Error analysis of the Strang splitting for the 3D semilinear wave equation with finite-energy data
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913138330304512 |
|---|---|
| author | Ruff, Maximilian |
| author_facet | Ruff, Maximilian |
| contents | 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 results, where the order is reduced by $1/2$ in both cases. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13126 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Error analysis of the Strang splitting for the 3D semilinear wave equation with finite-energy data Ruff, Maximilian Numerical Analysis Analysis of PDEs 65M15 (Primary) 35B33, 35L71, 65M12, 65M70 (Secondary) 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 results, where the order is reduced by $1/2$ in both cases. 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. |
| title | Error analysis of the Strang splitting for the 3D semilinear wave equation with finite-energy data |
| topic | Numerical Analysis Analysis of PDEs 65M15 (Primary) 35B33, 35L71, 65M12, 65M70 (Secondary) |
| url | https://arxiv.org/abs/2503.13126 |