A class of Rosenbrock-type schemes for second-order nonlinear systems of ordinary differential equations
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.
scientific article; zbMATH DE number 4032944
scientific article