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A higher order local linearization method for solving ordinary differential equations. (English) Zbl 1115.65080

This paper is concerned with the numerical solution of initial value problems for first order differential systems (1) \({\mathbf x}'(t)= {\mathbf f}( {\mathbf x}(t) )\). In a previous paper of some of the authors [J. C. Jimenez and F. Carbonell, Appl. Math. Comput. 171, 1282–1295 (2005; Zbl 1094.65074)] they propose to approximate locally the solution of (1) at each \((t_n, {\mathbf x}_n)\) by their linear approximation \({\mathbf x}_n (t), t \in [t_n, t_n+h]\) that satisfies \( {\mathbf x}_n'(t)= {\mathbf f}'( {\mathbf x}_n ) {\mathbf x}_n (t)+ ( {\mathbf f}( {\mathbf x}_n -{\mathbf f}'( {\mathbf x}_n ){\mathbf x}_n ),\) \( {\mathbf x}_n (t_n)= {\mathbf x}_n, \) and it is second order accurate in \(h\).
In the paper under consideration, the authors propose higher order methods by using additional terms in the Taylor series expansion of the local solution at the starting point. Some comments about the practical evaluation of the terms that appear in the Taylor expansion are given. Further, they show that for a method of order \(p\) the phase portrait of a dynamical system around an hyperbolic stationary point is reproduced by the numerical method with order \(p\).
Finally, some numerical experiments with a two dimensional system are presented showing that a third order method of the proposed family is able to reproduce the local manifolds of this system more accurately than the standard second order linearization and also than a explicit third order Runge-Kutta method.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems

Citations:

Zbl 1094.65074
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References:

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