Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws

TL;DR

This paper develops entropy-stable high-order continuous Galerkin methods for scalar conservation laws, ensuring entropy conservation and local bounds, with proven stability and optimal convergence demonstrated through numerical tests.

Contribution

It introduces entropy-preserving and entropy-dissipative stabilization techniques for high-order CG methods, including a novel subcell flux limiting procedure.

Findings

Standard CG is entropy conservative for square entropy.

Entropy stabilization achieves optimal convergence.

Numerical tests confirm stability and accuracy.

Abstract

This paper addresses the design of linear and nonlinear stabilization procedures for high-order continuous Galerkin (CG) finite element discretizations of scalar conservation laws. We prove that the standard CG method is entropy conservative for the square entropy. In general, the rate of entropy production/dissipation depends on the residual of the governing equation and on the accuracy of the finite element approximation to the entropy variable. The inclusion of linear high-order stabilization generates an additional source/sink in the entropy budget equation. To balance the amount of entropy production in each cell, we construct entropy-dissipative element contributions using a coercive bilinear form and a parameter-free entropy viscosity coefficient. The entropy stabilization term is high-order consistent, and optimal convergence behavior is achieved in practice. To enforce preservation of local bounds in addition to entropy stability, we use the Bernstein basis representation of the finite element solution and a new subcell flux limiting procedure. The underlying inequality constraints ensure the validity of localized entropy conditions and local maximum principles. The benefits of the proposed modifications are illustrated by numerical results for linear and nonlinear test problems.