An Adaptive Newton--Picard Algorithm with Subspace Iteration for Computing Periodic Solutions

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Abstract
This paper is concerned with the efficient computation of periodic orbits in large-scale dynamical systems that arise after spatial discretization of partial differential equations (PDEs). A hybrid Newton--Picard scheme based on the shooting method is derived, which in its simplest form is the recursive projection method (RPM) of Shroff and Keller [SIAM J. Numer. Anal., 30 (1993), pp. 1099--1120] and is used to compute and determine the stability of both stable and unstable periodic orbits. The number of time integrations needed to obtain a solution is shown to be determined only by the system's dynamics. This contrasts with traditional approaches based on Newton's method, for which the number of time integrations grows with the order of the spatial discretization. Two test examples are given to show the performance of the methods and to illustrate various theoretical points.