Very high-order time scheme for the finite volume method

Abstract

We present a new class of very-high-order finite volume schemes for the time dependent convection-diffusion problem providing a sixth-order accuracy both in space and time. The method is based on local polynomial reconstructions dedicated to the convective or the diffusive part of the operator where the Dirichlet boundary condition is explicitly integrated in the polynomial reconstruction. The study focus on the time discretization problem where both explicit and implicit accurate time schemes are considered whereas no operator splitting is performed to easily preserve the sixth-order accuracy in time. Numerical tests are proposed to demonstrate the efficiency of the method.