Frame-invariant directional vector limiters for discontinuous Galerkin methods

Abstract

Second and higher order numerical approximations of conservation laws for vector fields call for the use of limiting techniques based on generalized monotonicity criteria. In this paper, we introduce a family of directional vertexbased slope limiters for tensor-valued gradients of formally second-order accurate piecewise-linear discontinuous Galerkin (DG) discretizations. The proposed methodology enforces local maximum principles for scalar products corresponding to projections of a vector field onto the unit vectors of a frame-invariant orthogonal basis. In particular, we consider anisotropic limiters based on singular value decompositions and the Gram-Schmidt orthogonalization procedure. The proposed extension to hyperbolic systems features a sequential limiting strategy and a global invariant domain fix. The pros and cons of different approaches to vector limiting are illustrated by the results of numerical studies for the two-dimensional shallow water equations and for the Euler equations of gas dynamics.