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We propose a numerical method to solve general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on non-uniform grids. The properties of the method are based on the introduction of an artificial dissipation that is defined so that any convex invariant sets 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 [Hoff(1979), Hoff(1985], and [Frid (2001)].

A new second-order method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum principle on the specific entropy. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.1 γ−1 ρ −1 ) for a polytropic ideal gas. Using the notation s e (ρ, e) := ∂s ∂e (ρ, e) and s ρ (ρ, e) := ∂s ∂ρ (ρ, e), the equation of state then takes the form p := −ρ 2 s ρ s −1 e . To simplify the notation we introduce the Note that (2.9) implies j∈{1: I} c ij = 0. Furthermore, if either ϕ i or ϕ j is zero on ∂D, then c ij = −c ji . In particular we have i∈{1: I} c ij = 0 if ϕ j is zero on ∂D. This property will be used to establish conservation.

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This paper proposes an explicit, (at least) second-order, maximum principle satisfying, 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 [Comput. 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. Kružkov [20] and Bardos, le Roux, and Nédélec [2]). Mesh.Let {K h } h>0 be a mesh family that we assume to be affine, conforming (no hanging nodes), and shape-regular in the sense of Ciarlet. By convention, the elements in K h are closed in Ê d . We may have different reference elements, but always