Jacobian-dependent vs Jacobian-free discretizations for nonlinear differential problems
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Jacobian-dependent vs Jacobian-free discretizations for nonlinear differential problems Dajana Conte1 · Raffaele D’Ambrosio2 · Giovanni Pagano1 · Beatrice Paternoster1 Received: 3 December 2019 / Revised: 27 March 2020 / Accepted: 17 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract The paper provides a comparison between two relevant classes of numerical discretizations for stiff and nonstiff problems. Such a comparison regards linearly implicit Jacobiandependent Runge–Kutta methods and fully implicit Runge–Kutta methods based on Gauss–Legendre nodes, both A-stable. We show that Jacobian-dependent discretizations are more efficient than Jacobian-free fully implicit methods, as visible in the numerical evidence. Keywords Linearly implicit methods · Jacobian-dependent methods · Stiff problems Mathematics Subject Classification 65L04 · 65L05
1 Introduction The wide literature on the numerical solution for nonlinear differential problems
y (t) = f (t, y(t)), t ∈ [t0 , T ],
(1.1)
y(t0 ) = y0 ∈ Rd ,
Communicated by Jose Alberto Cuminato.
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Raffaele D’Ambrosio [email protected] Dajana Conte [email protected] Giovanni Pagano [email protected] Beatrice Paternoster [email protected]
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Department of Mathematics, University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, SA, Italy
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Department of Information Engineering and Computer Science and Mathematics, University of L’Aquila, Via Vetoio, Loc. Coppito, 67100 L’Aquila, Italy 0123456789().: V,-vol
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D. Conte et al.
where the vector field f is smooth enough to guarantee the well-posedness of the problem, is rich with general purpose methods, as well as on adapted methods, specially tuned to the problem under investigation. The numerical integration of (1.1) is important not only for itself but also for the time integration of spatially discretized time-dependent partial differential equations (Cardone et al. 2017a; Conte et al. 2016, 2020, D’Ambrosio et al. 2017; D’Ambrosio and Paternoster 2014a, 2016; Isaacson and Keller 1994; Schiesser 1991; Schiesser and Griffiths 2009; Smith 1985) and, therefore, an adaptation of the numerical schemes to the problems may be particularly favourable for an efficient computation of the solutions. A relevant example of adaptation has been provided by the literature on the socalled exponentially fitted methods (Cardone et al. 2017b; Conte et al. 2010, 2014, 2019; Conte and Paternoster 2016; D’Ambrosio et al. 2018, a; D’Ambrosio and Paternoster 2014b; Martán-Vaquero and Vigo-Aguiar 2008; Ixaru 2019; Ixaru and Vanden Berghe 2004; Ozawa 2001; Paternoster 2012), whose coefficients are dependent on parameters which characterize the problem, such as the frequency oscillation for oscillatory solutions. Clearly, this level of adaptation is feasible if a good approximation of the parameters is known in advance. If this is not the case, a competitive level of adaption is made possible by Jacobian-dependent discretizations, which are characteri
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