Properties of the Unified Coordinates

  • Wai-How Hui
  • Kun Xu

Abstract

We shall call the system of coordinates (λ, ξ, η, ς) defined in (6.1) unified in the sense that it unifies the Eulerian system when Q = 0 with the Lagrangian when Q = q, and also in the sense that the system of governing equations (6.19) unites the geometrical conservation laws with the physical ones to form a closed system of PDE in conservation form.

Keywords

Eulerian System Mesh Equation Monitor Function Mesh Movement Mesh Angle 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    L.G. MARGOLIN. Introduction to an arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys., 135: 198–202, 1997.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    D. J. BENSON. Computational methods in Lagrangian and Eulerian hydrocodes. Comput. Method. Appl. Mech. Engrg., 99: 235–394, 1992.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    W.H. HUI AND S. KUDRIAKOV. A unified coordinate system for solving the threedimensional Euler equations. J. Comput. Phys., 172: 235–260, 2001.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    W.H. HUI AND Y. HE. Hyperbolicity and optimal coordinates of the threedimensional steady Euler equations. SIAM Journal on Applied Mathematics, 57: 893–928, 1997.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    C. ROGERS, W.K. SCHIEF AND W.H. HUI. On complex-lamellar motion of a Prim gas. J. Math Anal. Appl., 266: 55–69, 2002.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    K. MILLER AND R.N. MILLER. Moving finite element. SIAM J. Numer Anal., 18: 1019–1032, 1981.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    E.A. DORFI AND L. O’C. DRURY. Simple adaptive grids for 1-D initial value problems. J. Comput. Phys., 69: 175–195, 1987.MATHCrossRefGoogle Scholar
  8. [8]
    W. HUANG, Y. REN AND R.D. RUSSELL. Moving mesh methods based upon moving mesh partial differential equations. J. Comput. Phys., 11: 279–290, 1994.MathSciNetCrossRefGoogle Scholar
  9. [9]
    W. HUANG AND R.D. RUSSELL. Adaptive mesh movement-the MMPDE approach and its applications. J. Comput. Appl. Math., 128: 383–398, 2001.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    W. HUANG, J. MA AND R.D. RUSSELL. A study of moving mesh PDE methods for numerical simulation of blowup in reaction diffusion equations. J. Comput. Phys., (2008), doi: 10.1016/j.jcp.2008.03.024.Google Scholar
  11. [11]
    S. ADJERID AND J.E. FLAHERTY. A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations. SIAM J. Numer. Anal., 23: 778–795, 1986.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    B.M. HERBST, S.W. SCHOOMBIE AND A.R. MITCHELL. Equidistributing principles in moving mesh finite element methods. J. Comput. Appl. Math., 9: 377–389, 1983.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    S.F. DAVIS AND J.E. FLAHERTY. An adaptive finite element method for initialboundary value problems for partial differential equations. SIAM J. Sci. Stat. Comput., 3: 6–27, 1982.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    R.G. HINDMAN AND J. SPENCER. A new approach to truly adaptive grid generation. AIAA Paper 83-0450, 1983.Google Scholar
  15. [15]
    Y. REN AND R.D. RUSSELL. Moving mesh techniques based upon equidistribution and their stability. SIAM J. Sci. Statis. Comput., 13: 1265–1286, 1992.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    G. NI, S. JIANG AND K. XU. Remapping-free ALE-type kinetic method for flow computations. J. Comput. Phys., 228: 3154–3171, 2009.MathSciNetCrossRefGoogle 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