Exponentially fitted two-step peer methods for oscillatory problems
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Exponentially fitted two-step peer methods for oscillatory problems Dajana Conte1
· Fakhrodin Mohammadi2 · Leila Moradi1 · Beatrice Paternoster1
Received: 21 October 2019 / Revised: 18 April 2020 / Accepted: 18 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This paper concerns the construction of a general class of exponentially fitted two-step implicit peer methods for the numerical integration of Ordinary Differential Equations (ODEs) with oscillatory solution. Exponentially fitted methods are able to exploit a-priori known information about the qualitative behaviour of the solution to efficiently furnish an accurate solution. Moreover, peer methods are very suitable for a parallel implementation, which may be necessary in the discretization of Partial Differential Equations (PDEs) when the number of spatial points increases. Examples of methods with 2 and 3 stages are provided. Numerical experiments are carried out in order to confirm theoretical expectations. Keywords Peer methods · Exponential fitting · Ordinary differential equations · Oscillatory problems Mathematics Subject Classification 65L04 · 65L05
Communicated by Jose Alberto Cuminato. This work is supported by GNCS-INDAM project and by PRIN2017-MIUR project.
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Dajana Conte [email protected] Fakhrodin Mohammadi [email protected] Leila Moradi [email protected] Beatrice Paternoster [email protected]
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Department of Mathematics, University of Salerno, 84084 Fisciano, Italy
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Department of Mathematics, University of Hormozgan, P. O. Box 3995, Bandar Abbas, Iran 0123456789().: V,-vol
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1 Introduction We are interested in the numerical solution of initial value problems for ODEs exhibiting oscillatory solution. Classical numerical integrators could require a very small stepsize to follow the oscillations, expecially when the frequency increases. To develop efficient and accurate numerical methods, we propose an adapted numerical integration based on exploiting a-priori known information about the behavior of the exact solution, by means of exponential fitting strategy (Ixaru and Vanden Berghe 2004). We combine this feature with the usage of peer methods, which represent a highly structured subclass of General Linear Methods (Jackiewicz 2009) and are identified with several distinct stages, such as Runge–Kutta methods. Peer methods have been introduced in linearly implicit form in Schmitt and Weiner (2004). Explicit peer methods have been derived in Kulikov and Weiner (2010), Schmitt and Weiner (2010), Schmitt et al. (2009) and Weiner et al. (2008), while implicit peer methods are described in Beck et al. (2012), Podhaisky et al. (2005), Schmitt et al. (2013, 2005a, b) and Soleimani and Weiner (2017). The attribute “peer” means that all s stages have the same good accuracy properties and a linearly implicit implementation using only one Newton-step is possible for implicit methods since accurate predictors are easily available (Schmitt and Weiner 2004). Mor
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