Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods
type: Item
instance of: scholarly article
title: Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods
zbMATH Open document ID: 1258.65068
author: Bin Wang
author: Xinyuan Wu
author: Jianlin Xia
published in: BIT
publication date: 2012-11-21T00:00:00Z
review text: The paper derives and investigates multidimensional exponential fitting modified Runge-Kutta-Nyström (MEFMRKN) methods for a system of oscillatory second order differential equations of the form \[ \begin{cases} q'(t)+Mq(t)=f(q(t)),\;t\in[0,T],\\ q(0)=q_0,\;q'(0)={q_0}' \end{cases} \tag{1} \] where \(M\) is a \(d\times d\) symmetric and positive semi-definite matrix implicitly containing the frequencies of the problem, \(q:{\mathbb R}\rightarrow{\mathbb R}^d\) is the solution of the system and \(f(q)\) is the gradient of a real-valued function \(U(q)\) whose second derivatives are continuous. \noindent Taking into account the fact that the oscillatory system is a separable Hamiltonian system with Hamiltonian \(H(p,q)=\frac{1}{2}p^Tp+\frac{1}{2}q^TMQ+U(q),\) where \[ \begin{cases} q'(t)=p\\ p'=-\nabla U(q)-Mq\\ q(0)=q_0,\;p(0)=p_0 \end{cases} \] one derives the symplecticity conditions for the MEFMRKN methods. The first section is an introduction in the nature. Through the second section one achieves the multidimensional exponential fitting modified RKN or MEFMRKN schemes for the multidimensional oscillatory system (1), showing the relationship between MEFMRKN and RKN integrators. The third section presents and proves the symplecticity conditions for multidimensional modified RKN methods and then for MEFMRKN methods. Taking into account the fact that the explicit symplectic integrators do not require the solution of large and complicate systems of nonlinear algebraic or transcedental equations when solving multidimensional problems, the fourth section presents two explicit symplectic MEFMRKN methods: a two-stage explicit symplectic MEFMRKN method and a three-stage explicit symplectic MEFMRKN method. The efficiency of the presented methods is demonstrated in the fifth section, by five numerical experiments corresponding to two kinds of the problems. The main conclusions are within the last section.
reviewed by: Romulus Militaru
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 65P10
Mathematics Subject Classification ID: 37M15
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 6109252
zbMATH Keywords: exponential fitiing
zbMATH Keywords: symplecticity conditions
zbMATH Keywords: oscillatory systems
zbMATH Keywords: Runge-Kutta-Nyström methods
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: numerical experiments
MaRDI profile type: Publication
full work available at URL: s10543-012-0379-z
OpenAlex ID: W2116724366
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DOI: 10.1007/S10543-012-0379-Z
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description: scientific article; zbMATH DE number 6109252