A class of Rosenbrock-type schemes for second-order nonlinear systems of ordinary differential equations
type: Item
instance of: scholarly article
title: A class of Rosenbrock-type schemes for second-order nonlinear systems of ordinary differential equations
zbMATH Open document ID: 0634.65055
DOI: 10.1016/0898-1221(87)90002-2
publication date: 1987-01-01T00:00:00Z
review text: The authors propose and analyze a class of Rosenbrock-type schemes for a special second-order system of the form \((1)\quad u''=g(u,t),\) \(du/dt=u'\), \(t\in [0,t]\), \(u(0)=u_ 0\), \(u'(0)=v_ 0\) where u:[0,T]\(\to X\) \((=Banach\) space, especially \(X={\mathbb{R}}^ n)\). The developed schemes are such that when \(g(u,t)=-Au\) for \(A\in L({\mathbb{R}}^ n,{\mathbb{R}}^ n)\) positive definite and symmetric, the methods of \textit{G. A. Baker} and \textit{J. H. Bramble} [RAIRO Anal. Numer. 13, 75-100 (1979; Zbl 0405.65057)] are reproduced. In order to analyze the schemes, the authors perform the usual conversion of the second-order problem (1) into first- order form. For this system the authors pose a Rosenbrock-type method using the square of the Jacobian to generate the linear systems which must be solved at each time step. This approach distinguishes the scheme from other Rosenbrock methods. Employing the concept of Butcher series [cf. \textit{E. Hairer} and \textit{G. Wanner}, Computing 11, 287-303 (1973; Zbl 0271.65048)] the authors develop order conditions for the scheme. Using some results of \textit{S. P. Nørsett} and \textit{G. Wanner} [BIT 19, 79-94 (1979; Zbl 0413.65011)] the authors obtain conditions on a parameter of the scheme which gives unconditionally stability. Some numerical examples conclude the note.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 4032944
zbMATH Keywords: second-order system
zbMATH Keywords: Banach space
zbMATH Keywords: Rosenbrock methods
zbMATH Keywords: Butcher series
zbMATH Keywords: unconditionally stability
zbMATH Keywords: numerical examples
author: Som Prakash Goyal
author: Steven M. Serbin
reviewed by: Hans Ade
MaRDI profile type: Publication
full work available at URL: 0898-1221(87)90002-2
OpenAlex ID: W1985640870
cites work: Implicit Runge-Kutta Processes
cites work: On Runge-Kutta processes of high order
cites work: An Algebraic Theory of Integration Methods
cites work: Stability Criteria for Implicit Runge–Kutta Methods
cites work: Semidiscrete and single step fully discrete approximations for second order hyperbolic equations
cites work: Some general implicit processes for the numerical solution of differential equations
cites work: On the choice of γ for singly-implicit RK or Rosenbrock methods
cites work: Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations
cites work: A study of Rosenbrock-type methods of high order
cites work: An Analysis of Rosenbrock Methods for Nonlinear Stiff Initial Value Problems
cites work: Two-step fourth order P-stable methods for second order differential equations
cites work: Unconditionally stable methods for second order differential equations
cites work: Efficient P-stable methods for periodic initial value problems
cites work: Phase properties of high order, almost P-stable formulae
cites work: The real-pole sandwich for rational approximations and oscillation equations
cites work: Multistep-multistage-multiderivative methods for ordinary differential equations
cites work: Order conditions for Rosenbrock type methods
cites work: On the Butcher group and general multi-value methods
cites work: Q4124398
published in: Computers & Mathematics with Applications
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description: scientific article; zbMATH DE number 4032944