Second-order MUSCL schemes based on Dual Mesh Gradient Reconstruction (DMGR)

¹ Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France.
vivien.desveaux@univ-nantes.fr
² Institut de Mathématiques de Bordeaux, CNRS UMR 5251, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France.
³ INRIA Sud-Ouest, 351 cours de la Libération, 33405 Talence Cedex, France .

Received: 5 July 2012
Revised: 24 May 2013

Abstract

We discuss new MUSCL reconstructions to approximate the solutions of hyperbolic systems of conservations laws on 2D unstructured meshes. To address such an issue, we write two MUSCL schemes on two overlapping meshes. A gradient reconstruction procedure is next defined by involving both approximations coming from each MUSCL scheme. This process increases the number of numerical unknowns, but it allows to reconstruct very accurate gradients. Moreover a particular attention is paid on the limitation procedure to enforce the required robustness property. Indeed, the invariant region is usually preserved at the expense of a more restrictive CFL condition. Here, we try to optimize this condition in order to reduce the computational cost.