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 nite volume method may give rise to new numerical diculties resulting into accuracy degradations. Indeed, for the simple onedimensional unstationary convection-diusi on equation for instance, a conicting 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 specic diculty and achieve fourth-, sixth- or eighth-order of accuracy schemes in space and time. To this end, a new nite volume method is designed based on specic 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.

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