On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems
type: Item
instance of: scholarly article
title: On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems
zbMATH Open document ID: 0882.65065
author: Sebastian Reich
published in: Numerische Mathematik
publication date: 1997-06-30T00:00:00Z
review text: The author introduces a new family of partitioned Runge-Kutta algorithms (PRK) for approximation to the solution of separable Hamiltonian systems with holonomic constraints. The new algorithms are based on modification of explicit PRK methods for unconstrained systems in the form given by \textit{D. Okunbor} and \textit{R. D. Skeel} [Math. Comput. 59, No. 200, 439-455 (1992; Zbl 0765.65079)]. They are semi-explicit and in applications require less computation than PRK programs in current use. An analysis is made of the order of error of the algorithms based on a more general study of the backward error for numerical integrators defined on a \(C^\infty\) manifold \(\mathcal M\) contained in \(\mathbb{R}^m\). It is proved that, except for orders one and two, the order of error in the constrained system may be less than the order of error of the corresponding unconstrained system. The author describes two extensions of the program which are designed so that the order of error of the constrained program is the same as the order of error of the unconstrained program. The first of these requires a special choice of coefficients in the approximating routine, and the second is based on composition of PRK algorithms which have error of first or second order. Numerical results are presented which result from applying the three programs to the equations for the motion of the double pendulum in a gravitational field. The effective order of error is found to be similar to error constants for unconstrained explicit PRK methods discussed by \textit{R. I. McLachlan} [SIAM J. Sci. Comput. 16, No. 1, 151-168 (1995; Zbl 0821.65048)].
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L70
zbMATH DE Number: 1027757
zbMATH Keywords: semi-explicit symplectic partitioned Runge-Kutta methods
zbMATH Keywords: numerical results
zbMATH Keywords: separable Hamiltonian systems
zbMATH Keywords: algorithms
zbMATH Keywords: motion of the double pendulum
reviewed by: John B. Butler jun.
MaRDI profile type: Publication
full work available at URL: s002110050261
OpenAlex ID: W2083752714
DOI: 10.1007/S002110050261
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description: scientific article; zbMATH DE number 1027757