On matrix-free computation of 2D unstable manifolds

TL;DR

This paper presents a matrix-free, stable algorithm for computing 2D unstable manifolds of periodic solutions in dynamical systems, using boundary value problems, multiple shooting, and Newton-Krylov continuation, demonstrated on fluid flow models.

Contribution

It introduces a scalable, matrix-free orbit continuation method for unstable manifolds using multiple shooting and Newton-Krylov techniques, improving computational efficiency and stability.

Findings

Number of integrations scales with shooting intervals, not system size.

Algorithm effectively computes manifolds for fluid flow models.

Conditioning maintained via stability-based shooting interval selection.

Abstract

Recently, a flexible and stable algorithm was introduced for the computation of 2D unstable manifolds of periodic solutions to systems of ordinary differential equations. The main idea of this approach is to represent orbits in this manifold as the solutions of an appropriate boundary value problem. The boundary value problem is under determined and a one parameter family of solutions can be found by means of arclength continuation. This family of orbits covers a piece of the manifold. The quality of this covering depends on the way the boundary value problem is discretised, as do the tractability and accuracy of the computation. In this paper, we describe an implementation of the orbit continuation algorithm which relies on multiple shooting and Newton-Krylov continuation. We show that the number of time integrations necessary for each continuation step scales only with the number of shooting intervals but not with the number of degrees of freedom of the dynamical system. The number of shooting intervals is chosen based on linear stability analysis to keep the conditioning of the boundary value problem in check. We demonstrate our algorithm with two test systems: a low-order model of shear flow and a well-resolved simulation of turbulent plane Couette flow.