Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 384, Issue -, Pages 308-325Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2019.01.032
Keywords
Hyperbolic conservation laws; Discontinuous Galerkin methods; Vector limiters; Objectivity; Shallow water equations; Euler equations
Funding
- German Research Association (DFG) [AI 117/2-1 (KU 1530/12-1)]
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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 vertex-based slope limiters for tensorvalued 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. (C) 2019 Elsevier Inc. All rights reserved.
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