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Godunov Methods

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Godunov Methods

Abstract

This paper reviews the class of numerical schemes, known as Godunov Methods, used for the solution of hyperbolic conservation laws. Such numerical schemes can be characterised by the solution (exact or approximate) of a Riemann Problem (classical or generalised) within computational cells in order to obtain the numerical fluxes.

Since the original first order scheme, proposed by Godunov in 1959, there has been much development of the idea; for example, the MUSCL scheme of van Leer in 1979, the PPM scheme of Woodward and Colella in 1984 and the Higher Order Godunov schemes of Bell, Colella and Trangenstein (1989).

As well as considering the original scheme and its later variants, we place these developments in historical context, making links with other work in the area.

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References

  • J. B. Bell, P. Colella, and J. A. Trangenstein. Higher-order Godunov methods for general systems of hyperbolic conservation-laws. J. Computational Phys., 82, pp 362–397,(1989).

    MathSciNet  MATH  Google Scholar 

  • M. Ben-Artzi and J. Falcovitz. A 2nd-order Godunov-type scheme for compressible fluid-dynamics. J. Computational Phys., 55, pp 1–32, (1984).

    MathSciNet  MATH  Google Scholar 

  • M. Ben-Artzi and J. Falcovitz. GRP - An analytic approach to high-resolution upwind schemes for compressible fluid-flow. Lecture Notes Phys., 218, pp 87–91, (1985).

    Google Scholar 

  • J.P. Boris and D.L. Book. Flux-corrected transport. I. SHASTA, a fluid-transport algorithm that works. J. Computational Phys., 11, pp 38–69, (1973).

    MATH  Google Scholar 

  • P. Colella and P. R. Woodward. The piecewise parabolic method (PPM) for gasdynamical simulations. J. Computational Phys., 54, pp 174–201, (1984).

    MATH  Google Scholar 

  • S. F. Davis. Simplified 2nd-order Godunov-type methods. Siam J. On Scientific Statistical Computing, 9, pp 445–473, (1988).

    MathSciNet  MATH  Google Scholar 

  • B. Einfeldt. On Godunov-type methods for gas-dynamics. Siam J. On Numerical Analysis, 25, pp 294–318, (1988).

    MathSciNet  MATH  Google Scholar 

  • B. Engquist and S. Osher. One-sided difference approximations for non-linear conservation-laws. Mathematics Computation, 36, pp 321–351, (1981).

    MathSciNet  MATH  Google Scholar 

  • P. Glaister. Flux difference splitting for the Euler equations with axial symmetry. J.Engineering Mathematics, 22, pp 107–121, (1988).

    MathSciNet  MATH  Google Scholar 

  • S. K. Godunov. A difference scheme for numerical computation of discontinuous solutions of equations of fluid dynamics. Math. Sbornik, 47, pp 271–306, (1959).

    MATH  Google Scholar 

  • J. B. Goodman and R. J. Le Veque. A geometric approach to high-resolution TVD schemes. Siam J. On Numerical Analysis, 25, pp 268–284, (1988).

    MathSciNet  MATH  Google Scholar 

  • S. Hancock. Physics International, San Leandro, California. Unpublished Private Communication to van Leer, (1980).

    Google Scholar 

  • A. Harten. High-resolution schemes for hyperbolic conservation-laws. J. Computational Phys., 49, pp 357–393, (1983).

    MathSciNet  MATH  Google Scholar 

  • A. Harten. ENO schemes with subcell resolution. J. Computational Phys., 83, pp 148–184, (1989).

    MathSciNet  MATH  Google Scholar 

  • A. Harten, B. Engquist, S. Osher, and S. R. Chakravarthy. Uniformly high-order accurate essentially nonoscillatory schemes, III. J. Computational Phys., 71, pp 231–303,(1987).

    MathSciNet  MATH  Google Scholar 

  • A. Harten and J. M. Hyman. Self-adjusting grid methods for one-dimensional hyperbolic conservation-laws. J. Computational Phys., 50, pp 235–269, (1983).

    MathSciNet  MATH  Google Scholar 

  • A. Harten, P. D. Lax, and B. Van Leer. On upstream differencing and Godunov-type schemes for hyperbolic conservation-laws. Siam Review, 25, pp 35–61, (1983).

    MathSciNet  MATH  Google Scholar 

  • A. Harten and S. Osher. Uniformly high-order accurate nonoscillatory schemes .I. Siam J. On Numerical Analysis, 24, pp 279–309, (1987).

    MathSciNet  MATH  Google Scholar 

  • A. Harten, S. Osher, B. Engquist, and S. R. Chakravarthy. Some results on uniformly high-order accurate essentially nonoscillatory schemes. Applied Numerical Mathematics, 2, pp 347–377, (1986).

    MathSciNet  MATH  Google Scholar 

  • P.W. Hemker and S.P. Spekreijse.Multiple grid and Osher’s scheme for the efficient solution of the steady Euler equations. Applied Numerical Mathematics, 2, pp 475–493, (1986).

    MathSciNet  MATH  Google Scholar 

  • G. S. Jiang, D. Levy, C. T. Lin, S. Osher, and E. Tadmor. High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws.Siam J. On Numerical Analysis, 35, pp 2147–2168, (1998).

    MathSciNet  MATH  Google Scholar 

  • P. D. Lax. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, volume 11 of Regional Conference Series in Applied Mathematics. SIAM, (1973).

