Séminaire de Mécanique d'Orsay

Le Jeudi 15 mai à 14h00 - Salle de conférences du LIMSI

A second-order maximum principle preserving continuous finite element technique for nonlinear scalar conservation equations

Jean-Luc Guermond
Texas A&M

In the first part of the talk I will introduce a first-order viscosity method for the explicit approximation of scalar conservation equations with Lipschitz fluxes using continuous finite elements on arbitrary grids in any space dimension. Provided the lumped mass matrix is positive definite, the method is shown to satisfy the local maximum principle under a usual CFL condition. The method is independent of the cell type; for instance, the mesh can be a combination of tetrahedra, hexahedra, and prisms in three space dimensions. An a priori convergence estimate is given provided the initial data is BV. In the second part of the talk I will extend the accuracy of the method to second-order (at least). The technique is based on mass-lumping correction, a high-order entropy viscosity method, and the Boris-Book-Zalesak flux correction technique. The algorithm works for arbitrary meshes in any space dimension and for all Lipschitz fluxes. The formal second-order accuracy of the method and its convergence properties are tested on a series of linear and nonlinear benchmark problems.