1-D Flow Computation Using the Unified Coordinates

  • Wai-How Hui
  • Kun Xu


The gas dynamics equations in Eulerian coordinates (t, x) are written in conservation PDE form as
$$ \frac{\partial } {{\partial t}}\left( \begin{gathered} \rho \hfill \\ \rho u \hfill \\ \rho e \hfill \\ \end{gathered} \right) + \frac{\partial } {{\partial x}}\left( \begin{gathered} \rho u \hfill \\ \rho u^2 + p \hfill \\ u\left( {\rho e + p} \right) \hfill \\ \end{gathered} \right) = 0, $$
$$ e = \frac{1} {2}u^2 + \frac{1} {{\gamma - 1\rho }}\frac{p} {\rho }. $$


Riemann Problem Cell Interface Riemann Solution Godunov Scheme Godunov Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    W.H. HUI AND S. KUDRIAKOV. The role of coordinates in the computation of discontinuities in one-dimensional flow. Computational Fluid Dynamics Journal, 8: 495–510, 2000.Google Scholar
  2. [2]
    D.H. WAGNER. Equivalence of Euler and Lagrangian equations of gas dynamics for weak solutions. J. Differ. Equations, 68: 118–136, 1987.zbMATHCrossRefGoogle Scholar
  3. [3]
    J.P. BORIS AND D.L. BOOK. Flux-corrected transport. I.SHASTA, a fluid transport algorithm that works. J. Comput. Phys., 11: 38–69, 1973.zbMATHCrossRefGoogle Scholar
  4. [4]
    B. VAN LEER. Towards the ultimate conservative difference scheme IV, a new approach to numerical convection. J. Comput. Phys., 23: 276–299, 1977.zbMATHCrossRefGoogle Scholar
  5. [5]
    A. HARTEN, B. ENGQUIST, S. OSHER AND S. R. CHAKRAVARTHY. Uniformly high order accuracy essentially non-oscillatory schemes III. J. Comput. Phys., 71: 231–303, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    G.S. JIANG AND C.W. SHU. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126: 202–228, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    W.H. HUI AND C.Y. LOH. A new Lagrangian method for steady supersonic flow computation, Part III: strong shocks. J. Comput. Phys., 103: 465–471, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    C.Y. LEPAGE AND W.H. HUI. A shock-adaptive Godunov scheme based on the generalized Lagrangian formulation. J. Comput. Phys., 122: 291–299, 1995.zbMATHCrossRefGoogle Scholar
  9. [9]
    A. HARTEN AND J.M. HYMAN. Self-adjusting grid method for one-dimensional hyperbolic conservation laws. J. Comput. Phys., 50: 235–269, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    P.D. LAX AND B. WENDROFF. Syetems of conservation laws. Comm. Pure and Applied Mathematics, 13: 217–237, 1960.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    H.W. HUI AND C.Y. LOH. A new Lagrangian method for steady supersonic flow computation, Part 2: slipline resolution. J. Comput. Phys., 103: 450–464, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    W.H. HUI AND S. KUDRIAKOV. On wall overheating and other computational difficulties of shock-capturing methods. Computational Fluid Dynamics J., 10: 192–209, 2001.Google Scholar
  13. [13]
    K. XU. Dissipative mechanism in Godunov method. Computational Fluid Dynamics for the 21st Century, M. Hafez, K. Morinishi, and J. Periaux (Eds), pp. 309–321, 2001.Google Scholar
  14. [14]
    H.Z. TANG. On the sonic point glitch. J. Comput. Phys., 202: 507–532, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    B. EINFELDT, C.D. MUNZ, P.L. ROE AND B. SJOGREEN. On Godunov-type methods near low densities. J. Comput. Phys., 92: 273–295, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    R.E. PEIERLS. Theory on von Neumann’s method of treating shocks. Technical Report LA-332, Los Alamos Scientific Laboratory, 1945.Google Scholar
  17. [17]
    R. LANDSHOFF. A numerical method for treating fluid flow in the presence of shocks. Technical Report LA-1930, Los Alamos Scientific Laboratory, 1955.Google Scholar
  18. [18]
    I.G. CAMERSON. An analysis of the errors caused by using artificial viscosity terms to represent steady state shock waves. J. Comput. Phys., 1: 1–20, 1966.CrossRefGoogle Scholar
  19. [19]
    L.G. MARGOLIN, H.M. RUPPEL AND R.B. DEMUTH. Gradient Scaling for a Nonuniform Meshes. in Numerical Methods for Laminar and Turbulent Flow, Pineridge Press, Swansea, UK, 1985.Google Scholar
  20. [20]
    W.F. NOH. Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux. J. Comput. Phys., 72: 78–120, 1987.zbMATHCrossRefGoogle Scholar
  21. [21]
    P. GLAISTER. An approximate linearized Riemann solver for the euler equations for real gases. J. Comput. Phys., 74: 382–408, 1988.zbMATHCrossRefGoogle Scholar
  22. [22]
    R. SANDERS AND A. WEISER. Higher resolution steggered mesh approach for nonlinear hyperbolic systems of conservation laws. J. Comput. Phys., 101: 314–329, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    R. MENIKOFF. Errors when shock waves interact due to numerical shock width. SIAM J. Sci. Comput., 15: 1227–1242, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    R. DONAT AND A. MARQUINA. Capturing shock reflections: An improved flux formula. J. Comput. Phys., 125: 42–58, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    M. GEHMEYR, B. CHENG AND D. MIHALAS. Noh’s constant-velocity shock problem revisited. Shock Waves, 7: 255–274, 1997.zbMATHCrossRefGoogle Scholar
  26. [26]
    E.J. CARAMANA, M.J. SHASHKOV AND P.P. WHALEN. Formulations of artificial voscosity for multi-dimensional shock wave computations. J. Comput. Phys., 144: 70–97, 1998.MathSciNetCrossRefGoogle Scholar
  27. [27]
    R.P. FEDKIW, A. MARQUINA AND R. MERRIMAN. An isobaric fix for the overheating problem in multimaterial compressible flows. J. Comput. Phys., 148: 545–578, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    W.J. RIDER. Revisiting wall heating. J. Comput. Phys., 162: 395–410, 2000.zbMATHCrossRefGoogle Scholar
  29. [29]
    H.Z. TANG AND T.G. LIU. A note on the conservative schemes for the Euler equations. J. Comput. Phys., 218: 451–459, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    S. KUDRIAKOV AND W.H. HUI. On a new defect of shock-capturing methods. J. Comput. Phy., 227: 2105–2117, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    J. VON NEUMANN AND R.D. RICHTMYER. A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys., 21: 232, 1950.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    S.K. GODUNOV. A Difference Scheme for Numerical Computation of Discontinuous Solutions of Hydrodynamic Equations. Math. Sbornik, 47: 271, 1959.MathSciNetGoogle Scholar
  33. [33]
    C.Y. LOH AND M.S. LIOU. Lagrangian solution of supersonic real gas flows. J. Comput. Phys., 104: 150–161, 1993.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Science Press Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Wai-How Hui
    • 1
  • Kun Xu
    • 1
  1. 1.Mathematics DepartmentHong Kong University of Science and TechnologyChina

Personalised recommendations