zbMATH — the first resource for mathematics

On maximum-principle-satisfying high order schemes for scalar conservation laws. (English) Zbl 1187.65096
Summary: We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization solving one-dimensional scalar conservation laws. Strong stability preserving high order time discretizations will keep the maximum principle. It is straightforward to extend the method to two and higher dimensions on rectangular meshes. We also show that the same limiter can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. Numerical tests for both the WENO finite volume scheme and the DG method are reported.

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35B50 Maximum principles in context of PDEs
Full Text: DOI
[1] Carpenter, M.H.; Gottlieb, D.; Abarbanel, S.; Don, W.-S., The theoretical accuracy of runge – kutta time discretizations for the initial boundary value problem: a study of the boundary error, SIAM journal on scientific computing, 16, 1241-1252, (1995) · Zbl 0839.65098
[2] Cockburn, B.; Johnson, C.; Shu, C.-W.; Tadmor, E., ()
[3] Cockburn, B.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of computation, 52, 411-435, (1989) · Zbl 0662.65083
[4] Crandall, M.; Majda, A., Monotone difference approximations for scalar conservation laws, Mathematics of computation, 34, 1-21, (1980) · Zbl 0423.65052
[5] Dafermos, C.M., Hyperbolic conservation laws in continuum physics, (2000), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0940.35002
[6] Gottlieb, S.; Ketcheson, D.I.; Shu, C.-W., High order strong stability preserving time discretizations, Journal of scientific computing, 38, 251-289, (2009) · Zbl 1203.65135
[7] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability preserving high order time discretization methods, SIAM review, 43, 89-112, (2001) · Zbl 0967.65098
[8] Harten, A., High resolution schemes for hyperbolic conservation laws, Journal of computational physics, 49, 357-393, (1983) · Zbl 0565.65050
[9] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, Journal of computational physics, 71, 231-303, (1987) · Zbl 0652.65067
[10] Helbing, D.; Hennecke, A.; Shvetsov, V.; Treiber, M., MASTER: macroscopic traffic simulation based on a gas-kinetic, non-local traffic model, Transportation research part B, 35, 183-211, (2001)
[11] Hesthaven, J.S.; Gottlieb, S.; Gottlieb, D., Spectral methods for time-dependent problems, (2007), Cambridge University Press · Zbl 1111.65093
[12] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, Journal of computational physics, 126, 202-228, (1996) · Zbl 0877.65065
[13] Jiang, G.-S.; Tadmor, E., Nonoscillatory central schemes for multidimensional hyperbolic conservative laws, SIAM journal on scientific computing, 19, 1892-1917, (1998) · Zbl 0914.65095
[14] Levy, D.; Tadmor, E., Non-oscillatory central schemes for the incompressible 2D Euler equations, Mathematical research letters, 4, 321-340, (1997) · Zbl 0883.76057
[15] Liu, J.-G.; Shu, C.-W., A high-order discontinuous Galerkin method for 2D incompressible flows, Journal of computational physics, 160, 577-596, (2000) · Zbl 0963.76069
[16] Liu, X.-D.; Osher, S., Non-oscillatory high order accurate self similar maximum principle satisfying shock capturing schemes, SIAM journal on numerical analysis, 33, 760-779, (1996) · Zbl 0859.65091
[17] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, Journal of computational physics, 115, 200-212, (1994) · Zbl 0811.65076
[18] Lu, Y.; Wong, S.C.; Zhang, M.; Shu, C.-W.; Chen, W., Explicit construction of entropy solutions for the lighthill – whitham – richards traffic flow model with a piecewise quadratic flow-density relationship, Transportation research part B, 42, 355-372, (2008)
[19] Osher, S.; Chakravarthy, S., High resolution schemes and the entropy condition, SIAM journal on numerical analysis, 21, 955-984, (1984) · Zbl 0556.65074
[20] Perthame, B.; Shu, C.-W., On positivity preserving finite volume schemes for Euler equations, Numerische Mathematik, 73, 119-130, (1996) · Zbl 0857.76062
[21] Sanders, R., A third-order accurate variation nonexpansive difference scheme for single nonlinear conservation law, Mathematics of computation, 51, 535-558, (1988) · Zbl 0699.65069
[22] 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
[23] Shu, C.-W., Total-variation-diminishing time discretizations, SIAM journal on scientific and statistical computing, 9, 1073-1084, (1988) · Zbl 0662.65081
[24] Shu, C.-W., High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM review, 51, 82-126, (2009) · Zbl 1160.65330
[25] 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
[26] X. Zhang, C.-W. Shu, A genuinely high order total variation diminishing scheme for one-dimensional scalar conservation laws, SIAM Journal on Numerical Analysis, submitted for publication. · Zbl 1226.65083
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.