×

zbMATH — the first resource for mathematics

Compact accurately boundary-adjusting high-resolution technique for fluid dynamics. (English) Zbl 1172.76034
Summary: A novel high-resolution numerical method is presented for one-dimensional hyperbolic problems based on the extension of the original upwind leapfrog scheme to quasi-linear conservation laws. The method is second-order accurate on non-uniform grids in space and time, has a very small dispersion error and computational stencil defined within one space-time cell. For shock-capturing, the scheme is equipped with a conservative nonlinear correction procedure which is directly based on the maximum principle. Plentiful numerical examples are provided for linear advection, quasi-linear scalar hyperbolic conservation laws and gas dynamics, and comparisons with other computational methods in the literature are discussed.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Bogey, C.; Bailly, C., A family of low dispersive and low dissipative explicit schemes for flow and noise computations, J. comput. phys., 194, 194-214, (2004) · Zbl 1042.76044
[2] Boris, J.P.; Book, D.L.; Hain, K., Flux-corrected transport: generalization of the method, J. comput. phys., 31, 335-350, (1975) · Zbl 0306.76004
[3] Cockburn, B.; Shu, C.-W., Runge – kutta discontinuous Galerkin methods for convection-dominated problems, SIAM J. sci. comput., 16, 173-261, (2001) · Zbl 1065.76135
[4] Colella, P.; Woodward, P.R., The piecewise parabolic method (PPM) for gas-dynamical simulations, J. comput. phys., 54, 174-201, (1984) · Zbl 0531.76082
[5] Colonius, T.; Lele, S.K., Computational aeroacoustics: progress on nonlinear problems of sound generation, Prog. aerospace sci., 40, 345-416, (2004)
[6] Godunov, S.K., A difference scheme for numerical computation of discontinuous solutions of equations of fluid dynamics, Math. sbornik, 47, 271-306, (1959) · Zbl 0171.46204
[7] Goloviznin, V.M.; Samarskii, A.A., Difference approximation of convective transport with spatial splitting of time derivative, Math. model., 10, 86-100, (1998) · Zbl 1189.76364
[8] Goloviznin, V.M.; Samarskii, A.A., Some properties of the CABARET scheme, Math. model., 10, 101-116, (1998) · Zbl 1189.76365
[9] Goloviznin, V.M.; Karabasov, S.A.; Kobrinski, I.M., Balanced-characteristic schemes with staggered conservative and transport variables, Math. model., 15, 29-48, (2003) · Zbl 1102.76328
[10] Goloviznin, V.M., Balanced characteristic method for systems of hyperbolic conservation laws, Doklady math., 72, 619-623, (2005) · Zbl 1125.65339
[11] Grinstein, F.F.; G Margolin, L.; Rider, W.J., Implicit large eddy simulation: computing turbulent dynamics, (2007), Cambridge University Press New York · Zbl 1135.76001
[12] Harten, A.; Engqist, B.; Osher, S.; Chakravarthy, S., Uniformly high order accurate essentially non-oscillatory schemes III, J. comput. phys., 71, 231-303, (1987) · Zbl 0652.65067
[13] Hirsch, C., Numerical computation of internal and external flows. the fundamentals of computational fluid dynamics, (2007), John Wiley & Sons, Ltd., Elsevier
[14] Iserles, A., Generalized leapfrog methods, IMA J. numer. anal., 6, 381-392, (1986) · Zbl 0637.65089
[15] G.-S. Jiang, D. Levy, C.-T. Lin, S. Osher, E. Tadmor, High-resolution non-oscillatory central schemes with non-staggered grids for hyperbolic conservation laws, SIAM J. Numer. Anal. 35 (1998) 2147-2168. · Zbl 0920.65053
[16] Karabasov, S.A.; Goloviznin, V.M., New efficient high-resolution method for nonlinear problems in aeroacoustics, Aiaa j., 45, 2861-2871, (2007)
[17] Keller, H.B., Computational fluid dynamics, (1978), AMS Bookstore
[18] Kim, S., High-order upwind leapfrog methods for multidimensional acoustic equations, Int J. numer. mech. fluids, 44, 505-523, (2004) · Zbl 1079.76598
[19] Kolgan, V.P., The application of a minimal solution slope principle for the development of finite-difference schemes for solving gas dynamics problems with discontinuities, Uch. zapiski tsagi, 3, 68-77, (1972)
[20] Lele, S.K., Compact finite-difference scheme with spectral-like resolution, J. comput. phys., 103, 16-42, (1992) · Zbl 0759.65006
[21] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. comput. phys., 115, 200-212, (1994) · Zbl 0811.65076
[22] Osher, S.; Chakravarthy, S., High resolution schemes and the entropy condition, SIAM J. numer. anal., 21, 984-995, (1984) · Zbl 0556.65074
[23] Ostapenko, V.V., On the monotonicity of the balanced-characteristic scheme, Math. model., 21, 29-42, (2009) · Zbl 1250.65114
[24] Qiu, J.; Shu, C.-W., Runge – kutta discontinuous Galerkin method using WENO limiters, SIAM J. sci. comput., 26, 907-929, (2003) · Zbl 1077.65109
[25] Roache, P.J., Computational fluid dynamics, (1982), Hermosa Albuquerque
[26] Roe, P.L., Linear bicharacteristic schemes without dissipation, SIAM J. sci. comput., 19, 1405-1427, (1998) · Zbl 0915.65106
[27] Safronov, A.V., A stabilization technique for characteristic finite-difference schemes for solving gas dynamics problems, Comput. methods and program., 8, 6-9, (2007)
[28] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439-471, (1988) · Zbl 0653.65072
[29] Sod, G.A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. comput. phys., 27, 1-31, (1978) · Zbl 0387.76063
[30] Tam, C.K.W.; Webb, J.C., Dispersion-relation-preserving finite difference schemes for computational acoustics, J. comput. phys., 107, 262-281, (1993) · Zbl 0790.76057
[31] Tang, H.; Liu, T., A note on the conservative schemes for the Euler equations, J. comput. phys., 218, 451-459, (2006) · Zbl 1103.76041
[32] Titarev, V.A.; Toro, E.F., WENO schemes based on upwind and centred TVD fluxes, Comput. fluids, 34, 705-720, (2005) · Zbl 1134.65361
[33] Toro, E., Rieman solvers and numerical methods for fluid dynamics, (1997), Springer Berlin, Heidelberg
[34] Tran, Q.H.; Scheurer, B., High-order monotonicity-preserving compact schemes for linear scalar advection on 2-D irregular meshes, J. comput. phys., 175, 454-486, (2002) · Zbl 1016.76055
[35] Van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to godunov’s method, J. comput. phys., 32, 101-136, (1979) · Zbl 1364.65223
[36] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115-173, (1984) · Zbl 0573.76057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.