EBDF-type methods based on the linear barycentric rational interpolants for stiff IVPs
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EBDF-type methods based on the linear barycentric rational interpolants for stiff IVPs Zahra Esmaeelzadeh1 · Ali Abdi1
· Gholamreza Hojjati1
Received: 28 September 2020 / Revised: 7 November 2020 / Accepted: 9 November 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract Linear barycentric rational interpolants, instead of customary polynomial interpolants, have been recently used to design the efficient numerical integrators for ODEs. In this way, the BDF-type methods based on these interpolants have been introduced as a general class of the methods in this type with better accuracy and stability properties. In this paper, we introduce an extension of them equipped to super-future point technique. The order of convergence and stability of the proposed methods are discussed and confirmed by some given numerical experiments. Keywords Linear barycentric rational interpolation · Barycentric rational finite differences · Stiff differential equations · BDF methods · Linear stability Mathematics Subject Classification 65L05
1 Introduction Linear multistep methods (LMMs) for the numerical solution of y (t) = f (y(t)), t ∈ [t0 , T ], y(t0 ) = y0 ,
(1)
where f : R D → R D and D is the dimensionality of the system, are widely used because of their low computational cost. Among the methods in this class, some of them
B
Ali Abdi [email protected] Zahra Esmaeelzadeh [email protected] Gholamreza Hojjati [email protected]
1
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
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Z. Esmaeelzadeh et al.
such as Adams methods [13,25] and backward differentiation formulae (BDF) [17], have good stability properties which make them to be suitable for stiff or mildly stiff systems. Furthermore, multistep methods have much room for manoeuvre to modify them in view point of the stability properties: Gragg and Stetter introduced generalized multistep methods in [24]; a year later Butcher introduced modified multistep methods [12]; also Gear published his work on hybrid methods in [23]; moreover adaptive BDF (A-BDF) was proposed in [22]. The Other possibility to modify multistep methods is incorporating the second derivative of the solution g(y) := y = (∂ f /∂ y) f (y) into the method. Some efficient methods have been designed in this direction such as second derivative multistep methods in [20] and second derivative BDF (SDBDF) in [25]. Moreover, super-future points technique has been used to overcome some proved stability barriers on LMMs: extended BDF (EBDF) [14] and its modification (MEBDF) [16], and adaptive EBDF (A-EBDF) [26]. Also, this technique has been used on LMMs incorporating second derivative of the solution: extended special class of second derivative LMMs (E2BD in two classes) [15] and extended SBDF [27]. The above-mentioned traditional multistep methods are based on the classical polynomial interpolation which are ill-conditioned and lead to Runge’s phenomenon if the interpolation nodes are equispaced [11]. In contrast, the linear ba
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