PARALLEL ODE-SOLVERS WITH STEPSIZE CONTROL

In this paper we propose a parallel implementation of one-step methods with stepsize control for the numerical solution of IVPs for ODEs of the form y'(t)=f(t, y(t)), y(t0)=y0. The proposed implementation is based on the fact that any one-step ODE-method on a mesh {t0<t1< ⋯ <tN} can be viewed as a first-order difference equation of the form yn+1=Fn+1(y>n), y0 known. In a previous paper (1989) we introduced a paral iterative algorithm for the approximation of the trajectory (y0, y1,..., yN), in which a block of guessed values (u00 := y0, u01,..., u0N is iterated, concurrently with respect to the index n, until an error proportional to a given iteration tolerance TOLit is reached. Here that parallel algorithm is developed further in order to perform the stepsize control strategy, according to a given step tolerance TOLst. Moreover, an analysis of the optimal ratio between TOLit and TOLst is given. The paper ends with numerical examples and estimations of the attainable speedup. © 1990.