    MATH  Google Scholar 

  • P. D. Lax and B. Wendroff. Systems of conservation laws. Comm. Pure & Appl. Math., 13, pp 217–237, (1960).

    MathSciNet  MATH  Google Scholar 

  • R. J. Le Veque. Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm.J. Computational Phys., 146, pp 346–365, (1998).

    MathSciNet  Google Scholar 

  • X. D. Liu, S. Osher, and T. Chan.Weighted essentially nonoscillatory schemes.J. Computational Phys., 115, pp 200–212, (1994).

    MathSciNet  MATH  Google Scholar 

  • H. Nessyahu and E. Tadmor. Non-oscillatory central differencing for hyperbolic conservation-laws. J. Computational Phys., 87, pp 408–463, (1990).

    MathSciNet  MATH  Google Scholar 

  • S. Osher and F. Solomon. Upwind difference-schemes for hyperbolic systems of conservation-laws. Mathematics Computation, 38, pp 339–374, (1982).

    MathSciNet  MATH  Google Scholar 

  • P. L. Roe. Approximate Riemann solvers, parameter vectors, and difference-schemes. J. Computational Phys., 43, pp 357–372, (1981).

    MathSciNet  MATH  Google Scholar 

  • P. L. Roe.Sonic flux formulas.Siam J. On Scientific Statistical Computing, 13, pp 611–630, (1992).

    MathSciNet  MATH  Google Scholar 

  • P.L. Roe. Numerical algorithms for the linear wave equation. Technical Report 81047, Royal Aircraft Establishment, (1981).

    Google Scholar 

  • P.L. Roe and J. Pike. Efficient constrauction and utilisation of approximate Riemann solutions. In Computing Methods in Applied Science and Engineering. North Holland, (1984).

    Google Scholar 

  • C. W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Computational Phys., 77, pp 439–471, (1988).

    MathSciNet  MATH  Google Scholar 

  • C. W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes .2. J. Computational Phys., 83, pp 32–78, (1989).

    MathSciNet  MATH  Google Scholar 

  • P. K. Sweby. High-resolution schemes using flux limiters for hyperbolic conservation-laws. Siam J. On Numerical Analysis, 21, pp 995–1011, (1984).

    MathSciNet  MATH  Google Scholar 

  • E. F. Toro. A weighted average flux method for hyperbolic conservation laws. Proceedings Royal Soc. London Series A-Mathematical Phys. Engineering Sciences, 423, pp 401–418, (1989).

    MATH  Google Scholar 

  • E. F. Toro. A linearized Riemann solver for the time-dependent Euler equations of gas-dynamics. Proceedings Royal Soc. London Series A-Mathematical Phys. Engineering Sciences, 434, pp 683–693, (1991).

    MathSciNet  MATH  Google Scholar 

  • E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Springer, (1997).

    MATH  Google Scholar 

  • E.F Toro, M. Spruce, and W. Speares. Restoration of the contact surface in the HLL-Riemann solver. Shock Waves, 4, pp 25–34, (1994).

    MATH  Google Scholar 

  • B. Van Leer. Towards the ultimate conservative difference scheme I. The quest of mono-tonicity. Springer Lecture Notes in Physics, 18, pp 163–168, (1973).

    Google Scholar 

  • B. Van Leer. Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme.J. Computational Phys., 14, pp 361–370, (1974).

    MATH  Google Scholar 

  • B. Van Leer. Towards the ultimate conservative difference scheme III. Upstream-centered finite-diference schemes for ideal compressible flow. J. Computational Phys., 23, pp 263–275, (1977).

    MATH  Google Scholar 

  • B. Van Leer. Towards the ultimate conservative difference scheme IV. A new approach to numerical convection. J. Computational Phys., 23, pp 276–299, (1977).

    MATH  Google Scholar 

  • B. Van Leer.Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method. J. Computational Phys., 32, pp 101–136, (1979).

    MATH  Google Scholar 

  • B. Van Leer.On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe. Siam J. On Scientific Statistical Computing, 5, pp 1–20, (1984).

    MathSciNet  MATH  Google Scholar 

  • B. Van Leer. Godunov’s method for gas-dynamics: Current applications and future developments. J. Computational Phys., 132, p 1, (1997).

    Google Scholar 

  • B. Van Leer. An introduction to the article “Reminiscences about difference schemes” by S. K. Godunov. J. Computational Phys., 153, pp 1–5, (1999).

    MathSciNet  MATH  Google Scholar 

  • P. Woodward and P. Colella.The numerical-simulation of two-dimensional fluid-flow with strong shocks. J. Computational Phys., 54, pp 115–173, (1984).

    MathSciNet  MATH  Google Scholar 

  • S.T. Zalesak. Fully dimensional flux corrected transport algorithms for fluids. J. Computational Phys., 31, pp 335–362, (1979).

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Reading, Whiteknights, P.O. Box 220, Reading, RG6 6AX, England

    Peter K. Sweby

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  1. Peter K. Sweby
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Editors and Affiliations

  1. Manchester Metropolitan University, Manchester, UK

    E. F. Toro

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© 2001 Springer Science+Business Media New York

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Sweby, P.K. (2001). Godunov Methods. In: Toro, E.F. (eds) Godunov Methods. Springer, New York, NY. https://doi.org/10.1007/978-1-4615-0663-8_85

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  • DOI: https://doi.org/10.1007/978-1-4615-0663-8_85

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-5183-2

  • Online ISBN: 978-1-4615-0663-8

  • eBook Packages: Springer Book Archive

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