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