High order preserving residual distribution schemes for advection-diffusion scalar problems on arbitrary grids

Remi Abgrall (1, 2) , Dante de Santis (1) , Mario Ricchiuto (1, 2)

1 BACCHUS - Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems
2 IMB - Institut de Mathématiques de Bordeaux

This paper deals with the construction of a class of high order accurate Residual Distribution schemes for advection-diffusion problems using conformal meshes. The problems we consider range from pure difusion 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.This paper deals with the construction of a class of high order accurate Residual Distribution schemes for advection-diffusion problems using conformal meshes. The problems we consider range from pure difusion 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.

Dans ce rapport, nous construisons une classe de schémas distribuant le résidu d'ordre très élevé adaptée aux problémes de convection diffusion. Les maillages employés sont non structurés arbitraires mais conformes. Les problèmes considérés vont de la diffusion pure à la convection pure. L'approximation numérique est obtenue en considérant des éléments finis de type Lagrange. Le résidu total est construit en prenant en compte simultanément les termes advectifs et diffusifs. Au travers des éléments, le gradient de l'approximation polynômiale est discontinu, ce qui conduit à considérer plusieurs types de reconstruction du gradient, afin d'en obtenir une approximation globalement continue avec le même type d'approximation polynômiale. Des variantes linéaires et non linéaires du schéma sont construites et testés sur des problèmes d'advection-diffusion linéaire, Burger visqueux, un problème de diffusion anisotrope et un probléme à viscosité évanescente. On montre que l'on obtient l'ordre trois dans toutes ces situations au moyen d'une méthode locale.

Mots clés: Higher order schemes; Residual distribution; Viscous term; Advection-diffusion problems; Gradient recovery

Domaines: Analyse numérique [math.NA]; Milieux fluides et réactifs

Remi Abgrall, Dante de Santis, Mario Ricchiuto. High order preserving residual distribution schemes for advection-diffusion scalar problems on arbitrary grids. [Research Report] RR-8157, INRIA. 2012, pp.46. ⟨hal-00758930v2⟩