The new class of multistep multiderivative hybrid methods for the numerical solution of chemical stiff systems of first
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The new class of multistep multiderivative hybrid methods for the numerical solution of chemical stiff systems of first order IVPs Mohammad Mehdizadeh Khalsaraei1 · Ali Shokri1 · Maryam Molayi1 Received: 24 February 2020 / Accepted: 11 July 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we present a general form of Nth derivative multistep methods. In these hybrid multistep multiderivative methods, additional stage points (or off-step points) have been used in the first derivative of the solution to improve the absolute stability regions. The accuracy and stability properties of these methods are investigated. We apply the new methods for the numerical integration of some famous stiff chemical problems such as Belousov–Zhabotinskii reaction, the Chapman atmosphere, chemical Akzo-Nobel problem, ROBER problem (suggested by Robertson) and some others which are widely used in numerical studies. Keywords Stiff IVPs Chemical reactions · Off-step points · A(𝛼)-stability · Multiderivative methods Mathematics Subject Classification 65L05; 65L06 · 65L20
1 Introduction Let us consider the system of ordinary differential equations (ODEs)
dy = f (x, y), dx
y(0) = y0 ,
(1)
* Mohammad Mehdizadeh Khalsaraei [email protected] Ali Shokri [email protected] Maryam Molayi [email protected] 1
Faculty of Mathematical Science, University of Maragheh, Maragheh, Iran
13
Vol.:(0123456789)
Journal of Mathematical Chemistry
on the bounded interval I = [x0 , XN ] , where y ∶ I → ℝm and f ∶ I × ℝm → ℝm+1 is N − 1 times continuously differentiable function. The numerical integration of (1) has attracted a lot of attention in the past decades since several physical and chemical problems can be modeled by these equations [11, 14, 20–113]. A few researchers have made an effort to explore and develop numerical approaches to integrate (1). These approaches including Runge–Kutta method, linear multistep method, Taylor, and extrapolation methods [49, 51–117]. Special multistep methods based on numerical integration, such as Adams–Bashforth methods, Adams–Moulton methods and methods built on numerical differentiation for solving (1) have been derived in [3]. Linear multistep method and Runge–Kutta methods had the disadvantage of requiring additional starting values and special procedures for changing steplength. To reduce these difficulties, researchers have sought other classes by lowering the stepnumber of the linear multistep methods without reducing their order. Such formulas which incorporate a function evaluation at an off-step point were proposed by Gragg, and Statter [52] and were christened ‘hybrid’ methods. High-order hybrid formulas with one or more off-step points were derived in [53]. The hybrid BDF methods with some off-step points were first introduced by Ebadi and Gokhale [47]. The hybrid Obrechkoff BDF methods with some off-step points were presented in [39]. We may, therefore affirm, the introduction of hybrid formulae as an important step into the no man’s land described by Kopal [
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