Invariant solutions: Newton–Krylov–hookstep¶
Equilibria, travelling waves, periodic orbits and relative periodic orbits are fixed points of a map built from the time stepper. Conatus finds them by a matrix-free Newton method with a trust region, and certifies each one by its residual.
Formulations¶
Write the discrete flow map over a time \(T\) as \(\Phi_T\) (the exact map of the time stepper, with \(dt = T/n\)), and the velocity field of the semi-discrete system as \(v(u) = M^{-1}(F(u) - L u)\).
Equilibrium: \(v(u) = 0\).
Travelling wave \(u(x - ct)\): \(v(u) + c\,\partial_x u = 0\), unknowns \((u, c)\), with the phase condition \(\langle \partial_x u^*, \delta u\rangle = 0\) that removes the free translation.
Periodic orbit: \(\Phi_T(u) = u\), unknowns \((u, T)\), phase condition \(\langle v(u^*), \delta u\rangle = 0\) against time shifts.
Relative or pre-periodic orbit: \(S\,\tau_\ell\,\Phi_T(u) = u\) with a shift \(\tau_\ell\) and optionally a reflection \(S\), unknowns \((u, T, \ell)\), with both phase conditions.
Because \(\Phi_T\) is the discrete map, the reported residual is that of the discrete problem, with no time-discretisation error hidden in it.
Newton–Krylov¶
The Newton correction solves \(J\,\delta z = -R(z)\) by GMRES. The Jacobian is never formed: its products with a vector use the exact tangent of the time stepper (Lyapunov exponents, covariant vectors and ensembles and the adjoint pages describe the same tangent), and the derivative with respect to \(T\) is the velocity at the end point.
Hookstep¶
GMRES gives an Arnoldi basis \(V_{m+1}\) and a Hessenberg matrix \(\bar H\). The Newton step minimises \(\lVert \beta e_1 - \bar H y\rVert\). If it is longer than the trust radius \(\Delta\), the hookstep (Viswanath 2007) replaces it by
from the singular value decomposition \(\bar H = U\,\mathrm{diag}(s)\,W^T\), with \(\mu\) found by bisection. The radius follows the ratio of actual to predicted reduction: halved and retried in the same Krylov space when the step is poor, doubled when it is good at the boundary. Every tolerance, budget and radius is an argument; a search that does not converge raises an error with its history.
Stability of a solution¶
Once found, an equilibrium’s stability follows from the eigenvalues of its Jacobian, and a periodic orbit’s from its Floquet multipliers (eigenvalues of \(D\Phi_T\), by Arnoldi on the tangent map). A solution computed in a symmetry subspace is declared stable only after the symmetry-breaking directions have been checked (Symmetry subspaces, slices and phase-space geometry).
Edge states and minimal seeds¶
Between laminar and turbulent behaviour lies the edge of chaos. Bisection on the amplitude of an initial condition, repeated along the trajectory, tracks the edge state; the converged edge trajectory is then a seed for Newton. The same bisection on the energy of an optimised initial condition brackets the transition threshold of a minimal seed.
References¶
D. Viswanath, Recurrent motions within plane Couette turbulence, J. Fluid Mech. 580 (2007) 339–358.
P. Cvitanović, R. L. Davidchack and E. Siminos, On the state space geometry of the Kuramoto–Sivashinsky flow in a periodic domain, SIAM J. Appl. Dyn. Syst. 9 (2010) 1–33.
G. Kawahara, M. Uhlmann and L. van Veen, The significance of simple invariant solutions in turbulent flows, Annu. Rev. Fluid Mech. 44 (2012) 203–225.
J. F. Gibson, J. Halcrow and P. Cvitanović, Visualizing the geometry of state space in plane Couette flow, J. Fluid Mech. 611 (2008) 107–130.
J. D. Skufca, J. A. Yorke and B. Eckhardt, Edge of chaos in a parallel shear flow, Phys. Rev. Lett. 96 (2006) 174101.
C. C. T. Pringle and R. R. Kerswell, Using nonlinear transient growth to construct the minimal seed for shear flow turbulence, Phys. Rev. Lett. 105 (2010) 154502.