TVD schemes for unstructured grids

A number of approaches have evolved over the last decade for the implementation of total variational diminishing (TVD) schemes within an unstructured grid finite volume method framework. Unfortunately none of these approaches has been comprehensive enough to permit the general implementation of TVD-based schemes in unstructured grids, and/or accurate enough to recover the exact TVD formulation in structured grids. In this paper we propose a simple method that allows the implementation of the full spectrum of TVD schemes in unstructured grids, while recovering their exact formulation on structured grids. Four schemes implemented using this approach, TVD-MINMOD, TVD-MUSCL (monotonic upstream-centered scheme for conservation laws, MUSCL), TVD-SUPERBEE, TVD-OSHER, are tested and compared to Bruner's TVD formulation [Parallelization of the Euler equations on unstructured grids, AIAA paper 97-1894, 1995], and to the Barth and Jesperson linear reconstruction scheme [The design and application of upwind schemes on unstructured meshes, AIAA paper 89-0366, 1989] by solving four pure advection problems. Results indicate that the Bruner formulation yields, for the same original TVD scheme, overly diffusive results when compared to the current method. The BJ-MUSCL and TVD-MUSCL are shown to be comparable and more accurate than the OSHER scheme. The SUPERBEE performs best though showing tendency for stepping the modeled profile. In all tests the current method is found to retain the behavior of the structured grid TVD formulation. (C) 2002 Elsevier Science Ltd. All rights reserved.