M. Darwish and F. Moukalled (Lebanon)
TVD schemes, unstructured grid, convection schemes.
The implementation of TVD schemes within a multi dimensional structured grid environment is rather well understood. With the increasing importance of unstructured grid for CFD applications, a number of approaches have evolved for the implementation of these 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, TVD-SUPERBEE, TVD-OSHER, are tested and compared to Bruner’s TVD formulation [1], and to the Barth and Jesperson linear reconstruction scheme [2] by solving two problems involving advection of a step and a sinusoidal profile. 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.
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