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

TL;DR

This paper introduces limiter-based entropy stabilization techniques for finite element schemes solving nonlinear hyperbolic problems, ensuring maximum principles and entropy conditions are satisfied for accurate, stable solutions.

Contribution

It develops novel entropy fixes using limiters that enforce entropy inequalities in semi-discrete and fully discrete schemes for scalar and system hyperbolic equations.

Findings

Entropy-dissipative schemes converge to correct weak solutions.

The proposed limiters preserve maximum principles and entropy stability.

Numerical tests validate the effectiveness of the entropy fixes.

Abstract

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. 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.