# 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.