Parallel Numerical Picard Iteration Methods

Abstract

In this work, we propose a new class of parallel time integrators for initial-value problems based on the well-known Picard iteration. To this end, we first investigate a class of sequential integrators, known as numerical Picard iteration methods, which falls into the general framework of deferred correction methods. We show that the numerical Picard iteration methods admit a $\min (J,M+1)$-order rate of convergence, where J denotes the number of Picard iterations and $M+1$ is the number of collocation points. We then propose a class of parallel solvers so that J Picard iterations can be proceeded simultaneously and nearly constantly. We show that the parallel solvers yield the same convergence rate as that of the numerical Picard iteration methods. The main features of the proposed parallelized approach are as follows. (1) Instead of computing the solution point by point [as in revisionist integral deferred correction (RIDC) methods], the proposed methods proceed segment by segment. (2) The proposed approach leads to a higher speedup; the speedup is shown to be $J(M+1)$ (while the speedup of the Jth order RIDC is, at most, J). (3) The approach is applicable for non-uniform points, such as Chebyshev points. The stability region of the proposed methods is analyzed in detail, and we present numerical examples to verify the theoretical findings.