Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems

Abstract

We propose a numerical method for solving general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on nonuniform grids. The properties of the method are based on the introduction of an artificial dissipation that is defined so that any convex invariant set containing the initial data is an invariant domain for the method. The invariant domain property is proved for any hyperbolic system provided a CFL condition holds. The solution is also shown to satisfy a discrete entropy inequality for every admissible entropy of the system. The method is formally first-order accurate in space and can be made high-order in time by using strong stability preserving algorithms. This technique extends to continuous finite elements the work of [D. Hoff, Math. Comp., 33 (1979), pp. 1171--1193], [D. Hoff, Trans. Amer. Math. Soc., 289 (1985), pp. 591--610], and [H. Frid, Arch. Ration. Mech. Anal., 160 (2001), pp. 245--269].