Order-of-Magnitude Speedup for Steady States and Traveling Waves via Stokes Preconditioning in Channelflow and Openpipeflow

Abstract

Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton's method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier-Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10 to 50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.

Stably stratified exact coherent structures in shear flow: the effect of Prandtl number

“…(2.1a)-(2.1c), using the standard Newton-Raphson method and LU decomposition to invert the discretised system Jacobian. The second was a modified version of the Channelflow 2.0 DNS software (Gibson et al 2019) that employs a matrix-free iterative solver to find invariant solutions via two standard methods: Stokes preconditioning (Tuckerman 1989;Tuckerman et al 2019) and time integration (Viswanath 2007;Gibson et al 2008). The former method was used to compute Pr 10 states and the latter was used for higher Pr.…”

“…The additional solutions in § 3.2.3 were obtained using a modified version of the Channelflow 2.0 DNS software (Gibson et al 2019) that employs a matrixfree iterative solver to find invariant solutions via two standard methods: Stokes preconditioning (Tuckerman 1989) and time integration (Viswanath 2007;Gibson et al 2008). The former method was used to the compute Pr 10 solutions in § 3.2.3 and the latter was used at higher Pr, where the Stokes-preconditioned system suffers the effects of poorer numerical conditioning more acutely (Tuckerman, Langham & Willis 2019).…”

“…2008). The former method was used to the compute solutions in § 3.2.3 and the latter was used at higher , where the Stokes-preconditioned system suffers the effects of poorer numerical conditioning more acutely (Tuckerman, Langham & Willis 2019).…”

Continuation and stability of rotating waves in the magnetized spherical Couette system: secondary transitions and multistability

“…In [34], RW were obtained as a zeros of a nonlinear system, requiring the use of preconditioning techniques, to accelerate the convergence of the linear solver associated with the Newton's method. By contrast, the Stokes preconditioning method used in [49] only relies on time integration of the large nonlinear system over a single time step and thus is easier to implement if a time-stepping code is already available. However, as mentioned in [30], RW can also be found as periodic orbits which is the approach followed in the present study.…”

Video: Life and fate of a bubble in a constricted Hele-Shaw channel

“…A difficulty in applying these ideas in fluid dynamics is that the invariant objects of interest can be extremely high dimensional and cannot be easily determined. Nevertheless, recent advances in both computer power and efficient algorithms (Tuckerman et al 2019;Farano et al 2019) have allowed ever increasing access to such objects in studies of turbulence (Budanur et al 2017), convection (Sánchez Umbría & Net 2019) and droplet break up (Gallino et al 2018). In our chosen system, the bubble exhibits a variety of complex behaviours including symmetry breaking, bistability, steady and periodic states as well as non-trivial transient evolution (Franco-Gómez et al 2018).…”