Second derivative backward differentiation formulae for ODEs based on barycentric rational interpolants
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Second derivative backward differentiation formulae for ODEs based on barycentric rational interpolants Ali Abdi1 · Gholamreza Hojjati1 Received: 14 September 2019 / Accepted: 22 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract For their several attractive features from the viewpoint of the numerical computations, linear barycentric rational interpolants have been recently used to construct various numerical methods for solving different classes of equations. In this paper, we introduce a family of linear multistep second derivative methods together with a starting procedure based on barycentric rational interpolants. The order of convergence and linear stability properties of the proposed methods have been investigated. To validate the theoretical results and efficiency of the methods, some numerical experiments have been provided. Keywords Ordinary differential equations · Stiff problems · Barycentric rational interpolation · Barycentric rational finite differences · Second derivative methods · Linear stability Mathematics Subject Classification 2010 65L05
1 Introduction The most famous linear multistep methods (LMMs) for the numerical solution of the stiff autonomous ordinary differential equations (ODEs): y (t) = f (y(t)), y(t0 ) = y0 ,
t ∈ [t0 , T ],
Gholamreza Hojjati
[email protected] Ali Abdi a [email protected] 1
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
(1)
Numerical Algorithms
where f : RD → RD and D is the dimension of the system, have been constructed based on customary polynomial interpolants. Among the first derivative LMMs, the class of backward differentiation formulae (BDF) [24] and its extensions EBDF (extended BDF) [16], MEBDF (modified EBDF) [18], MF–MEBDF (matrix free MEBDF) [28], A–BDF method [23], and A–EBDF method [26] have been known as more efficient methods for solving stiff problems of the form (1). The nature of multivalue and/or multistage of these methods falls them into the large family of so-called general linear methods (GLMs) introduced by Butcher [13] (see also [14, 29]). In the construction of the methods of high orders with extensive stability regions, the methods incorporating the second derivative of the solution, g(y) := y = (∂f/∂y)f (y), have been more attracted. For such methods in the class of LMMs, one can refer to second derivative extended BDF [17], second derivative multistep methods [21, 27], and second derivative BDF (SDBDF) [25] which all belong to second derivative GLMs (SGLMs) [4, 5, 15]. It should be noted that in the implementation of the implicit methods, ∂f/∂y is usually required to solve non-linear algebraic equations; therefore, computation of g(y) = (∂f/∂y)f (y) in the second derivative schemes does not impose additional computational cost which makes such schemes to be more practical and efficient [1, 6, 17, 21]. The structure of the proposed methods in the present paper is inspired from that of SDBDF. A k-step SDBDF takes the form: k
αj ym+j = hβk f (ym+k )
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