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Error analysis of the Strang splitting for the 3D semilinear wave equation with finite-energy data

Maximilian Ruff

TL;DR

The article analyzes a Strang-splitting time integrator for the 3D semilinear wave equation on $\mathbb{T}^3$ with finite-energy data, establishing sharp convergence rates for cubic and quartic nonlinearities under minimal regularity. The authors develop continuous and discrete Strichartz estimates on the torus, introduce a Fourier-filter to manage nonlinear interactions, and derive detailed semi-discrete error recursions, followed by fully discrete results using a Fourier pseudo-spectral spatial discretization. For $\alpha=3$, they obtain almost second-order convergence in $L^2$ and almost first-order in the energy norm, while for $\alpha=4$ the temporal order is reduced by a half. They also provide fully discrete error bounds combining temporal and spatial discretizations and corroborate the theory with numerical experiments that demonstrate sharpness and the role of the filter. Overall, the work advances low-regularity, structure-preserving time integration for nonlinear wave equations on bounded domains with finite-energy data, and offers practical guidance for implementing high-accuracy simulations on $\mathbb{T}^3$.

Abstract

We study a variant of the Strang splitting for the time integration of the semilinear wave equation under the finite-energy condition on the torus $\mathbb{T}^3$. In the case of a cubic nonlinearity, we show almost second-order convergence in $L^2$ and almost first-order convergence in $H^1$. If the nonlinearity has a quartic form instead, we show an analogous convergence result with an order reduced by 1/2. To our knowledge these are the best convergence results available for the 3D cubic and quartic wave equations under the finite-energy condition. Our approach relies on continuous- and discrete-time Strichartz estimates. We also make use of the integration and summation by parts formulas to exploit cancellations in the error terms. Moreover, error bounds for a full discretization using the Fourier pseudo-spectral method in space are given. Finally, we discuss a numerical example indicating the sharpness of our theoretical results.

Error analysis of the Strang splitting for the 3D semilinear wave equation with finite-energy data

TL;DR

The article analyzes a Strang-splitting time integrator for the 3D semilinear wave equation on with finite-energy data, establishing sharp convergence rates for cubic and quartic nonlinearities under minimal regularity. The authors develop continuous and discrete Strichartz estimates on the torus, introduce a Fourier-filter to manage nonlinear interactions, and derive detailed semi-discrete error recursions, followed by fully discrete results using a Fourier pseudo-spectral spatial discretization. For , they obtain almost second-order convergence in and almost first-order in the energy norm, while for the temporal order is reduced by a half. They also provide fully discrete error bounds combining temporal and spatial discretizations and corroborate the theory with numerical experiments that demonstrate sharpness and the role of the filter. Overall, the work advances low-regularity, structure-preserving time integration for nonlinear wave equations on bounded domains with finite-energy data, and offers practical guidance for implementing high-accuracy simulations on .

Abstract

We study a variant of the Strang splitting for the time integration of the semilinear wave equation under the finite-energy condition on the torus . In the case of a cubic nonlinearity, we show almost second-order convergence in and almost first-order convergence in . If the nonlinearity has a quartic form instead, we show an analogous convergence result with an order reduced by 1/2. To our knowledge these are the best convergence results available for the 3D cubic and quartic wave equations under the finite-energy condition. Our approach relies on continuous- and discrete-time Strichartz estimates. We also make use of the integration and summation by parts formulas to exploit cancellations in the error terms. Moreover, error bounds for a full discretization using the Fourier pseudo-spectral method in space are given. Finally, we discuss a numerical example indicating the sharpness of our theoretical results.

Paper Structure

This paper contains 22 sections, 28 theorems, 218 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results in the semi-discrete setting
  3. The fully discrete scheme
  4. Organization of the paper
  5. Notations
  6. Linear estimates
  7. Continuous and discrete Strichartz estimates on the full space
  8. Strichartz estimates for the half-wave group on the torus
  9. Application to the wave equation on $\mathbb{T}^3$
  10. Some properties of the filter operator $\pi_K$
  11. Nonlinear Wave equation
  12. Review of local wellposedness theory
  13. Nonlinear estimates
  14. Error analysis of the semi-discretization in time
  15. Error recursion
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $U = (u,\partial_tu) \in C([0,T],H^1(\mathbb{T}^3) \times L^2(\mathbb{T}^3))$ solve the semilinear wave equation NLW2. Then there are a constant $C>0$ and a maximum step size $\tau_0>0$, such that the approximations $U_n$ obtained by the Strang splitting scheme Strang satisfy the following error and for $\alpha=4$, Theses bounds are uniform in $\tau \in (0,\tau_0]$ and $n \in \mathbb{N}_0$ wi

Figures (3)

  • Figure 1: Errors for $\alpha=3$
  • Figure 2: Errors for $\alpha=4$
  • Figure 3: Errors for $\alpha=5$

Theorems & Definitions (61)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 51 more