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WENO schemes based on upwind and centred TVD fluxes. (English) Zbl 1134.65361
Summary: We propose to use second-order TVD fluxes, instead of first-order monotone fluxes, in the framework of finite-volume weighted essentially non-oscillatory (WENO) schemes. We call the new improved schemes the WENO-TVD schemes. They include both upwind and centred schemes on non-staggered meshes. Numerical results suggest that our schemes are superior to the original schemes used with first-order monotone fluxes. This is especially so for long time evolution problems containing both smooth and non-smooth features.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
Full Text: DOI
[1] Balsara, D.S.; Shu, C.W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. comput. phys., 160, 405-452, (2000) · Zbl 0961.65078
[2] Chen, G.Q.; Toro, E.F., Centred schemes for nonlinear hyperbolic equations, Journal of hyperbolic differential equations, 1, 3, 531-566, (2004) · Zbl 1063.65076
[3] Godunov, S.K., A finite difference method for the computation of discontinious solutions of the equation of fluid dynamics, Mat. sb., 47, 357-393, (1959)
[4] Jiang, G.S.; Shu, C.W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202-212, (1996) · Zbl 0877.65065
[5] Levy, D.; Puppo, G.; Russo, G., A fourth order central WENO scheme for multi-dimensional hyperbolic systems of conservation laws, SIAM J. sci. comput., 22, 480-506, (2002) · Zbl 1014.65079
[6] Qiu, J.; Shu, C.W., On the construction, comparison, and local characteristic decomposition for high order central WENO schemes, J. comput. phys., 183, 187-209, (2002) · Zbl 1018.65106
[7] Roe, P.L., Some contributions to the modelling of discontinuous flows, Lect. appl. math., 22, (1985) · Zbl 0609.76075
[8] Shu, C.W., Total-variation-diminishing time discretizations, SIAM J. sci. stat. comput., 9, 1073-1084, (1988) · Zbl 0662.65081
[9] Suresh, A.; Huynh, T., Accurate monotonicity preserving scheme with Runge-Kutta time stepping, J. comput. phys., 136, 83-99, (1997) · Zbl 0886.65099
[10] Toro EF. On Glimm-related schemes for conservation laws. Technical Report MMU-9602, Department of Mathematics and Physics, Manchester Metropolitan University, UK, 1996
[11] Toro, E.F.; Billett, S.J., Centred TVD schemes for hyperbolic conservation laws, IMA J. numer. anal., 20, 47-79, (2000) · Zbl 0943.65100
[12] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer-Verlag · Zbl 0923.76004
[13] Toro EF. A weighted average flux method for hyperbolic conservation laws. Proc Roy Soc Lond A 1989;423:401-18 · Zbl 0674.76060
[14] Toro, E.F., The weighted average flux method applied to the Euler equations, Philos. trans. roy. soc. lond. A, 341, 499-530, (1992) · Zbl 0767.76043
[15] Toro, E.F.; Spruce, M.; Speares, W., Restoration of the contact surface in the harten-Lax-Van leer Riemann solver, J. shock waves, 4, 25-34, (1994) · Zbl 0811.76053
[16] Toro EF, Titarev VA. TVD fluxes for the high-order ADER schemes. J Sci Comput 2004, in press
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