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 TOL it is reached. Here that parallel algorithm is developed further in order to perform the stepsize control strategy, according to a given step tolerance TOL st. Moreover, an analysis of the optimal ratio between TOL it and TOL st is given. The paper ends with numerical examples and estimations of the attainable speedup.