Advertisement

Modified Extended BDF Time-Integration Methods, Applied to Circuit Equations

  • Sandra Allaart-Bruin
  • Jan ter Maten
  • Sjoerd Verduyn Lunel
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 4)

Abstract

Electric circuits designers are frequently interested in the transient behaviour of the designed circuit. A common method for time integration of the Differential Algebraic circuit Equations (DAE) is the Backward Differentiation Formula (BDF) method. In 1983, J. Cash proposed the Modified Extended BDF (MEBDF) method, which combines better stability properties and higher order of convergence than BDF, but requires more computations per step. We prove reduction of convergence order for MEBDF when applied to DAE’s with higher DAE-index. However, because in practice, in circuit analysis, the DAE-index does not exceed 2, the reduction is quite moderate and it equals the BDF-order in that case. One gains better, or even unconditional, stability. One also obtains consistent solutions.

Keywords

Consistent Solution Inverse Function Theorem Initial Valu Backward Differentiation Formula Algebraic Part 
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.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical solution of initial-value problems in differential-algebraic equations, SIAM — Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1996).Google Scholar
  2. 2.
    Bruin, S.M.A.: Modified Extended BDF applied to circuit equations. Philips Research Laboratories, Nat.Lab. Unclassified Report 2001/826, MSc-Thesis Free University Amsterdam (2001). http://www.extra.research.philips.com/publ/rep/nl-ur/ur2001-826.pdf/publ/rep/nl-ur/ur2001-826.pdf
  3. 3.
    Cash, J.R.: The integration of stiff initial value problems in ODEs using Modified Extended Backward Differentiation Formulae, Comp. & Maths. with Appls, Vol 9, No 5, (1983) pp. 645–657.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cash, J.R.: Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs, Journal of Computational and Applied Mathematics, 125, (2000) pp. 117–130.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Estévez Schwarz, D., Tischendorf, C.: Structural analysis of electric circuits and consequences for MNA, Int. J. of Circuit Theory and Applications 28, (2000) pp. 131–162.CrossRefzbMATHGoogle Scholar
  6. 6.
    Frank, J.E., van der Houwen, P.J.: Diagonalizable extended backward differential formulas, BIT, Vol 40, No 3, (2000) pp. 497–512.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Guenther, M., Rentrop, P., Feldmann, U.: CHORAL — A One Step Method as Numerical Low Pass Filter in Electrical Network Analysis. In: van Rienen, U., Günther M., Hecht D. (Eds.): Scientific Computing in Electrical Engineering, Lecture Notes in Computational Science and Engineering, No 18, (SCEE-2000, Warnemünde), Springer Verlag (2001) pp. 199–215.CrossRefGoogle Scholar
  8. 8.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, Springer Berlin (1987).Google Scholar
  9. 9.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, Springer Berlin (1996).CrossRefzbMATHGoogle Scholar
  10. 10.
    Hosea, M.E., Shampine, L.F.: Analysis and implementation of TR-BDF2, Applied Numerical Mathematics 20, (1996) pp. 21–37.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    de Swart, J.J.B.: Parallel software for implicit differential equations, PhD-Thesis Univ. of Amsterdam (1997).Google Scholar
  12. 12.
    Tischendorf, C.: Solution of index-2 differential algebraic equations and its application in circuit simulation, Logos Verlag Berlin, PhD-Thesis Humboldt Univ. zu Berlin (1996).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Sandra Allaart-Bruin
    • 1
  • Jan ter Maten
    • 1
  • Sjoerd Verduyn Lunel
    • 2
  1. 1.Philips Research LaboratoriesEindhoven University of TechnologyEindhoven
  2. 2.Leiden UniversityLeiden

Personalised recommendations