Regular Article: 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.

[1]  B. Vanleer,et al.  On the Relation Between the Upwind-Differencing Schemes of Godunov, Engquist–Osher and Roe , 1984 .

[2]  Pilar García-Navarro,et al.  A HIGH-RESOLUTION GODUNOV-TYPE SCHEME IN FINITE VOLUMES FOR THE 2D SHALLOW-WATER EQUATIONS , 1993 .

[3]  M. Hanif Chaudhry,et al.  Explicit Methods for 2‐D Transient Free Surface Flows , 1990 .

[4]  Derek M. Causon,et al.  POSITIVELY CONSERVATIVE HIGH‐RESOLUTION CONVECTION SCHEMES FOR UNSTRUCTURED ELEMENTS , 1996 .

[5]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[6]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[7]  S. Osher,et al.  Triangle based adaptive stencils for the solution of hyperbolic conservation laws , 1992 .

[8]  Bram van Leer,et al.  On the Relation Between the Upwind-Differencing Schemes of Godunov, Engquist–Osher and Roe , 1984 .

[9]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[10]  Xu-Dong Liu,et al.  A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws , 1993 .

[11]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[12]  Antony Jameson,et al.  Positive schemes and shock modelling for compressible flows , 1995 .

[13]  Timothy J. Barth,et al.  The design and application of upwind schemes on unstructured meshes , 1989 .

[14]  Matthew Hubbard,et al.  Conservative Multidimensional Upwinding for the Steady Two-Dimensional Shallow Water Equations , 1997 .

[15]  Benoît Perthame,et al.  A variant of Van Leer's method for multidimensional systems of conservation laws , 1994 .

[16]  C. Hirsch,et al.  Fundamentals of numerical discretization , 1988 .

[17]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[18]  P. Sweby.  High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws , 1984 .

[19]  S. Spekreijse.  Multigrid solution of monotone second-order discretizations of hyperbolic conservation laws , 1986 .

[20]  Charles Hirsch,et al.  Numerical computation of internal & external flows: fundamentals of numerical discretization , 1988 .