Two-Dimensional Slope Limiters for Finite Volume Schemes on Non-Coordinate-Aligned Meshes

Abstract

Abstract. In this paper we develop a new limiter for linear reconstruction on non-coordinatealigned meshes in two space dimensions, with focus on Cartesian embedded boundary grids. Our limiter is inherently two-dimensional and linearity preserving. It separately limits the x and y components of the gradient, as opposed to a scalar limiter which limits all components simultaneously with one scalar. The limiter is based on solving a tiny linear program (LP) on each cell, using a very efficient version of the simplex method. A variety of computational results on triangular and embedded boundary meshes are presented. They demonstrate that the LP limiter successfully removes oscillations and significantly increases solution accuracy compared to a scalar limiter.

Anisotropic slope limiting for discontinuous Galerkin methods

“…While the results of Berger et al [14,16] indicate that anisotropic slope limiting is a better way to enforce inequality constraints of the form (12), the cost of an optimization-based algorithm is clearly higher than that of a closed-form expression such as given in (13).…”

“…In this section, we consider 2D examples that illustrate the performance of the anisotropic slope limiters and compare their results to those of the 'standard' vertex-based slope limiter [11,13] and of the optimization-based techniques [14,16]. In addition to a grid convergence study on uniform meshes, we also evaluate selected limiting techniques on nonuniform meshes.…”

“…An anisotropic slope limiter based on solving linear programming (LP) problems was introduced in [16]. Because this limiter will serve as one of the main references when evaluating methods proposed in this work, a brief description of the LP limiter is in order.…”

“…In contrast to [16], we place the control points at the vertices of element K: This is performed to provide a fairer comparison to limiting schemes proposed here. Following [16], the limiter is formulated as a constrained optimization method based on solving small LP problems with the goal of minimizing the l 1 -norm of the difference between limited and unlimited gradient. That is, the correction factors˛x and˛y are chosen so as to minimize the objective function ˛x ju x j ˛yˇu yˇ ( 16) subject to (15).…”

“…We introduce an anisotropic version of the vertex-based slope limiter developed in [11] and evaluate it using test problems that illustrate the need for an anisotropic limiting strategy. The proposed algorithm extends the optimization approach presented in [13] and is based on the same set of inequality constraints as optimization-based anisotropic limiters [14][15][16]; however, it avoids solving explicitly the optimization problems that compute the correction factors for gradient components or directional derivatives of the solution. Our limiter relies on simple operator splitting techniques instead.…”