A SECOND-ORDER MAXIMUM PRINCIPLE PRESERVING LAGRANGE FINITE ELEMENT TECHNIQUE FOR NONLINEAR SCALAR CONSERVATION EQUATIONS ∗

This paper proposes an explicit, (at least) second-order, maximum principle sat- isfying, Lagrange finite element method for solving nonlinear scalar conservation equations. The technique is based on a new viscous bilinear form introduced in Guermond and Nazarov (Com- put. Methods Appl. Mech. Engrg., 272 (2014), pp. 198-213), 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.