×

zbMATH — the first resource for mathematics

New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. (English) Zbl 0987.65085
The authors first review some central difference schemes and then propose a new fully discrete second order central difference scheme for systems of one dimensional hyperbolic conservation laws. This scheme has the merit that its numerical viscosity is of order \({\mathcal O}(\Delta x)^{2r-1})\) which is independent of \(1\Delta t\). (The parameter \(r\) has never been defined in the paper, but it is presumably the order of the scheme.) Furthermore, this scheme becomes semi-discrete when \(\Delta t \rightarrow 0\). In this case the scheme satisfies the scalar total-variation diminishing property. Extensions of the semi-discrete scheme to convection-diffusion equations and to multi-dimensions are also discussed. Fully discrete schemes can also be obtained by applying Runge-Kutta methods to the semi-discrete formula. A number of numerical examples is solved by the method to justify the theoretical results obtained.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
Software:
HLLE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. M. Anile, V. Romano, and, G. Russo, Extended hydrodynamical model of carrier transport in semiconductors, SIAM J. Appl. Math, in press. · Zbl 0966.35076
[2] P. Arminjon, D. Stanescu, and M.-C. Viallon, A two-dimensional finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for compressible flow, in Proc. 6th Int. Symp. on CFD, Lake Tahoe, 1995, M. Hafez and K. Oshima, editors, Vol. IV, pp. 7-14.
[3] Arminjon, P.; Viallon, M.-C., Généralisation du schéma de nessyahu – tadmor pour une équation hyperbolique à deux dimensions d’espace, C. R. acad. sci. Paris Sér. I, 320, 85, (1995) · Zbl 0831.65091
[4] Bianco, F.; Puppo, G.; Russo, G., High order central schemes for hyperbolic systems of conservation laws, SIAM J. sci. comput., 21, 294, (1999) · Zbl 0940.65093
[5] P. Colella, private communication.
[6] Dahle, H.K., Adaptive characteristic operator splitting techniques for convection-dominated diffusion problems in one and two space dimensions, (1988)
[7] Engquist, B.; Runborg, O., Multi-phase computations in geometrical optics, J. comput. appl. math., 74, 175, (1996) · Zbl 0947.78001
[8] S. Evje, K. H. Karlsen, K.-A. Lie, and, N. H. Risebro, Front tracking and operator splitting for nonlinear degenerate convection – diffusion equations, Institut Mittag-Leffler Report, Stockholm, Sweden, 1997. · Zbl 0961.65073
[9] A. C. Fowler, Glaciers and ice sheets, in, The Mathematics of Models for Climatology and Environment, edited by, J. I. Diaz, NATO ASI Series, Vol, 48, Springer-Verlag, Berlin/New York, 1996, p, 302.
[10] Friedrichs, K.O., Symmetric hyperbolic linear differential equations, Comm. pure appl. math., 7, 345, (1954) · Zbl 0059.08902
[11] Godunov, S.K., A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. sb., 47, 271, (1959) · Zbl 0171.46204
[12] Goodman, J.; Kurganov, A.; Rosenau, P., Breakdown of Burgers-type equations with saturating dissipation fluxes, Nonlinearity, 12, 247, (1999) · Zbl 0946.35039
[13] Harten, A., High resolution schemes for hyperbolic conservation laws, J. comput. phys., 49, 357, (1983) · Zbl 0565.65050
[14] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S.R., Uniformly high order accurate essentially non-oscillatory schemes, III, J. comput. phys., 71, 231, (1987) · Zbl 0652.65067
[15] Harten, A.; Lax, P.D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM rev., 25, 35, (1983) · Zbl 0565.65051
[16] Jameson, A., Internat. J. comput. fluid dynamics, 4, 171, (1995)
[17] A. Jameson, 5, 1, 1995.
