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Anti-diffusive flux corrections for high order finite difference WENO schemes. (English) Zbl 1087.76080
Summary: We generalize a technique of anti-diffusive flux corrections, recently introduced by B. Després and F. Lagoutière [J. Sci. Comput. 16, No. 4, 479–524 (2001; Zbl 0999.76091)] for first-order schemes, to high order finite difference weighted essentially non-oscillatory (WENO) schemes. The objective is to obtain sharp resolution for contact discontinuities, close to the quality of discrete traveling waves which do not smear progressively for longer time, while maintaining high order accuracy in smooth regions and non-oscillatory property for discontinuities. Numerical examples for one and two space dimensional scalar problems and systems demonstrate the good quality of this flux correction. High order accuracy is maintained and contact discontinuities are sharpened significantly compared with the original WENO schemes on the same meshes.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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[1] Balsara, D.; Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, Journal of computational physics, 160, 405-452, (2000) · Zbl 0961.65078
[2] Bouchut, F., An antidiffusive entropy scheme for monotone scalar conservation law, Journal of scientific computing, 21, 1-30, (2004) · Zbl 1091.65084
[3] Després, B.; Lagoutière, F., Contact discontinuity capturing schemes for linear advection, compressible gas dynamics, Journal of scientific computing, 16, 479-524, (2001) · Zbl 0999.76091
[4] B. Després, F. Lagoutière, Numerical resolution of a two component compressible fluid model with interfaces and mixing zones, preprint
[5] Glimm, J.; Grove, J.; Li, X.; Oh, W.; Tan, D.C., The dynamics of bubble growth for Rayleigh-Taylor unstable interfaces, Physics of fluids, 31, 447-465, (1988) · Zbl 0641.76099
[6] Goodman, J.B.; LeVeque, R.J., On the accuracy of stable schemes for 2D scalar conservation laws, Mathematics of computation, 45, 15-21, (1985) · Zbl 0592.65058
[7] Harten, A., The artificial compression method for computation of shocks and contact discontinuities: III, Mathematics of computation, 32, 363-389, (1978) · Zbl 0409.76057
[8] Harten, A., ENO schemes with subcell resolution, Journal of computational physics, 83, 148-184, (1989) · Zbl 0696.65078
[9] Harten, A.; Engquist, B.; Osher, S.; Chakravathy, S., Uniformly high order accurate essentially non-oscillatory schemes: III, Journal of computational physics, 71, 231-303, (1987) · Zbl 0652.65067
[10] Hu, C.; Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, Journal of computational physics, 150, 97-127, (1999) · Zbl 0926.65090
[11] Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, Journal of computational physics, 126, 202-228, (1996) · Zbl 0877.65065
[12] Lax, P.D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Communications on pure and applied mathematics, 7, 159-193, (1954) · Zbl 0055.19404
[13] Liu, X.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, Journal of computational physics, 115, 200-212, (1994) · Zbl 0811.65076
[14] Quirk, J.J., A contribution to the great Riemann solver debate, International journal for numerical methods in fluids, 18, 555-574, (1994) · Zbl 0794.76061
[15] Roe, P.L., Some contribution to the modelling of discontinuous flows, Lectures in applied mathematics, 22, 163-193, (1985)
[16] Shi, J.; Hu, C.; Shu, C.-W., A technique of treating negative weights in WENO schemes, Journal of computational physics, 175, 108-127, (2002) · Zbl 0992.65094
[17] Shi, J.; Zhang, Y.-T.; Shu, C.-W., Resolution of high order WENO schemes for complicated flow structures, Journal of computational physics, 186, 690-696, (2003) · Zbl 1047.76081
[18] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (), 325-432 · Zbl 0927.65111
[19] Shu, C.-W., High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD, International journal of computational fluid dynamics, 17, 107-118, (2003) · Zbl 1034.76044
[20] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of computational physics, 77, 439-471, (1988) · Zbl 0653.65072
[21] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, Journal of computational physics, 83, 32-78, (1989) · Zbl 0674.65061
[22] Siddiqi, K.; Kimia, B.; Shu, C.-W., Geometric shock-capturing ENO schemes for subpixel interpolation computation and curve evolution, Graphical models and image processing (CVGIP:GMIP), 59, 278-301, (1997)
[23] Sod, G., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, Journal of computational physics, 43, 1-31, (1978) · Zbl 0387.76063
[24] Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM journal on numerical analysis, 21, 995-1011, (1984) · Zbl 0565.65048
[25] V.A. Titarev, E.F. Toro, ENO and WENO schemes based on upwind and centered TVD fluxes, Computers and Fluids, to appear · Zbl 1134.65361
[26] E.F. Toro, V.A. Titarev, TVD fluxes for the high order ADER schemes, Journal of Scientific Computing, to appear · Zbl 1096.76029
[27] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of computational physics, 54, 115-173, (1984) · Zbl 0573.76057
[28] Yang, H., An artificial compression method for ENO schemes: the slope modification method, Journal of computational physics, 89, 125-160, (1990) · Zbl 0705.65062
[29] Young, Y.-N.; Tufo, H.; Dubey, A.; Rosner, R., On the miscible Rayleigh-Taylor instability: two and three dimensions, Journal of fluid mechanics, 447, 377-408, (2001) · Zbl 0999.76056
[30] Zhang, Y.-T.; Shi, J.; Shu, C.-W.; Zhou, Y., Numerical viscosity and resolution of high-order weighted essentially nonoscillatory schemes for compressible flows with high Reynolds numbers, Physical review E, 68, 046709, (2003)
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