A very high-order finite volume method for the time-dependent convectiondiffusion 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 convectiondiffusion equation for instance, a conflicting situation between the source term time discretization and the boundary conditions may arise when using the standard RungeKutta 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.
very high-order finite volume convectiondiffusion butcher tableau polynomial reconstruction