High order preserving residual distribution schemes for advection-diffusion like problems on arbitrary grids, application to the Navier Stokes equations

Remi Abgrall
Institut de Mathematiques de Bordeaux
University of Bordeaux
Talence, France


In this talk, we deal with the construction of a class of high order accurate Residual Distribution schemes for advection-diffusion-like problems using conformal meshes. We start by considering scalar problems. For this, we consider problems that range from pure diffusion to pure advection. The approximation of the solution is obtained using standard Lagrangian finite elements and the total residual of the problem is constructed taking into account both the advective and the diffusive terms in order to discretize with the same scheme both parts of the governing equation. To cope with the fact that the normal component of the gradients of the numerical solution is discontinuous across the faces of the elements, the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution. Linear and non-linear schemes are constructed and their accuracy is tested with the discretization of advection-diffusion and anisotropic diffusion problems. Then, by a formal extension of this method, we show its efficiency on the Navier-Stokes equations.