Finite-volume WENO schemes for three-dimensional conservation laws. (English) Zbl 1059.65078

Summary: We firstly carry out an extension of the finite-volume weighted essentially non-oscillatory (WENO) schemes to three space dimensions and higher orders of accuracy. Secondly, we propose to use more accurate fluxes as the building block. These are the HLLC and MUSTA fluxes of E. F. Toro [Multi-stage predictor-corrector fluxes for hyperbolic equations. Preprint (2003)] and of E. F. Toro, M. Spruce, and W. Speares [Restoration of the contact surface in the HLL Riemann solver, Report CoA 9204, June 1992; J. Shock Waves 4, No. 1, 25–34 (1994; Zbl 0811.76053)]. The numerical results suggest that the new WENO-HLLC and WENO-MUSTA schemes compare satisfactorily with the state-of-the-art finite-volume scheme of J. Shi, C. Hu, and C.-W. Shu [J. Comput. Phys. 175, No. 1, 108–127 (2002; Zbl 0992.65094)].


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws


Full Text: DOI


[1] Balsara, D.S.; Shu, C.W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. comput. phys, 160, 405-452, (2000) · Zbl 0961.65078
[2] Casper, J.; Atkins, H., A finite-volume high order ENO scheme for two dimensional hyperbolic systems, J. comput. phys, 106, 62-76, (1993) · Zbl 0774.65066
[3] G.Q. Chen, E.F. Toro, Centred schemes for nonlinear hyperbolic equations, Preprint NI03046-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 2003, 31p
[4] Cockburn, B.; Shu, C.-W., Runge – kutta discontinuous Galerkin methods for convection-dominated problems, J. sci. comput, 16, 173-261, (2001) · Zbl 1065.76135
[5] Godunov, S.K., A finite difference method for the computation of discontinuous solutions of the equation of fluid dynamics, Mat. sb, 47, 357-393, (1959)
[6] Grasso, F.; Pirozzoli, S., Shock wave-thermal inhomogeneity interactions: analysis and numerical simulations of sound generation, Phys. fluids, 12, 1, 205-219, (2000) · Zbl 1149.76390
[7] Jiang, G.S.; Shu, C.W., Efficient implementation of weighted ENO schemes, J. comput. phys, 126, 202-212, (1996) · Zbl 0877.65065
[8] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S.R., Uniformly high order accurate essentially non-oscillatory schemes III, J. comput. phys, 71, 231-303, (1987) · Zbl 0652.65067
[9] Hu, C.; Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, J. comput. phys, 150, 97-127, (1999) · Zbl 0926.65090
[10] Kolgan, N.E., Application of the minimum-derivative principle in the construction of finite-difference schemes for numerical analysis of discontinuous solutions in gas dynamics, Uchenye zapiski tsagi [sci. notes central inst. aerodyn.], 3, 6, 68-77, (1972), (in Russian)
[11] Kolgan, N.E., Finite-difference schemes for computation of three dimensional solutions of gas dynamics and calculation of a flow over a body under an angle of attack, Uchenye zapiski tsagi [sci. notes central inst. aerodyn.], 6, 2, 1-6, (1975), (in Russian)
[12] Liu, X.D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. comput. phys, 115, 200-212, (1994) · Zbl 0811.65076
[13] Rusanov, V.V., Calculation of interaction of non-steady shock waves with obstacles, USSR J. comput. math. phys, 1, 267-279, (1961)
[14] Shi, J.; Hu, C.; Shu, C.-W., A technique for treating negative weights in WENO schemes, J. comput. phys, 175, 108-127, (2002) · Zbl 0992.65094
[15] Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. sci. stat. comput, 9, 1073-1084, (1988) · Zbl 0662.65081
[16] E.F. Toro, On Glimm-related schemes for conservation laws, Technical Report MMU-9602, Department of Mathematics and Physics, Manchester Metropolitan University, UK, 1996
[17] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer Berlin · Zbl 0923.76004
[18] E.F. Toro, Multi-stage predictor – corrector fluxes for hyperbolic equations, Preprint NI03037-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK
[19] E.F. Toro, M. Spruce, W. Speares, Restoration of the Contact Surface in the HLL Riemann Solver, Report CoA 9204, June 1992, Department of Aerospace Science, College of Aeronautics, Cranfield Institute of Technology, England · Zbl 0811.76053
[20] Toro, E.F.; Spruce, M.; Speares, W., Restoration of the contact surface in the harten-lax – van leer Riemann solver, J. shock waves, 4, 25-34, (1994) · Zbl 0811.76053
[21] van Leer, B., Towards the ultimate conservative difference scheme V: a second order sequel to Godunov’ method, J. comput. phys, 32, 101-136, (1979) · Zbl 1364.65223
[22] Zhou, T.; Li, Y.; Shu, C.-W., Numerical comparison of WENO finite volume and runge – kutta discontinuous Galerkin methods, J. sci. comput, 16, 145-171, (2001) · Zbl 0991.65083
[23] 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
[24] 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, Phys. rev. E, 68, 1-16, (2003), article number 046709
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