Hierarchical slope limiting in explicit and implicit discontinuous Galerkin methods

In this paper, we present a collection of algorithmic tools for constraining high-order discontinuous Galerkin (DG) approximations to hyperbolic conservation laws. We begin with a review of hierarchical slope limiting techniques for explicit DG methods. A new interpretation of these techniques leads to an unconditionally stable implicit algorithm for steady-state computations. The implicit global problem for the mean values (coarse scales) has the computational structure of a finite volume method. The constrained derivatives (fine scales) are obtained by solving small local problems. The interscale transfer operators provide a two-way iterative coupling between the solutions to the global and local problems. Another highlight of this paper is a new approach to compatible gradient limiting for the Euler equations of gas dynamics. After limiting the conserved quantities, the gradients of the velocity and energy density are constrained in a consistent manner. Numerical studies confirm the accuracy and robustness of the proposed algorithms.