×

zbMATH — the first resource for mathematics

A technique of treating negative weights in WENO schemes. (English) Zbl 0992.65094
Summary: High-order accurate weighted essentially nonoscillatory (WEND) schemes have recently been developed for finite difference and finite volume methods both in structured and in unstructured meshes. A key idea in WENO scheme is a linear combination of lower-order fluxes or reconstructions to obtain a higher-order approximation. The combination coefficients, also called linear weights, are determined by local geometry of the mesh and order of accuracy and may become negative, such as in the central WENO schemes using staggered meshes, high-order finite volume WEND schemes in two space dimensions, and finite difference WEND approximations for second derivatives. WENO procedures cannot be applied directly to obtain a stable scheme if negative linear weights are present. The previous strategy for handling this difficulty is either by regrouping of stencils or by reducing the order of accuracy to get rid of the negative linear weights.
In this paper we present a simple and effective technique for handling negative linear weights without a need to get rid of them. Test cases are shown to illustrate the stability and accuracy of this approach.

MSC:
65M06 Finite difference 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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Balsara, D.; Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. comput. phys., 160, 405, (2000) · Zbl 0961.65078
[2] Casper, J.; Shu, C.-W.; Atkins, H.L., Comparison of two formulations for high-order accurate essentially nonoscillatory schemes, Aiaa j., 32, 1970, (1994) · Zbl 0827.76049
[3] Del Zanna, L.; Velli, M.; Londrillo, P., Dynamical response of a stellar atmosphere to pressure perturbations: numerical simulations, Astron. astrophys., 330, L13, (1998)
[4] Friedrichs, O., Weighted essentially non-oscillatory schemes for the interpolation of Mean values on unstructured grids, J. comput. phys., 144, 194, (1998)
[5] Grasso, F.; Pirozzoli, S., Shock-wave-vortex interactions: shock and vortex deformations, and sound production, Theor. comput. fluid dyn., 13, 421, (2000) · Zbl 0972.76051
[6] Grasso, F.; Pirozzoli, S., Shock wave-thermal inhomogeneity interactions: analysis and numerical simulations of sound generation, Phys. fluids, 12, 205, (2000) · Zbl 1149.76390
[7] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, J. comput. phys., 71, 231, (1987) · Zbl 0652.65067
[8] Hu, C.; Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, J. comput. phys., 150, 97, (1999) · Zbl 0926.65090
[9] Jiang, G.; Peng, D.-P., Weighted ENO schemes for hamilton – jacobi equations, SIAM J. sci. comput., 21, 2126, (2000) · Zbl 0957.35014
[10] Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202, (1996) · Zbl 0877.65065
[11] Jiang, G.; Wu, C.-C., A high order WENO finite difference scheme for the equations of ideal magnetohydrodynamics, J. comput. phys., 150, 561, (1999) · Zbl 0937.76051
[12] Levy, D.; Puppo, G.; Russo, G., Central WENO schemes for hyperbolic systems of conservation laws, Math. modelling numer. anal. (M^2 AN), 33, 547, (1999) · Zbl 0938.65110
[13] Levy, D.; Puppo, G.; Russo, G., Compact central WENO schemes for multidimensional conservation laws, SIAM J. sci. comput., 22, 656, (2000) · Zbl 0967.65089
[14] Levy, D.; Puppo, G.; Russo, G., A third order central WENO scheme for 2D conservation laws, Appl. numer. math., 33, 415, (2000) · Zbl 0965.65106
[15] Liang, S.; Chen, H., Numerical simulation of underwater blast-wave focusing using a high-order scheme, Aiaa j., 37, 1010, (1999)
[16] Liska, R.; Wendroff, B., Composite schemes for conservation laws, SIAM J. numer. anal., 35, 2250, (1998) · Zbl 0920.65054
[17] Liska, R.; Wendroff, B., Two-dimensional shallow water equations by composite schemes, Int. J. numer. meth. fluids., 30, 461, (1999) · Zbl 0946.76061
[18] 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
[19] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. comput. phys., 115, 200, (1994) · Zbl 0811.65076
[20] Montarnal, P.; Shu, C.-W., Real gas computation using an energy relaxation method and high order WENO schemes, J. comput. phys., 148, 59, (1999) · Zbl 0931.76061
[21] Noelle, S., The mot-ICE: a new high-resolution wave-propagation algorithm for multi-dimensional systems of conservation laws based on Fey’s method of transport, J. comput. phys., 164, 283, (2000) · Zbl 0967.65100
[22] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, edited by B. Cockburn, C. Johnson, C.-W. Shu, and E. Tadmor, Lecture Notes in Mathematics, Springer-Verlag, Berlin/New York, 1998, pp. 325-432. · Zbl 0927.65111
[23] Shu, C.-W., (), 439-582
[24] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. comput. phys., 77, 439, (1988) · Zbl 0653.65072
[25] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, II, J. comput. phys., 83, 32, (1989) · Zbl 0674.65061
[26] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115, (1984) · Zbl 0573.76057
[27] Yang, J.; Yang, S.; Chen, Y.; Hsu, C., Implicit weighted ENO schemes for the three-dimensional incompressible navier – stokes equations, J. comput. phys., 146, 464, (1998) · Zbl 0931.76066
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.