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Multidimensional Systems and Signal Processing

, Volume 16, Issue 4, pp 369–396 | Cite as

Automatic Generation of Wave Digital Structures for Numerically Integrating Linear Symmetric Hyperbolic PDEs

  • Michael Vollmer
Article

Abstract

The numerical integration of partial differential equations (PDEs) resulting from passive physical problems can be performed by simulation of the actual system with multidimensional (MD) passive wave digital filters. Due to the principle of action at proximity, physical systems are usually massively parallel and only locally connected. Beyond these properties, the wave digital filters are well-known for their excellent numerical stability behavior and their high robustness. The new result of this paper is a synthesis procedure for automatic generation of the algorithms, based on MD wave digital filters for the numerical integration of PDEs describing linear hyperbolic passive systems. The proposed procedure permits a fully automated software development, based on the given partial differential equation.

Keywords

numerical integration automatic code generation partial differential equations wave digital filters parallel computing 

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References

  1. 1.
    V. Belevitch, Classical Network Theory, Holden-Day Inc., 1968.Google Scholar
  2. 2.
    S.D. Bilbao, “Wave and Scattering Methods for the Numerical Integration of Partial Differential Equations,” Doctoral Dissertation, Stanford University, 2001.Google Scholar
  3. 3.
    Bilbao, S.D. 2004Wave and Scattering Methods for the Numerical SimulationWileyChichester, UKGoogle Scholar
  4. 4.
    N.K. Bose, Applied Multidimensional System Theory, Van Nordstrand Reinhold Company, 1982.Google Scholar
  5. 5.
    Dahlquist, G. 1963“A Special Stability Problem for Linear Multistep Methods”BIT32743CrossRefGoogle Scholar
  6. 6.
    A. Fettweis, “Multidimensional Wave Digital Filters (invited contribution),” European Conference on Circuit Theory and Design, vol. II, pp. 409–416, Genova, Italy, 7–10. September, 1976.Google Scholar
  7. 7.
    A. Fettweis, “Multidimensional Circuit and Systems Theory,” Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS ’84), Montreal, Canada, 07–10 May, 1984, pp. 951–957.Google Scholar
  8. 8.
    A. Fettweis, “Wave Digital Filters: Theory and Practice,” (invited paper) Proceedings of the IEEE (The Institute of Electrical and Electronics Engineers), vol. 74, no. 2, 1986, pp. 270–327, (Correction to “...”, no. 75, vol. 5, p. 729).Google Scholar
  9. 9.
    A. Fettweis, “Discrete Passive Modelling of Physical Systems Described by PDEs,” in (EUSIPCO ’92), Brussels, Belgium, 24–27. August 1992.Google Scholar
  10. 10.
    Fettweis, A. 1993“Discrete Passive Modelling of Physical Systems Described by Partial Differential Equations”Rao, C.R. eds. Multivariate Analysis: Future DirectionsElsevier Science PublishersAmsterdam, The Netherlands115130Google Scholar
  11. 11.
    A. Fettweis, “Numerische Integration nach dem Wellendigitalfilterprinzip,” Course at the Ruhr-Universität Bochum, 1999.Google Scholar
  12. 12.
    A. Fettweis and G. Nitsche, “Numerical Integration of Partial Differential Equations using Principles of Multidimensional Wave Digital Filter” in J. A. Nossek, (Hrsg.). Parallel Processing on VLSI Arrays, Kluwer Academic Publishers, 1991, 7–24.Google Scholar
  13. 13.
    Fettweis, A., Nitsche, G. 1991“Transformation Approach to Numerically Integrating PDEs by Means of WDF Principles”Multidimensional Systems and Signal Processing2127159CrossRefGoogle Scholar
  14. 14.
    Fränken D., “Passive Systeme zur Verarbeitung komplexer zeitdiskreter Signale,” Doctoral Dissertation, Universität-Gesamthochschule Paderborn, 1997.Google Scholar
  15. 15.
    Friedrichs, K.O. 1954“Symmetric Hyperbolic Differential Equations”Comm. Pure Appl. Math.7354392Google Scholar
  16. 16.
    M. Fries, “Numerical Integration of Euler Flow by Means of Multidimensional wave Digital Principles,” Doctoral dissertation, Ruhr-Universität Bochum, 1995.Google Scholar
  17. 17.
    A. Fettweis and G.A. Seraji, “New Results in Numerically Integrating PDEs by the Wave Digital Approach,” Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS ’99), vol. 5, Orlando, FL, USA, 30 May–2 June, 1999, pp. 17–20.Google Scholar
  18. 18.
    G. Hemetsberger, “Numerische Integration Hyperbolischer Partieller Differentialgleichungen unter Verwendung Mehrdimensionaler Wellendigitalfilter,” Doctoral dissertation, Ruhr-Universität Bochum, 1995.Google Scholar
  19. 19.
    E. Hairer, S.P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I, Springer-Verlag, 1993.Google Scholar
  20. 20.
    G. Linnenberg, “Über die diskrete Verarbeitung mehrdimensionaler Signale unter Verwendung mehrdimensionaler Wellendigitalfilter,” Doctoral dissertation, Ruhr-Universität Bochum, 1984.Google Scholar
  21. 21.
    K. Meerkötter and D. Fränken, “Digital Realization of Connection Networks by Voltage-Wave Two-Port Adaptors”, in International Journal of Electronics and Communications (AEÜ), vol. 50, no.6, 1996, pp. 362–367.Google Scholar
  22. 22.
    G. Nitsche, “Numerische Lösung partieller Differentialgleichungen mit Hilfe von Wellendigitalfiltern”, Doctoral Dissertation, Ruhr-Universität Bochum, 1993.Google Scholar
  23. 23.
    Ochs, K. 2001“Passive Integration Methods : Fundamental Theory”Archiv für Elektronik und Übertragungstechnik55153163Google Scholar
  24. 24.
    C. Rao and S.K. Mitra, Generalized Inverse of Matrices and its Applications, Wiley Inc., 1971.Google Scholar
  25. 25.
    B.D.H. Tellegen, Theorie der Electrische Netwerken, Nordhoff, 1954.Google Scholar
  26. 26.
    R. Zurmühl, Matrizen 4.Auflage, Springer-Verlag, 1964.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Chair of communications engineeringRuhr-University BochumBochumGermany

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