[18] Jiang, G.-S.; Levy, D.; Lin, C.-T.; Osher, S.; Tadmor, E., High-resolution non-oscillatory central schemes with non-staggered grids for hyperbolic conservation laws, SIAM J. numer. anal., 35, 2147, (1998) · Zbl 0920.65053
[19] Jiang, G.-S.; Tadmor, E., Non-oscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. sci. comput., 19, 1892, (1998) · Zbl 0914.65095
[20] Jin, S.; Xin, Z., The relaxation schemes for hyperbolic systems of conservation laws in arbitrary space dimensions, Cpam, 48, 235, (1995) · Zbl 0826.65078
[21] Karlsen, K.H.; Brusdal, K.; Dahle, H.K.; Evje, S.; Lie, K.-A., The corrected operator splitting approach applied to a nonlinear advection – diffusion problem, Comput. methods appl. mech. eng., 167, 239, (1998) · Zbl 0942.76047
[22] Karlsen, K.H.; Lie, K.-A., An unconditionally stable splitting for a class of nonlinear parabolic equations, IMA J. numer. anal., 19, 609, (1999) · Zbl 0949.65089
[23] Karlsen, K.H.; Risebro, N.H., An operator splitting method for nonlinear convection – diffusion equations, Numer. math., 77, 365, (1997) · Zbl 0882.35074
[24] Karlsen, K.H.; Risebro, N.H., Corrected operator splitting for nonlinear parabolic equations, SIAM J. numer. anal., 37, 980, (2000) · Zbl 0951.35080
[25] R. Kupferman, Simulation of viscoelastic fluids: Couette-Taylor flow, J. Comput. Phys, to appear. · Zbl 0935.76058
[26] Kupferman, R., A numerical study of the axisymmetric couette – taylor problem using a fast high-resolution second-order central scheme, SIAM J. sci. comput., 20, 858, (1998) · Zbl 0922.76256
[27] Kupferman, R.; Tadmor, E., A fast high-resolution second-order central scheme for incompressible flows, Proc. nat. acad. sci., 94, 4848, (1997) · Zbl 0875.76386
[28] Kurganov, A., Conservation laws: stability of numerical approximations and nonlinear regularization, (1997)
[29] Kurganov, A.; Rosenau, P., Effects of a saturating dissipation in Burgers-type equations, Comm. pure appl. math., 50, 753, (1997) · Zbl 0888.35097
[30] Lax, P.D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. pure appl. math., 7, 159, (1954) · Zbl 0055.19404
[31] van Leer, B., Towards the ultimate conservative difference scheme. III. upstream-centered finite-difference schemes for ideal compressible flow, J. comput. phys., 23, 263, (1977) · Zbl 0339.76039
[32] 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
[33] Levy, D.; Puppo, G.; Russo, G., Central WENO schemes for hyperbolic systems of conservation laws, Math. model. numer. anal., 33, 547, (1999) · Zbl 0938.65110
[34] Levy, D.; Tadmor, E., Non-oscillatory central schemes for the incompressible 2-D Euler equations, Math. res. lett., 4, 1, (1997)
[35] Liu, X.-D.; Lax, P.D., Positive schemes for solving multidimensional systems of hyperbolic conservation laws, Comput. fluid dynamics J., 5, 1, (1996)
[36] Liu, X.-D.; Osher, S., Convex ENO high order multi-dimensional schemes without field by field decomposition or staggered grids, J. comput. phys., 142, 304, (1998) · Zbl 0941.65082
[37] Liu, X.-D.; Tadmor, E., Third order nonoscillatory central scheme for hyperbolic conservation laws, Numer. math., 79, 397, (1998) · Zbl 0906.65093
[38] Medovikov, A.A., High order explicit methods for parabolic equations, Bit, 38, 372, (1998) · Zbl 0909.65060
[39] Nessyahu, H.; Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. comput. phys., 87, 408, (1990) · Zbl 0697.65068
[40] Nessyahu, H.; Tadmor, E., The convergence rate of approximate solutions for non-linear scalar conservation laws, SIAM J. numer. anal., 29, 1505, (1992) · Zbl 0765.65092
[41] Osher, S., Convergence of generalized MUSCL schemes, Sinum, 22, 947, (1984) · Zbl 0627.35061
[42] Osher, S.; Tadmor, E., On the convergence of difference approximations to scalar conservation laws, Math. comput., 50, 19, (1988) · Zbl 0637.65091
[43] Roe, P., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[44] V. Romano, and, G. Russo, Numerical solution for hydrodynamical models of semiconductors, M^3AS, preprint. · Zbl 1012.82027
[45] Shu, C.-W., Total-variation-diminishing time discretizations, Sissc, 6, 1073, (1988) · Zbl 0662.65081
[46] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439, (1988) · Zbl 0653.65072
[47] Sod, G., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. comput. phys., 22, 1, (1978) · Zbl 0387.76063
[48] Tadmor, E., Convenient total variation diminishing conditions for nonlinear difference schemes, SIAM J. numer. anal., 25, 1002, (1988) · Zbl 0662.65082
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.