Hajduk, HennesKuzmin, DmitriAizinger, Vadym2018-06-122018-06-122018-062190-1767http://hdl.handle.net/2003/3690610.17877/DE290R-18905Second 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.enErgebnisberichte des Instituts für Angewandte Mathematik;588hyperbolic conservation lawsdiscontinuous Galerkin methodsvector limitersobjectivityshallow water equationsEuler equations610preprint