Theory in the $L^{2}$-norm
  7. Two-Step Formulation via Chebyshev Polynomials
  8. Main Convergence Results
  9. Towards Optimal $H^{1}$ Convergence
  10. Numerical Experiments
  11. Optimal $L^2$ Convergence
  12. Optimal $H^{1}$-Convergence and weighted transition
  13. Space-time local forcing
  14. Spatially constant solution
  15. Appendix

Key Result

Lemma 1

Let $r=0,1,\ldots,p-1$. Then, the polynomials $P_{p,\nu,k}$ defined in (defPp) satisfy the recurrence relation while $P_{p,\nu,k}^{\Delta t}$ satisfies

Figures (7)

  • Figure 1: Snapshots of the LF-LTS($\nu$) solution of \ref{['model problem']} with $\nu=0.01$ and $f$ as in \ref{['SourceEx1']} (top left, top right and bottom left). Relative $L^2$-error vs. $h = h_{\operatorname*{c}}$ for $\mathbb{P}_1$ finite elements (bottom right).
  • Figure 2: $L^2$-error vs. $h = h_{\operatorname*{c}}$: LF-LTS($\nu$) method \ref{['eq:StabLFLTSrhs']} (dash-dotted red line) and alternative "split LFC" approach from CarleHochbruck3 -- see Remark \ref{['rem:LFCLTScomparison']} (solid blue line).
  • Figure 3: Left: Source $f(x,t)$ for different times $t$, inside locally refined mesh with $\Omega_{\operatorname*{f}}=[1.6,2.4]$. Right: Relative $H^1$-error with (pink solid line), or without (blue dash-dotted line), weighted transition vs. mesh size $h = h_{\operatorname*{c}}$.
  • Figure 4: Left: Source $f(x,t)$ for different times $t$, nonzero across coarse-to-fine mesh interface with $\Omega_{\operatorname*{f}}=[2,2.4]$. Right: Relative $H^1$-error with (pink solid line), or without (blue dash-dotted line), weighted transition vs. mesh size $h = h_{\operatorname*{c}}$.
  • Figure 5: Left: Source $f(x,t)$ for different times $t$, outside locally refined region with $\Omega_{\operatorname*{f}}=[2.2,2.4]$. Right: Relative $H^1$-error with (pink solid line), or without (blue dash-dotted line), weighted transition vs. mesh size $h = h_{\operatorname*{c}}$.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Example 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Corollary 4
  • Lemma 5
  • Remark 3
  • Theorem 6
  • ...and 7 more