Variational Formulations for Explicit Runge-Kutta Methods
Variational space-time formulations for partial di fferential equations have been of great interest in the last decades, among other things, because they allow to develop mesh-adaptive algorithms. Since it is known that implicit time marching schemes have variational structure, they are often employ...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2019 |
| País: | España |
| Institución: | Basque Center for Applied Mathematics (BCAM) |
| Repositorio: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/1008 |
| Acceso en línea: | http://hdl.handle.net/20.500.11824/1008 https://doi.org/10.1016/j.finel.2019.06.007 |
| Access Level: | acceso abierto |
| Palabra clave: | linear diffusion equation discontinuous Petrov-Galerkin formulations dynamic meshes Runge-Kutta methods |
| Sumario: | Variational space-time formulations for partial di fferential equations have been of great interest in the last decades, among other things, because they allow to develop mesh-adaptive algorithms. Since it is known that implicit time marching schemes have variational structure, they are often employed for adaptivity. Previously, Galerkin formulations of explicit methods were introduced for ordinary di fferential equations employing speci fic inexact quadrature rules. In this work, we prove that the explicit Runge-Kutta methods can be expressed as discontinuous-in-time Petrov-Galerkin methods for the linear di ffusion equation. We systematically build trial and test functions that, after exact integration in time, lead to one, two, and general stage explicit Runge-Kutta methods. This approach enables us to reproduce the existing time-domain (goal-oriented) adaptive algorithms using explicit methods in time. |
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