The initial value problem for stiff linear differential-algebraic equations is considered. A block variant of multistep difference schemes is proposed to solve these problems. Sufficient conditions for the methods to converge to the exact solution are formulated, and an estimate of the convergence rate is obtained. Results of numerical calculations for test examples are given.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer-Verlag, Berlin, 1996).
V. F. Chistyakov, Differential-Algebraic Operators with a Finite-Dimensional Kernel (Nauka, Novosibirsk, 1996) [in Russian].
K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (SIAM, Philadelphia, 1996).
Yu. E. Boyarintsev, Regular and Singular Systems of Linear Ordinary Differential Equations (Nauka, Novosibirsk, 1980) [in Russian].
R. Lamour, R. März, and C. Tischendorf, Differential-Algebraic Equations: A Projector Based Analysis (Springer, Berlin, 2013).
P. Kunkel and V. Mehrmann, “Stability properties of differential-algebraic equations and spin-stabilized discretizations,” Electron. Trans. Numer. Anal. 26, 385–420 (2007).
M. V. Bulatov and L. S. Solovarova, “Loss of L-stability of an Euler method for a linear problem,” Izv. Irkutsk. Gos. Univ., Ser. Mat. 12, 3–11 (2015).
V. F. Chistyakov, “Preservation of stability type of difference schemes when solving stiff differential algebraic equations,” Numer. Anal. Appl. 4 (4), 363–375 (2011).
M. V. Bulatov, “Numerical solution of differential-algebraic equations by block methods,” Computational Science—ICCS 2003: International Conference (2003), pp. 516–522.
M. V. Bulatov, Lee Ming-Gong, and L. S. Solovarova, “On first- and second-order difference schemes for differential-algebraic equations of index at most two,” Comput. Math. Math. Phys. 50 (11), 1808–1817 (2010).
M. V. Bulatov, V. H. Linh, and L. S. Solovarova, “On BDF-based multistep schemes for some classes of linear differential-algebraic equations of index at most 2,” Acta Math. Vietnam 41 (4), 715–730 (2016).
N. S. Bakhvalov, Numerical Methods: Analysis, Algebra, Ordinary Differential Equations (Nauka, Moscow, 1975; Mir, Moscow, 1977).
The study of M.V. Bulatov and L.S. Solovarova was supported in part by the Russian Foundation for Basic Research, project no. 18-51-54001. The study of V.H. Linh was supported in part by NAFOSTED, project no. 101.02-2017.314.
Translated by N. Berestova
About this article
Cite this article
Bulatov, M.V., Linh, V.H. & Solovarova, L.S. Block Difference Schemes of High Order for Stiff Linear Differential-Algebraic Equations. Comput. Math. and Math. Phys. 59, 1049–1057 (2019). https://doi.org/10.1134/S0965542519070042
- differential-algebraic equations
- stiff problems
- block difference schemes