ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2018. Vol. 25

On the Stability of the Spline-Collocation Difference Scheme for a Semilinear Differential-Algebraic Index System (1,0)

S. V. Svinina

In the paper, a semi-linear differential-algebraic system of partial differential equations of index (1, 0) with a rectangular domain of definition and compatible initial-boundary conditions is considered. It is assumed that the matrix pencil con- structed from the coefficients of a differential-algebraic system is smoothly similar to the special canonical form. A uniform grid, in the rectangular domain of definition, for a numerical solving of the system, is constructed. On the grid, a rectangular elementary sub-region is allocated with a fixed number of nodes in each direction. The solution of the system, in each such sub-domain, is sought in the form of the Newton polyno- mial. The values of polynomial on the joint lines of the elementary sub-regions must coincide. A differential-algebraic system is written in the inner nodes of an elementary sub-region. Derivatives entering the system at each node of the elementary sub-region are approximated by the corresponding derivatives of the Newton polynomial. As a result, a nonlinear spline-collocation difference scheme the order of approximation of which coincides with the order of the spline for each independent variable is written out. Using the transformation of the matrix pencil of the system and the properties of the interpolation spline, the spline-collocation difference scheme is transformed to a matrix-difference equation. It is shown, in the paper, that the matrix-difference equation can be written in normal form. This form of writing of the difference scheme makes it possible to apply the method of simple iterations to it. Using the simple iteration method, an iterative process is written and it is proved that the corresponding transition operator is a compression operator and maps the grid space into itself. Incidentally, it is proved that the difference scheme has a unique solution and is stable in the grid space. To justify the last statement, the results of the author’s previous work are used. As a result, in the work, the existence and stability of a unique solution of a spline-collocation difference scheme with an arbitrary order of approximation are justified. The stability of the difference scheme in the present work is understood in the sense of the definition by A.A. Samarskii. The results of a numerical solving of a semi-linear differential-algebraic system of partial differential equations are demonstrated in the test example.

About the Authors
Svetlana V. Svinina, Cand. Sci. (Phys.–Math.), Senior Research Scientist, Matrosov Institute for System Dynamics and Control Theory SB RAS, 134, Lermontov st., Irkutsk, 664033, Russian Federation, tel.: (3952)453034, e-mail: svinina@icc.ru
For citation
Svinina S. V. On the Stability of the Spline-Collocation Difference Scheme for a Semilinear Differential-Algebraic Index System (1,0). The Bulletin of Irkutsk State  University. Series Mathematics, 2018, vol. 25, pp. 93-108. (in Russian) https://doi.org/10.26516/1997-7670.2018.25.93
differential-algebraic system, index, semilinear system, difference scheme, spline

1. Berezin M.V., Zhidkov N.P. Metody vychislenii [Computing Methods]. Vol. 1. Moscow, Nauka Publ., 1966, 632 p. (in Russian)

2. Gaidomak S.V. On the stability of an implicit spline collocation difference scheme for linear partial differential algebraic equations. Comput. Math. Math. Phys., 2013, vol. 53, no. 9, pp. 1272–1291. (in Russian) https://doi.org/10.7868/S004446691309007X

3. Gaidomak S.V. The canonical structure of a pencil of degenerate matrix functions. Russian Math, 2012, vol. 56, no. 2, pp. 19–28. (in Russian)

4. Gaidomak S.V. Boundary value problem for a first-order linear parabolic system. Comput. Math. Math. Phys., 2014, vol. 54, no. 4, pp. 620–630. (in Russian) https://doi.org/10.7868/S0044466914040061

5. Gaidomak S.V. Numerical solution of linear differential-algebraic systems of partial differential equations. Comput. Math. Math. Phys., 2015, vol. 55, no. 9, pp. 1501–1514. (in Russian) https://doi.org/10.7868/S0044466915060058

6. Demidenko G.V., Uspenskii S.V. Uravneniya i sistemy ne razreshennye otnositel’no starshej proizvodnoj [Equations and systems unsolved for the highest derivative]. Novosibirsk, Nauchnaya Kniga Publ., 1998, 438 p. (in Russian)

7. Kantorovich L.V., Akilov G.P. Funktsional’nyy analiz [Functional Analysis]. Moscow, Nauka Publ., 1977, 744 p. (in Russian)

8. Lancaster P. Teoriya matric [The theory of matrices]. Moscow, Nauka Publ., 1982, 272 p. (in Russian)

9. Oleinik O.A., Venttsel’ T.D. The first boundary problem and the Cauchy problem for quasi-linear equations of parabolic type. Mat. Sb. (N.S.), 1957, vol. 41(83), no. 1, pp. 105–128. (in Russian)

10. Rozhdestvensky B.L., Yanenko N.N. Sistemy kvazilinejnyh uravnenij i ih prilozheniya k gazovoj dinamike [Systems of quasilinear equations and their application to gas dynamics]. Moscow, Nauka Publ., 1978. 668 p. (in Russian)

11. Rushchinskii V.M. Prostranstvennye linejnye i nelinejnye modeli kotlogeneratorov [Space linear and nonlinear models of copper generators]. Voprosy identifikatsii i modelirovaniya [Problems of Identification and Modeling], 1968, pp. 8-15. (in Russian)

12. Samarskii A.A., Gulin A.V. Ustojchivost’ raznostnyh skhem [Stability of difference schemes]. Moscow, Editorial URSS Publ., 2009, 384 p. (in Russian)

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