Quadratically convergent numerical schemes for nonstandard initial value problems

摘要

Differential equations of the form y'=f(t, y, y') where f is not necessarily linear in its arguments represent certain physical phenomena and have been known for quite some time. Clairut's and Chrystal's equations, jerky oscillations of some electronic circuits, shift in singularity problems, isoenergetic cylindrical shock waves, and the one dimensional earth-moon-spaceship problem are a few of the many physical phenomena which are governed by the above type of equations. Earlier we established the existence of a unique solution of the nonstandard initial value problem y'=f (t, y, y'), y(t0) = y0 under certain natural hypotheses on f and developed some linearly convergent numerical schemes for the solution. In this paper we develop some quadratically convergent numerical schemes for the solution. A striking feature of these schemes is that the roundoff error terms are uniformly bounded and hence by taking smaller step lengths we get better approximations.