Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems

The algebraic flux correction (AFC) schemes presented in this work constrain a standard continuous finite element discretization of a nonlinear hyperbolic problem to satisfy relevant maximum principles and entropy stability conditions. The desired properties are enforced by applying a limiter to antidiffusive fluxes that represent the difference between the high-order baseline scheme and a property-preserving approximation of Lax–Friedrichs type. In the first step of the limiting procedure, the given target fluxes are adjusted in a way that guarantees preservation of local and/or global bounds. In the second step, additional limiting is performed, if necessary, to ensure the validity of fully discrete and/or semi-discrete entropy inequalities. The limiter-based entropy fixes considered in this work are applicable to finite element discretizations of scalar hyperbolic equations and systems alike. The underlying inequality constraints are formulated using Tadmor’s entropy stability theory. The proposed limiters impose entropy-conservative or entropy-dissipative bounds on the rate of entropy production by antidiffusive fluxes and Runge–Kutta (RK) time discretizations. Two versions of the fully discrete entropy fix are developed for this purpose. The first one incorporates temporal entropy production into the flux constraints, which makes them more restrictive and dependent on the time step. The second algorithm interprets the final stage of a high-order AFC-RK method as a constrained antidiffusive correction of an implicit low-order scheme (algebraic Lax–Friedrichs in space + backward Euler in time). In this case, iterative flux correction is required, but the inequality constraints are less restrictive and limiting can be performed using algorithms developed for the semi-discrete problem. To motivate the use of limiter-based entropy fixes, we prove a finite element version of the Lax–Wendroff theorem and perform numerical studies for standard test problems. In our numerical experiments, entropy-dissipative schemes converge to correct weak solutions of scalar conservation laws, of the Euler equations, and of the shallow water equations.