Abstract

This paper is devoted to the study and approximation of systems ofordinary differential equations based on an analysis of a certainerror functional associated, in a natural way, with the originalproblem. We prove that in seeking to minimize the error by usingstandard descent schemes, the procedure can never get stuck inlocal minima, but will always and steadily decrease the erroruntil getting to the original solution. One main step in theprocedure relies on a very particular linearization of theproblem: in some sense, it is like a globally convergent Newtontype method. Although our objective here is not to perform a rigorousnumerical study of the method, we illustrate its potential forapproximation by considering some stiff systemsof equations. The performance is astonishingly very good due to the fact that we can use very robust methods to approximatelinear stiff problems like implicit collocation schemes. We also include a couple of typicaltest models for the Lorentz system and the Kepler problem, again confirming a very goodperformance. Webelieve that this approach can be used in a systematic way toexamine other situations and other types of equations due to itsflexibility and its simplicity.

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