Computational Mathematics and Mathematical Physics

, Volume 58, Issue 11, pp 1775–1791 | Cite as

Stability of a Spline Collocation Difference Scheme for a Quasi-Linear Differential Algebraic System of First-Order Partial Differential Equations

  • S. V. SvininaEmail author


A quasi-linear differential algebraic system of partial differential equations with a special structure of the pencil of Jacobian matrices of small index is considered. A nonlinear spline collocation difference scheme of high approximation order is constructed for the system by approximating the required solution by a spline of arbitrary in each independent variable. It is proved by the simple iteration method that the nonlinear difference scheme has a solution that is uniformly bounded in the grid space. Numerical results are demonstrated using a test example.


differential algebraic systems partial differential equations spline collocation method difference scheme matrix pencil 



This work was supported by the Siberian Branch of the Russian Academy of Sciences within the framework of project no. 0348-216-0009 “Qualitative Theory and Numerical Analysis of Differential Algebraic Equations.”


  1. 1.
    B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations (Nauka, Moscow, 1978) [in Russian].Google Scholar
  2. 2.
    V. M. Rushchinskii, “Linear and nonlinear three-dimensional models of boiler generators,” Issues of Identification and Modeling (Moscow, 1968), pp. 8–15 [in Russian].Google Scholar
  3. 3.
    S. V. Gaidomak, “On the stability of an implicit spline collocation difference scheme for linear partial differential algebraic equations,” Comput. Math. Math. Phys. 53 (9), 1272–1291 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    S. V. Gaidomak, “The canonical structure of a pencil of degenerate matrix functions,” Russ. Math. 56 (2), 19–28 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    V. F. Chistyakov, Algebraic Differential Operators with a Finite-Dimensional Kernel (Nauka, Novosibirsk, 1996) [in Russian].zbMATHGoogle Scholar
  6. 6.
    M. V. Berezin and N. P. Zhidkov, Computing Methods (Pergamon, Oxford, 1965; Nauka, Moscow, 1966), Vol. 1.Google Scholar
  7. 7.
    A. A. Samarskii and A. V. Gulin, Stability of Difference Schemes (Librokom, Moscow, 2009) [in Russian].zbMATHGoogle Scholar
  8. 8.
    O. A. Oleinik and T. D. Venttsel’, “The first boundary problem and the Cauchy problem for quasi-linear equations of parabolic type,” Mat. Sb. 41 (1), 105–128 (1957).MathSciNetzbMATHGoogle Scholar
  9. 9.
    P. Lancaster, Theory of Matrices (Academic, New York, 1969).zbMATHGoogle Scholar
  10. 10.
    L. V. Kantorovich and G. P. Akilov, Functional Analysis (Pergamon, Oxford, 1982; Nauka, Moscow, 1977).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Matrosov Institute for System Dynamics and Control Theory, Siberian Branch, Russian Academy of SciencesIrkutskRussia

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