Shock Waves

, Volume 4, Issue 1, pp 25–34

Restoration of the contact surface in the HLL-Riemann solver

  • E. F. Toro
  • M. Spruce
  • W. Speares


The missing contact surface in the approximate Riemann solver of Harten, Lax, and van Leer is restored. This is achieved following the same principles as in the original solver. We also present new ways of obtaining wave-speed estimates. The resulting solver is as accurate and robust as the exact Riemann solver, but it is simpler and computationally more efficient than the latter, particulaly for non-ideal gases. The improved Riemann solver is implemented in the second-order WAF method and tested for one-dimensional problems with exact solutions and for a two-dimensional problem with experimental results.

Key words

Finite difference scheme Numerical simulation Riemann solver 


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  1. Davis SF (1988) Simplified second-order Godunov-type methods. SIAM J Sci and Stat Comput 9:445CrossRefGoogle Scholar
  2. Einfeldt B (1988) On Godunov-type methods for the Euler equations with general equation of state. In: Proc Second Internat Conference on Hyperbolic Problems, Aachen, Germany.Google Scholar
  3. Godunov SK (1959) A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat Sb 47:357Google Scholar
  4. Harten A, Lax PD, van Leer B (1983) On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review 25:35–61CrossRefGoogle Scholar
  5. Roe PL (1981) Approximate Riemann solvers, parameter vectors, and difference schemes. J Comput Physics 43:357CrossRefGoogle Scholar
  6. Toro EF (1989a) A fast Riemann solver with constant covolume applied to the random choice method. Int J Numer Methods in Fluids 9:1145Google Scholar
  7. Toro EF (1989b) A weighted average flux method for hyperbolic conservation laws. Proc Royal Soc London A 423:401Google Scholar
  8. Toro EF (1991) A linearized Riemann solver for the Euler equations of gas dynamics. Proc Roy Soc London A 434:683Google Scholar
  9. Toro EF (1992a) Riemann problems and the WAF method for solving the two-dimensional shallow water equations. Phil Trans Royal Soc London A 338:43Google Scholar
  10. Toro EF (1992b) The weighted average flux method applied to the Euler equations. Phil Trans Roy Soc London A 341:499Google Scholar
  11. Woodward P, Colella PJ (1984) The numerical simulation of two-dimensional fluid flow with strong shocks. J Comput Phys 54:115CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • E. F. Toro
    • 1
  • M. Spruce
    • 1
  • W. Speares
    • 1
  1. 1.Department of Aerospace Sciences, College of AeronauticsCranfield Institute of TechnologyCranfieldUK

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