The great majority of research work in CFD, especially those in the first several decades, treats it as numerical solution to nonlinear hyperbolic partial differential equations (PDEs). For a good summary, see Hirsch[1]. Most part of this monograph also treats CFD as numerical solution to nonlinear hyperbolic PDEs. But it is concerned mainly about the role of coordinates in CFD and, in particular, will base all CFD study on the newly discovered unified coordinates. To put it in perspective we shall first give an overview of the major developments of CFD as numerical solution to the initial value problem of nonlinear hyperbolic PDEs as follows.


Computational Fluid Dynamics Riemann Problem Contact Discontinuity Lagrangian System Godunov Scheme 
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]
    C. HIRSCH. Numerical Computation of Internal and External Flows, Vol. II: Computational Methods for Inviscid and Viscous Flows. Wiley, 1990.Google Scholar
  2. [2]
    G.F.B. RIEMANN. Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Abh. König. Gesell. Wiss. Göttingen, 8, 43, 1860.Google Scholar
  3. [3]
    R. COURANT, K. O. FRIEDRICHS AND H. LEWY. Über die Partiellen differenzengleichugen der mathematischen Physik. Math. Ann., 100, 32, 1928.MathSciNetzbMATHGoogle Scholar
  4. [4]
    J. VON NEUMANN AND R.D. RICHTMYER. A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys., 21: 232, 1950.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    S.K. GODUNOV. A difference scheme for numerical computation of discontinuous solutions of hydrodynamic equations. Math. Sbornik, 47:271, 1959.MathSciNetGoogle Scholar
  6. [6]
    J. GLIMM. Solution in the large for nonlinear hyperbolic systems of equations. Comm. Pure. Appl. Math., 18:697–715, 1965.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    A.J. CHORIN. Random Choice Solutions of Hyperbolic Systems. J. Comput. Phys., 22:517–533, 1976.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    G. MORETTI. Thoughts and Afterthoughts About Shock Computations. Polytechnic Institute of Brooklyn PIBAL Report, 72, 1972.Google Scholar
  9. [9]
    P.D. LAX AND B. WENDROFF. Syetems of conservation laws. Comm. Pure and Applied Mathematics, 13:217–237, 1960.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    W.H. HUI, P.Y. LI AND Z.W. LI. A unified coordinate system for solving the twodimensional Euler equations. J. Comput. Phys., 153:596–637, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    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
  12. [12]
    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
  13. [13]
    F.H. HARLOW. LAMS-1956, Los Alamos Scientic Laboratory Report, 1955.Google Scholar
  14. [14]
    C. W. HIRT, A. A. AMSDEN AND J. L. COOK. An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys., 14:227–253, 1974.zbMATHCrossRefGoogle Scholar
  15. [15]
    J. U. BRACKBILL AND J. S. SALTZMAN. Adaptive zoning for singular problems in two dimensions. J. Comput. Phys., 46, 342, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    W.H. HUI, G.P. ZHAO, J.J. HU AND Y. ZHENG. Flow-generated-grids — gridless computation using the unified coordinates. in Proceedings of the 9th International Conference on Numerical Grid Generation, 111–121, 2005.Google Scholar
  17. [17]
    D.H. WAGNER. Equivalence of Euler and Lagrangian equations of gas dynamics for weak solutions. J. Differ. Equations, 68:118–136, 1987.zbMATHCrossRefGoogle Scholar
  18. [18]
    W.H. HUI AND S. KUDRIAKOV. A unified coordinate system for solving the threedimensional Euler equations. J. Comput. Phys., 172:235–260, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    H. LAMB. Hydrodynamics. 6th Ed., New York: Dover publication, 1991.Google Scholar
  20. [20]
    G. K. BATCHELOR. An Introduction to Fluid Dynamics. Cambridge University Press, 1973.Google Scholar
  21. [21]
    C.S. YIH. Fluid Mechanics: A Concise Introduction to the Theory. Ann Arbor, Michigan, West River Press, 1988.Google Scholar
  22. [22]
    L. D. LANDAU AND E.M. LIFSHITZ. Fluid Mechanics. Addison-Wesley Pub. Co., 1959.Google Scholar
  23. [23]
    M. SHASHKOV. Conservative Finite-difference Methods on General Grids. CRC Press, 1996.Google Scholar
  24. [24]
    W.H. HUI, Z.N. WU AND B. GAO. Preliminary extension of the unified coordinate approach to computation of viscous flows. J. Sci. Comput., 30:301–344, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    W. H. HUI AND C.Y. LOH. A new Lagrangian method for steady supersonic flow computation, Part II: Slipline resolution. J. Comput. Phys., 103:450–464, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    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
  27. [27]
    C.Y. LOH AND W.H. HUI. A new Lagrangian method for steady supersonic flow computation Part I: Godunov scheme. J. Comput. Phys., 89:207–240, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    W.H. HUI. The unified coordinate system in computational fluid dynamics. Communications in Computational Physics, 2:577–610, 2007.MathSciNetGoogle Scholar
  29. [29]
    Z. N. WU. A note on the unified coordinate system for computing shock waves. J. Comput. Phys., 180:110–119, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    R. M. C. SO, Y. LIU AND Y. G. LAI. Mesh shape preservation for flow-induced vibration problems. Journal of Fluids and Structures, 18:287–304, 2003.CrossRefGoogle Scholar
  31. [31]
    W. H. HUI AND G. P. ZHAO. Capturing contact discontinuities using the unified coordinates, in Proceedings of the 2nd MIT Conference on Computational Fluid and Solid Mechanics (Ed: K J Bathe), Volume 2:2000–2005, 2003.Google Scholar
  32. [32]
    Y. C. TAI AND C. Y. KUO. A new model of granular flows over general topography with erosion and deposition. Acta Mech., 199:71–96, 2008.zbMATHCrossRefGoogle Scholar
  33. [33]
    Y. Y. NIU, Y. H. LIN, W. H. HUI AND C. C. CHANG. Development of a moving artificial compressibility solver on unified coordinates. 65:1029–1052, 2009.MathSciNetGoogle Scholar
  34. [34]
    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
  35. [35]
    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
  36. [36]
    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
  37. [37]
    R.J. LEVEQUE. Nonlinear conservation laws and finite volume methods for astrophysical fluid flow. Computational Methods for Astrophysical Flow, Edited by O. Steiner and A. Gautschy. Springer-Verlag, 1998.Google Scholar
  38. [38]
    E. GODLEWSKI AND P.A. RAVIART. Numerical Approximation of Hyperbolic Systems of Conservation Laws. New York, Springer-Verlag, 1996.zbMATHGoogle Scholar
  39. [39]
    E.F. TORO. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, 1999.Google 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