Numerical integration of ordinary differential equations on manifolds
type: Item
instance of: scholarly article
title: Numerical integration of ordinary differential equations on manifolds
zbMATH Open document ID: 0798.34012
DOI: 10.1007/BF02429858
published in: Journal of Nonlinear Science
publication date: 1994-05-30T00:00:00Z
review text: This article considers systems of differential equations \(\dot z = \sum^ n_{i=1} f_ i(z) E_ i(z)\), where \(f_ i(z)\) are scalar functions and \(E_ i(z)\) are vector fields on \(\mathbb{R}^ N\). Typically the solutions are constrained to some submanifold of \(\mathbb{R}^ N\). When the coefficients \(f_ i(z)\) are replaced by constants (``frozen coefficients'') it is assumed that the problem can be solved analytically. Runge-Kutta methods and linear multistep methods are generalized in such a way that only evaluations of the \(f_ i(z)\) are needed and that the numerical approximation evolves in the solution manifold. The main part of this paper studies the order of the methods. It is shown that, besides the usual order conditions, additional relations have to be satisfied by the parameters of the method. New algorithms of order 3 are presented.
Mathematics Subject Classification ID: 65J99
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 37-XX
zbMATH DE Number: 563604
zbMATH Keywords: frozen coefficients
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: linear multistep methods
zbMATH Keywords: numerical approximation
zbMATH Keywords: solution manifold
author: Peter E. Crouch
author: Robert L. Grossman
reviewed by: Ernst Hairer
MaRDI profile type: Publication
cites work: Almost Poisson integration of rigid body systems
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cites work: Spacecraft attitude control and stabilization: Applications of geometric control theory to rigid body models
cites work: Q4234226
cites work: Q4237414
cites work: Canonical transformations depending on a small parameter
cites work: Lie series and invariant functions for analytic symplectic maps
cites work: Q3780530
cites work: Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators
cites work: Symbolic computation of derivations using labelled trees
cites work: The numerical solution of differential-algebraic systems by Runge-Kutta methods
cites work: Q3779682
cites work: Q5553606
cites work: Nonlinear controllability and observability
cites work: Order Results for Implicit Runge–Kutta Methods Applied to Differential/Algebraic Systems
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description: scientific article; zbMATH DE number 563604