A Finite Volume Extension of the Lax-Friedrichs and Nessyahu-Tadmor Schemes for Conservation Laws on Unstructured Grids

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Abstract
The non-oscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution or 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. Some preliminary numerical experiments suggest that two-dimensional problems can be handled very efficiently by the method presented here.

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