On BDF-Based Multistep Schemes for Some Classes of Linear Differential-Algebraic Equations of Index at Most 2
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On BDF-Based Multistep Schemes for Some Classes of Linear Differential-Algebraic Equations of Index at Most 2 Mikhail Valeryanovich Bulatov1,2 · Vu Hoang Linh3 · Liubov Stepanovna Solovarova1
Received: 6 May 2015 / Accepted: 29 November 2015 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016
Abstract A family of efficient multistep difference schemes for solving some classes of linear non-autonomous differential-algebraic equations of index at most 2 is proposed. It is shown that if the popular backward differentiation formulas (BDFs) are applied to a reformulated form of the original problem, then the methods preserve the stability property and the convergence order that the corresponding BDF methods possess in the ODE case. Further issues such as numerical differentiation that may be involved in the implementation and computational errors are also discussed. Finally, several numerical experiments are given which confirm the theoretical results. Keywords Linear differential-algebraic equations · Index · Strangeness-free form · Multistep difference schemes · BDF methods · Convergence · Stability function Mathematics Subject Classification (2010) 65L07 · 65L80
Mikhail Valeryanovich Bulatov
[email protected] Vu Hoang Linh [email protected] Liubov Stepanovna Solovarova [email protected] 1
Matrosov Institute of System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Lermontov St., 134, Irkutsk, Russia
2
Irkutsk National Research Technical University, Lermontov St., 83, Irkutsk, Russia
3
Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam
M. V. Bulatov et al.
1 Introduction Coupled systems of differential and algebraic equations often occur as mathematical models in various areas of science and engineering, for instance, in multibody mechanics, electrical networks, chemical processes, and optimal control. Real-life examples of such problems can be found in [1, 3, 4, 6, 14, 17]. If the equations are linear, setting them in one system, we obtain a system of the form A(t)x (t) + B(t)x(t) = f (t),
t ∈ [t0 , tf ],
(1)
where A and B are n by n matrix functions, f is a n−dimensional vector function, and x is the unknown vector function. We assume that the data functions A, B, and f are sufficiently smooth and det A(t) ≡ 0. Such systems are called linear differential-algebraic equations (DAEs). Without loss of generality, we assume in addition that [t0 , tf ] = [0, 1]. For DAE (1), we assign initial condition x(0) = x0 ,
(2)
that is supposed to be consistent with the right hand side of (1). A continuously differentiable vector-function that satisfies (1) pointwise for t ∈ [0, 1] and also the initial condition (2) is called a solution of the initial value problem (1)–(2). It is known that the qualitative theory and the numerical analysis of DAEs are more difficult than ODEs, see, e.g., [6, 12, 14, 17, 19]. Difficulties that arise with DA
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