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

Abstract

a b s t r a c tThe 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.

A Very High-Order Accurate Staggered Finite Volume Scheme for the Stationary Incompressible Navier–Stokes and Euler Equations on Unstructured Meshes

“…In [11], a new class of polynomial reconstructions have been designed to provide very accurate approximations of the solution for the convection-diffusion problem. An extension for the three-dimensional case with curved boundaries [6] has been developed while the non-stationary problem with time-dependent Dirichlet or Neumann conditions has been studied in [8]. The main ingredient is a differentiated polynomial reconstruction depending on the operator we are dealing with.…”

Direct Arbitrary-Lagrangian–Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws

“…Recently, in [82] it has been successfully substituted to the WENO reconstruction within a high order 3D ADER finite volume scheme designed for fixed grids and solving different systems of hyperbolic conservation laws. Even more recently the a posteriori MOOD concept has been successfully used as an a posteriori subcell limiter for high order accurate Discontinuous Galerkin schemes in [129], or as an efficient high-order finite volume solver for convection-diffusion problems [32,31,101] or, lastly, to construct all-entropy finite volume schemes [124,123]. Note that in an indirect ALE context, for the so-called remap phase, an a posteriori slope limiter has been proposed in [99].…”

Design and analysis of ADER-type schemes for model advection-diffusion-reaction equations

“…An introduction to ADER schemes is found in Chapters 19 and 20 of [37]. Further developments and applications are found, for example, in [34], [38], [45], [35], [33], [36], [10], [32], [44], [5], [49], [4], [20], [43], [26], [27], [17], [8], [9], [28], [12], [13], [11], [18].…”