Lax-Friedrichs and Nessyahu-Tadmor Schemes for

Conservation Laws on Unstructured Grids

The non-oscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of Riemann problems at the cell interfaces is by-passed thanks to the use of the staggered Lax-Friedrichs scheme, is extended here to a two-step, two-dimensional non-oscillatory centered scheme in finite volume formulation. The construction of the scheme rests on a finite volume extension of the Lax-Friedrichs scheme, in which the finite volume cells are the barycentric cells constructed around the nodes of an FEM triangulation, for odd time steps, and some quadrilateral cells associated with this triangulation, for even time steps.

Piecewise linear cell interpolants using least-squares gradients combined with a van Leer-type slope limiting allow for an oscillation-free second-order resolution.

The method is illustrated by an air flow calculation around a 2-D double ellipse; and airfoil NACA0012. For more detail see [1], [2], [3].

The authors will now focus their efforts on applications to problems with the Navier-Stokes equations in conservation form for compressible flows [4] and three-dimensional transonic and supersonic flow problems (see Figure 2).

**Figure 1Mesh (4318 vertices) and Mach contours (Mach _{} = 0.85 and 1° of attack)**

**Figure 3Euler flow around a double ellipse. Original grid, barycentric cells C _{i} and quadrilateral cells L_{ij}**

**Figure 4Double ellipse: Initial mesh (1558 vertices) and solution (pressure and Mach contours)**

**Figure 5Double ellipse: Final mesh (5055 vertices) and solution (pressure and Mach contours)**

- P. Arminjon, M.C. Viallon and A. Madrane

*From Lax-Friedrichs to a multi-dimensional finite volume extension of the Nessyahu-Tadmor scheme for compressible flows*

Proc. Int. Conf. on Numerical Methods for the Euler and Navier-Stokes Equations, Centre de recherches mathématiques, Montréal, September 14-16, 1995, P. Arminjon and A. Dervieux, editors. To be published in the AMS-CRM Series. - P. Arminjon ,M.C. Viallon A.Madrane (1996).

*A finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor scheme for Conservation Laws on unstructured grids*

CRM-2457 (abstract, résumé), Int. J. for Comp. Fluid Dynamics. - P. Arminjon, A. Madrane, H. Kaddouri and M.C. Viallon

*Discontinuous finite elements and a 2-dimensional finite volume generalization of the Lax-Friedrichs and Nessyahu-Tadmor schemes for compressible flows on unstructured grids*

to appear in CFD Review, John Wiley, 1997, M.Hafez and K.Oshima, Editors. - A. Madrane and P. Arminjon

*Non-Oscillatory Central Schemes for compressible 2-D Navier-Stokes equations using unstructured meshes*

En préparation.

**Table des matières de la petite galerie du CRM**

**28 January 1998, webmaster@CRM.UMontreal.CA**