Implementation of second derivative general linear methods
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Implementation of second derivative general linear methods Ali Abdi1 · Dajana Conte2 Received: 27 January 2020 / Revised: 6 July 2020 / Accepted: 9 July 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract In this paper, the implementation of second derivative general linear methods (SGLMs) in a variable stepsize environment using Nordsieck technique is discussed and various implementation issues are investigated. All coefficients of a method of order four together with its error estimate are obtained. The method is derived with the aim of good zero-stability properties for a large range of ratios of sequential stepsizes to implement in a variable stepsize mode. The numerical experiments indicate that the constructed error estimate is very reliable in a variable stepsize environment and beautifully confirm the efficiency and robustness of the proposed scheme based on SGLMs. Moreover, the results verify that the proposed scheme outperforms the code ode15s from Matlab ODE suite on systems whose Jacobian has eigenvalues which are close to the imaginary axis. Keywords General linear methods · Second derivative methods · Local error estimation · Nordsieck technique · Implementation issues Mathematics Subject Classification 65L05
1 Introduction Butcher[11] introduced general linear methods (GLMs) for the numerical solution of initial value problems (IVPs) for systems of ordinary differential equations (ODEs)
* Ali Abdi [email protected] Dajana Conte [email protected] 1
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
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Dipartimento di Matematica, Università di Salerno, 84084 Fisciano, SA, Italy
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A. Abdi, D. Conte
y� (x) = f (y(x)), y(x0 ) = y0 ,
x ∈ [x0 , x],
(1)
where f ∶ ℝm → ℝm and m is the dimensionality of the system. GLMs as a large family of the methods include traditional numerical methods such as Runge–Kutta (RK) methods and linear multistep methods (LLMs). Furthermore, these methods opened the possibility of obtaining essentially new methods which were neither RK methods nor LMMs and nor slight variations of these methods (see, for instance,[12, 13, 17–22, 39–41]. The efficient experimental variable-stepsize/variable-order codes dim18[15], dim13s[39], and irks14[9] have been developed for nonstiff and stiff ODEs, where the two former ones are based on diagonally implicit multistage integration methods (DIMSIMs) and the latter one is based on GLMs with inherent RK stability (IRKS) property. Since in the implementation of all implicit methods, the Jacobian matrix 𝜕f ∕𝜕y is usually required to approximate the stage values by means of a variant of Newton method, it lets us consider the possibility of developing a class of formulas that uses the Jacobian matrix which can be of high order of accuracy and at the same time satisfy the requirement of good stability properties. Many methods have been introduced for the numerical solution of stiff systems which incorporate the second derivative of the solution, g(y) ∶= y�� = (𝜕f
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