Parallel algorithms for initial - value problems for difference and differential equations *

Let { y”} be a trajectory defined sequentially by a nonlinear vector difference equation y,,+ 1 = F”+ ,( y,) with y0 known. A Steffensen-like iterative method is proposed which starts from a guessed sequence {u:‘)} for the approximation of { yn) and allows certain computations to be performed in parallel. The sequence {u:““‘} is (k) obtained from {u, } by means of a formula of the form u,+, (kt’) = vAk+:” + Acj:ir)(~!,k+l) u:“)). Here the vectors ‘?n+l (kil) and the matrices A(f:ii) each require a function evaluation but can be computed in parallel with respect to n in a suitable interval [ Nk, Mk], the length of which depends on the number of processors available. Therefore, for the algorithm to serve effectively, the Steffensen iteration must converge quickly and the machine used must possess a large number of processors. Finally, note that the theory includes the case that the sequence { y,} is given by a one-step ODE solver.