# 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

### Abstract:

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.