Numerical Analysis and Applications

, Volume 4, Issue 4, pp 363–375 | Cite as

Preservation of stability type of difference schemes when solving stiff differential algebraic equations



Implicit methods applied to the numerical solution of systems of ordinary differential equations (ODEs) with an identically singular matrix multiplying the derivative of the sought-for vector-function are considered. The effects produced by losing L-stability of a classical implicit Euler scheme when solving such stiff systems are discussed.


differential algebraic equations index solution space implicit Euler scheme 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brenan, K.E., Campbell, S.L., and Petzold, L.R., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Philadelphia: SIAM, 1996.MATHGoogle Scholar
  2. 2.
    Boyarintsev, Yu.E. and Chistyakov, V.F., Algebro-differentsialnye sistemy. Metody resheniya i issledovaniya (Algebro-Differential Systems: Methods of Solution and Investigation), Novosibirsk: Nauka, 1998.Google Scholar
  3. 3.
    Chistyakov, V.F., Use of Difference Methods to Solve Linear Systems not Written for Derivatives, in Metody optimizatsii i ikh prilozheniya (Optimization Methods with Applications), Irkutsk: SEI SO AN SSSR, 1982, pp. 56–60.Google Scholar
  4. 4.
    Loginov, A.A., An Approach to the Construction of a Software Package of Numerical Integration for Systems of Ordinary Differential Equations, in Pakety prikladnykh programm. Metody i razrabotki (Application Packages: Methods and Developments), Novosibirsk: Nauka, 1981, pp. 112–119.Google Scholar
  5. 5.
    Maerz, R., Differential Algebraic Systems Anew, Appl. Num. Math., 2002, no. 42, pp. 327–338.Google Scholar
  6. 6.
    Hairer, E. and Wanner, G., Reshenie obyknovennykh differentsial’nykh uravnenii. Zhestkie i differentsial’no-algebraicheskie zadachi (Solving Ordinary Differential Equations. Stiff and Differential-Algebraic Problems), Moscow: Mir, 1990.Google Scholar
  7. 7.
    Godunov, S.K. and Ryabenkii, V.S., Raznostnye skhemy (Difference Schemes), Moscow: Nauka, 1977.Google Scholar
  8. 8.
    Bakhvalov, N.S., Zhidkov, N.P., and Kobel’kov, G.M., Chislennye metody (Numerical Methods), Moscow: Nauka, 1987.MATHGoogle Scholar
  9. 9.
    Boyarintsev, Yu.E. and Korsukov, V.M., Application of Difference Methods to Solving Regular Systems of Ordinary Differential Equations, in Voprosy prikladnoi matematiki (Problems in Applied Mathematics), Irkutsk: SEI SO AN SSSR, 1975, pp. 140–152.Google Scholar
  10. 10.
    Fedorenko, R.P., Vvedenie v vychislitel’nuyu fiziku (Introduction to Computational Physics), 2nd ed., Dolgoprudny: Intellekt, 2008.Google Scholar
  11. 11.
    Chistyakov, V.F., On Methods of Numerical Solution and Investigation of Singular Systems of Ordinary Differential Equations, Can. Sci. (Phys.-Math.) Dissertation, Novosibirsk, 1985.Google Scholar
  12. 12.
    Chistyakov, V.F., On Singular Systems of Ordinary Differential Equations and Their Integral Analogs, in Funktsii Lyapunova i ikh primenenie (Lyapunov Functions with Applications), Novosibirsk: Nauka, 1987, pp. 231–239.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Institute of System Dynamics and Control Theory, Siberian BranchRussian Academy of SciencesIrkutskRussia

Personalised recommendations