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Numerical Engineering: Experiences in Designing PDE Software with Selfadaptive Variable Step Size/Variable Order Difference Methods

  • W. Schönauer
  • E. Schnepf
  • K. Raith
Part of the Computing Supplementum book series (COMPUTING, volume 5)

Abstract

The basic ideas in designing software for the numerical solution of nonlinear systems of elliptic and parabolic PDE’s with variable step size/variable order difference methods are presented. The error is estimated by the difference of difference formulae, using members of families of difference formulae. Basic solution methods are developed for the solution of the BVP and the IVP for ODE’s. These methods are extended and combined to solution methods for elliptic and parabolic PDE’s. The nonlinear equations are solved by a robust Newton-Raphson method. The method tends to balance all the relevant errors according to a prescribed relative tolerance. For the final solution an estimate of the error of the solution is computed which means e. g. a global error for the IBVP’s.

Keywords

Error Equation Global Error Discretization Error Rectangular Domain Difference Formula 
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.

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References

  1. [1]
    Stetter, H. J.: Modular Analysis of Numerical Software (Lecture Notes in Mathematics, Vol. 773), pp. 133–145. Berlin-Heidelberg-New York: Springer 1980.Google Scholar
  2. [2]
    Schönauer, W., Raith, K., Glotz, G.: The SLDGL-program package for the selfadaptive solution of nonlinear systems of elliptic and parabolic PDE’s. In: Advances in Computer Methods—IV (Vichnevetsky, R., Stepleman, R. S., eds.), pp. 117–125. IMACS 1981.Google Scholar
  3. [3]
    Schönauer, W., Raith, K., Glotz, G.: The principle of the difference of difference quotients as a key to the selfadaptive solution of nonlinear partial differential equations. Computer Methods in Applied Mech. and Eng. 28, 327–359 (1981).MATHCrossRefGoogle Scholar
  4. [4]
    Schönauer, W., Schnepf, E., Raith, K.: The redesign and vectorization of the SLDGL-program package for the selfadaptive solution of non-linear systems of elliptic and parabolic PDE’s. To appear in the Proceedings of a working conference of the IFIP Working Group 2.5 on Numerical Software “PDE Software: Modules, Interfaces and Systems”, Söderköping, August 22–26, 1983.Google Scholar
  5. [5]
    Raith, K., Schnepf, E., Schönauer, W.: A new automatic mesh selection strategy for the solution of boundary value problems with self-adaptive difference methods. In: Proceedings of the Fourth GAMM-Conference on Numerical Methods in Fluid Mechanics (Viviand, H., ed.), Notes on Numerical Fluid Mechanics, Vol. 5, pp. 261–270. Vieweg 1982.Google Scholar
  6. [6]
    Lentini, M., Pereyra, V.: An adaptive finite difference solver for nonlinear two-point boundary problems with mild boundary layers. SIAM J. Numer. Anal. 14, 91–111 (1977).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • W. Schönauer
    • 1
  • E. Schnepf
    • 1
  • K. Raith
    • 1
  1. 1.RechenzentrumUniversität KarlsruheKarlsruhe 1Federal Republic of Germany

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