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This paper addresses the design of linear and nonlinear stabilizationprocedures for high-order continuous Galerkin (CG) finite elementdiscretizations of scalar conservation laws. We prove that the standard CGmethod is entropy conservative for the square entropy. In general, the rate ofentropy production/dissipation depends on the residual of the governingequation and on the accuracy of the finite element approximation to the entropyvariable. The inclusion of linear high-order stabilization generates anadditional source/sink in the entropy budget equation. To balance the amount ofentropy production in each cell, we construct entropy-dissipative elementcontributions using a coercive bilinear form and a parameter-free entropyviscosity coefficient. The entropy stabilization term is high-order consistent,and optimal convergence behavior is achieved in practice. To enforcepreservation of local bounds in addition to entropy stability, we use theBernstein basis representation of the finite element solution and a new subcellflux limiting procedure. The underlying inequality constraints ensure thevalidity of localized entropy conditions and local maximum principles. Thebenefits of the proposed modifications are illustrated by numerical results forlinear and nonlinear test problems.

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