A very high-order finite volume method for the time-dependent convection-diffusion problem with Butcher tableau extension

The time discretization of a very high-order finite volume method may give rise to new numerical difficulties resulting into accuracy degradations. Indeed, for the simple one-dimensional unstationary convection-diffusion equation for instance, a conflicting situation between the source term time dis...

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Detalhes bibliográficos
Autor principal: Clain, Stéphane (author)
Outros Autores: Machado, Gaspar J. (author)
Formato: article
Idioma:eng
Publicado em: 2014
Assuntos:
Texto completo:http://hdl.handle.net/1822/30783
País:Portugal
Oai:oai:repositorium.sdum.uminho.pt:1822/30783
Descrição
Resumo:The time discretization of a very high-order finite volume method may give rise to new numerical difficulties resulting into accuracy degradations. Indeed, for the simple one-dimensional unstationary convection-diffusion equation for instance, a conflicting situation between the source term time discretization and the boundary conditions may arise when using the standard Runge-Kutta method. We propose an alternative procedure by extending the Butcher Tableau to overcome this specific difficulty and achieve fourth-, sixth- or eighth-order of accuracy schemes in space and time. To this end, a new finite volume method is designed based on specific polynomial reconstructions for the space discretization, while we use the Extended Butcher Tableau to perform the time discretization. A large set of numerical tests has been carried out to validate the proposed method.