Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids

A framework is presented for the construction of multidimensional slope limiting operators for two-dimensional MUSCL-type finite volume schemes on triangular grids. A major component of this new viewpoint is the definition of multidimensional "maximum principle regions." These are defined by local constraints on the linear reconstruction of the solution which guarantee that an appropriate maximum principle is satisfied. This facilitates both the construction of new schemes and the improvement of existing limiters. It is the latter which constitutes the bulk of this paper. Numerical results are presented for the scalar advection equation and for a nonlinear system, the shallow water equations. The extension to systems is carried out using Roe's approximate Riemann solver. All the techniques presented are readily generalised to three dimensions. (C) 1999 Academic Press.