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

Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 50)


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 10–50 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.


Traveling Wave Plane Couette Flow Timestep Pipe Flow Krylov Vectors 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank Dwight Barkley and John Gibson for their contributions. We acknowledge the support of TRANSFLOW, provided by the Agence Nationale de la Recherche (ANR).


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Authors and Affiliations

  1. 1.Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI ParisPSL Research University, Sorbonne Université, Univ. Paris DiderotParisFrance
  2. 2.School of MathematicsUniversity of BristolBristolUK
  3. 3.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK

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