There is a growing interest in the study of periodic phenomena in
large-scale nonlinear dynamical systems. Often the high-dimensional
system has only low-dimensional dynamics, e.g., many reaction-diffusion
systems or Navier-Stokes flows at low Reynolds number. We present an
approach that exploits this property in order to compute branches of
periodic solutions of the large system of ordinary differential
equations (ODEs) obtained after a space discretisation of the PDE. We
call our approach the Newton-Picard method. Our method is based on the
recursive projection method (RPM) of Shroff and Keller but extends this
method in many different ways. Our technique tries to combine the
performance of straightforward time integration with the advantages of
solving a nonlinear boundary value problem using Newton's method and a
direct solver. Time integration works well for very stable limit
cycles. Solving a boundary value problem is expensive, but works also
for unstable limit cycles.
\\
\\
We will present some background material on RPM. Next we will explain
the basic features of the Newton-Picard method for single shooting. The
linearised system is solved by a combination of direct and iterative
techniques. First, we isolate the low-dimensional subspace of unstable
and weakly stable modes (using orthogonal subspace iteration) and
project the linearised system on this subspace and on its
(high-dimensional) orthogonal complement. In the high-dimensional
subspace we use iterative techniques such as Picard iteration or GMRES.
In the low-dimensional (but "hard") subspace, direct methods such as
Gaussian elimination or a least-squares are used. While computing the
projectors, we also obtain good estimates for the dominant,
stability-determining Floquet multipliers. We will present a framework
that allows us to monitor and steer the convergence behaviour of the
method.
\\
\\
RPM and the Newton-Picard technique have been developed for PDEs that
reduce to large systems of ODEs after space discretisation. In fact,
both methods can be applied to any large system of ODEs. We will
indicate how these methods can be applied to the discretisation of the
Navier-Stokes equations for incompressible flow (which reduce to an
index-2 system of differential-algebraic equations after space
discretisation when written in terms of velocity and pressure.)
\\
\\
The Newton-Picard method has already been extended to the computation
of bifurcation points on paths of periodic solutions and to multiple
shooting. Extension to certain collocation and finite difference
techniques is also possible.