Anisotropic slope limiting for discontinuous Galerkin methods

作者：Aizinger Vadym; Kosik Adam^*; Kuzmin Dmitri; Reuter Balthasar

来源：International Journal for Numerical Methods in Fluids, 2017, 84(9): 543-565.

DOI：10.1002/fld.4360

• hyperbolic conservation laws
• discontinuous Galerkin methods
• anisotropic slope limiting
• inequality-constrained optimization
• Taylor basis

In this paper, we present an anisotropic version of a vertex-based slope limiter for discontinuous Galerkin methods. The limiting procedure is carried out locally on each mesh element utilizing the bounds defined at each vertex by the largest and smallest mean value from all elements containing the vertex. The application of this slope limiter guarantees the preservation of monotonicity. Unnecessary limiting of smooth directional derivatives is prevented by constraining the x and y components of the gradient separately. As an inexpensive alternative to optimization-based methods based on solving small linear programming problems, we propose a simple operator splitting technique for calculating the correction factors for the x and y derivatives. We also provide the necessary generalizations for using the anisotropic limiting strategy in an arbitrary rotated frame of reference and in the vicinity of exterior boundaries with no Dirichlet information. The limiting procedure can be extended to elements of arbitrary polygonal shape and three dimensions in a straightforward fashion. The performance of the new anisotropic slope limiter is illustrated by two-dimensional numerical examples that employ piecewise linear discontinuous Galerkin approximations.

• 出版日期2017-7-30