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Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. (English) Zbl 0934.65109
The improvement of existing limiters in MUSCL-type numerical methods for the numerical solution of two-dimensional hyperbolic conservation laws is the central theme of the present paper. The framework of MUSCL-type schemes on triangular grids is exploited. Local constraints on the linear reconstruction of the solution are imposed so that an appropriate maximum principle is satisfied. This opens way to the development of new MUSCL-type schemes. Numerical results are given for a scalar problem as well as for the shallow water equation.

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76M20 Finite difference methods applied to problems in fluid mechanics
35L65 Hyperbolic conservation laws
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI
[1] Alcrudo, F.; Garcia-Navarro, P., A high resolution Godunov-type scheme in finite volumes for the 2d shallow water equations, Int. J. numer. methods fluids, 16, 489, (1993) · Zbl 0766.76067
[2] Batten, P.; Lambert, C.; Causon, D.M., Positively conservative high-resolution convection schemes for unstructured elements, Int. J. numer. methods eng., 39, 1821, (1996) · Zbl 0884.76048
[3] Barth, T.J.; Jespersen, D.C., The design and application of upwind schemes on unstructured meshes, (1989)
[4] Durlofsky, L.J.; Engquist, B.; Osher, S., Triangle based adaptive stencils for the solution of hyperbolic conservation laws, J. comput. phys., 98, 64, (1992) · Zbl 0747.65072
[5] Fennema, R.J.; Chaudry, M.H., Explicit methods for 2-D transient free-surface flows, J. hydraul. eng. ASCE, 116, 1013, (1990)
[6] Goodman, J.B.; LeVeque, R.J., On the accuracy of stable schemes for 2D conservation laws, Math. comp., 45, 15, (1985) · Zbl 0592.65058
[7] Hirsch, C., Fundamentals of numerical discretization, numerical computation of internal and external flows, 1, (1990)
[8] Hubbard, M.E., Numerical analysis report 2/98, Multidimensional slope limiters for MUSCL-type finite volume schemes, (1998)
[9] Hubbard, M.E.; Baines, M.J., Conservative multidimensional upwinding for the steady two-dimensional shallow water equations, J. comput. phys., 138, 419, (1997) · Zbl 0902.76065
[10] Jameson, A., Positive schemes and shock modeling for compressible flows, Int. J. numer. methods fluids, 20, 743, (1995) · Zbl 0837.76055
[11] LeVeque, R.J., Numerical methods for conservation laws, (1992) · Zbl 0847.65053
[12] Liu, X.-D., A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws, SIAM J. numer. anal., 30, 701, (1993) · Zbl 0791.65068
[13] Perthame, B.; Qiu, Y., A variant of Van Leer’s method for multidimensional systems of conservation laws, J. comput. phys., 112, 370, (1994) · Zbl 0816.65055
[14] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[15] Spekreijse, S.P., Multigrid solution of monotone second-order discretizations of hyperbolic conservation laws, Math. comp., 49, 135, (1987) · Zbl 0654.65066
[16] Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. numer. anal., 21, 995, (1984) · Zbl 0565.65048
[17] van Leer, B., Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov’s method, J. comput. phys., 32, 101, (1979) · Zbl 1364.65223
[18] van Leer, B., On the relation between the upwind-differencing schemes of Godunov, engquist-osher and roe, SIAM J. sci. statist. comput., 5, (1984) · Zbl 0547.65065
[19] Woodward, P.R.; Collela, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115, (1984)
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