Numerical Techniques

Part of the Scientific Computation book series (SCIENTCOMP)


The finite difference equations presented here follow the format and notation used by von Neumann [2.1] for the solutions of the differential equations that describe fluid dynamics in one space dimension. The material is divided into a Lagrange grid that moves with the flow. The space between consecutive grid lines is referred to as a zone. For multidimensional problems a zone is defined as the interior space of intersecting grid lines. The intersections are called zone node points. Subscripts define the Lagrange coordinates and superscripts the corresponding times. See Refs. [2.2, 3] for complete descriptions of mesh generators. For a one-dimensional network, X j n represents the X position of Lagrange coordinate j at time t n . Intermediate points are given by X j n+1/2 = 1/2(X j n+1 + X j n ) and X j+1/2 n = 1/2(X j+1 n + X j n ). A dot over a parameter represents a time derivative. Thus, X j n+1/2 represents the velocity of node point j at time t n+1/2 The complete set of equations for one-dimensional calculations in gas dynamics as well as elastic—plastic flow are given in Appendix B.


Shock Wave Finite Difference Scheme Artificial Viscosity Finite Difference Equation Grid Distortion 
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  1. [2.1]
    J. von Neumann, R.D. Richtmyer: A Method for the Numerical Calculation of Hydrodynamic Shocks, J. Appl. Phys. 21, 232–237 (1950)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. [2.2]
    D. Giroux: HEMP User’s Manual, Lawrence Livermore National Laboratory, UCRL-51079 Rev. 1 (1973)Google Scholar
  3. [2.3]
    K.H. Warren: HEMP DS User’s Manual, UCID-18075 Rev. 1 (1983)Google Scholar
  4. [2.4]
    M.L. Wilkins: Use of Artificial Viscosity in Multi-dimensional Fluid Dynamic Calculations, J. Comp. Phys. 36 281–303 (1980)MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  1. 1.Lawrence Livermore National LaboratoryLivermoreUSA

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