Synchronized slope limiting in discontinuous Galerkin methods for the equations of gas dynamics

This paper presents a new approach to synchronized limiting of density, energy, and pressure jumps in discontinuous Galerkin (DG) methods for the Euler equations of gas dynamics. A vertex-based version of the Barth-Jespersen limiter for scalar quantities is generalized to constrain the gradients of the conservative variables in a way which guarantees that all quantities of interest remain in the range of admissible values. The bounds for the corresponding inequality constraints are designed to enforce local maximum principles in regions of strong density variations and become less restrictive in smooth regions. The proposed limiting strategy guarantees positivity preservation and leads to closed-form expressions for the synchronized gradient correction factors without the need to solve inequality-constrained optimization problems. A numerical study is performed for two-dimensional test problems.