Construction of the Nordsieck second derivative methods with RK stability for stiff ODEs
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Construction of the Nordsieck second derivative methods with RK stability for stiff ODEs B. Behzad1,2 · B. Ghazanfari1 · A. Abdi2
Received: 14 June 2017 / Revised: 1 January 2018 / Accepted: 9 April 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract In this paper, we study the construction and implementation of special Nordsieck second derivative general linear methods of order p and stage order q = p in which the number of input and output values is r = p rather than r = p + 1. We will construct A- and L-stable methods of orders three and four in this form with Runge–Kutta stability properties. The efficiency of the constructed methods and reliability of the proposed error estimates are shown by implementing of the methods in a variable stepsize environment on some well-known stiff problems. Keywords Stiff differential equations · Second derivative methods · Nordsieck methods · Runge–Kutta stability · A and L stability · Variable stepsize Mathematics Subject Classification 65L05
1 Introduction General linear methods (GLMs) for the numerical solution of an initial value problem (IVP) y (x) = f (y(x)), x ∈ [x0 , x], y(x0 ) = y0 ,
(1)
Communicated by Frederic Valentin.
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A. Abdi [email protected] B. Behzad [email protected] B. Ghazanfari [email protected]
1
Department of Mathematics, Lorestan University, Khorramabad, Iran
2
Present Address: Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
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B. Behzad et al.
where f : Rm → Rm is a sufficiently smooth function and m is the dimensionality of the system, as a framework to including the multivalue and multistage methods was introduced in Butcher (1996). The features and some subclasses of these methods was studied in Bra´s et al. (2013), Butcher (1993, 2006, 2008, 2016), Butcher and Jackiewicz (1993, 1996, 1998, 1997), Butcher and Wright (2003a, b), Conte et al. (2010), Jackiewicz (2009), Wright (2002). Recently, GLMs were also considered for the numerical solution of second-order ordinary differential equations (D’Ambrosio et al. 2012, 2016; D’Ambrosio and Paternoster 2014). In the implementation of implicit methods, usually a modified Newton iteration is used to solve nonlinear algebraic equations which require the Jacobian matrix ∂ f /∂ y. Therefore, it is natural to seek a class of methods which uses the Jacobian matrix. It has been led to introduce the class of second derivative methods which can be of high order of accuracy and satisfy the requirement of good stability properties, simultaneously. For these methods, to save computational effort, (∂ f /∂ y)2 can be used as an approximation to the Jacobian matrix ∂g/∂ y, where g(y) := y . Although GLMs as a very big family of the numerical methods include many traditional methods such as linear multistep methods, Runge–Kutta methods, and twostep Runge–Kutta methods, they do not cover second derivative methods. Therefore, an extension of GLMs which includes second derivative methods such as two-derivative Runge– Kutta (TDRK) me
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