On superconvergence of Runge-Kutta convolution quadrature for the wave equation
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866917731338551296 |
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| author | Melenk, Jens Markus Rieder, Alexander |
| author_facet | Melenk, Jens Markus Rieder, Alexander |
| contents | The semidiscretization of a sound soft scattering problem modelled by the wave equation is analyzed. The spatial treatment is done by integral equation methods. Two temporal discretizations based on Runge-Kutta convolution quadrature are compared: one relies on the incoming wave as input data and the other one is based on its temporal derivative. The convergence rate of the latter is shown to be higher than previously established in the literature. Numerical results indicate sharpness of the analysis. The proof hinges on a novel estimate on the Dirichlet-to-Impedance map for certain Helmholtz problems. Namely, the frequency dependence can be lowered by one power of $\abs{s}$(up to a logarithmic term for polygonal domains) compared to the Dirichlet-to-Neumann map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_00847 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On superconvergence of Runge-Kutta convolution quadrature for the wave equation Melenk, Jens Markus Rieder, Alexander Numerical Analysis 65N38, 65N12, 65N15 The semidiscretization of a sound soft scattering problem modelled by the wave equation is analyzed. The spatial treatment is done by integral equation methods. Two temporal discretizations based on Runge-Kutta convolution quadrature are compared: one relies on the incoming wave as input data and the other one is based on its temporal derivative. The convergence rate of the latter is shown to be higher than previously established in the literature. Numerical results indicate sharpness of the analysis. The proof hinges on a novel estimate on the Dirichlet-to-Impedance map for certain Helmholtz problems. Namely, the frequency dependence can be lowered by one power of $\abs{s}$(up to a logarithmic term for polygonal domains) compared to the Dirichlet-to-Neumann map. |
| title | On superconvergence of Runge-Kutta convolution quadrature for the wave equation |
| topic | Numerical Analysis 65N38, 65N12, 65N15 |
| url | https://arxiv.org/abs/1904.00847 |