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====== findsoln ======

find an equilibrium, traveling wave, periodic orbit, or relative periodic orbit of plane Couette or channel flow,
using a [[docs:math:newton_krylov_hookstep| Newton-Krylov-hookstep search algorithm]].

====== main options ======

<code>
  -eqb      --equilibrium                                          search for equilibrium or relative equilibrium (trav wave)
  -orb      --periodicorbit                                        search for periodic orbit or relative periodic orbit
  -xrel     --xrelative                                            search over x phase shift for relative orbit or eqb
  -zrel     --zrelative                                            search over z phase shift for relative orbit or eqb
  -T        --maptime           <real>      default == 20          initial guess for orbit period or time of eqb/reqb map f^T(u)
  -R        --Reynolds          <real>      default == 400         Reynolds number
  -sigma    --sigma             <string>                           file containing symmetry of relative solution (default == identity)
  -symms    --symmetries        <string>                           file containing generators of isotropy group for symmetry-constrained search
  ... 
  -o        --outdir            <string>    default == ./          output directory
  -log      --logfile           <string>    default == stdout      output log (filename or "stdout")
  <flowfield>      (trailing arg 1)                                initial guess for solution
</code>

====== solution type ======

''findsoln'' finds solutions of //σ f^T(u) - u = 0// where //f^T// is the time-//T// forward map of the Navier-Stokes equations plus boundary conditions and σ is a symmetry of the flow. This equation has several kinds of solutions

  * equilibrium: //σ = 1///, all //T//
  * relative equilibrium: //σ = // a translation or phase shift that scales linearly in //T//, all //T// (also called a traveling wave)
  * periodic orbit: //σ = 1// and some finite value of //T// (and integer multiples of it)
  * preperiodic orbit: //σ ≠ 1// but //σ^n = 1// for some //n>1//, and some finite value of //T// (and integer multiples)
  * relative periodic orbit: some element //σ// of the appropriate symmetry group, and some finite value of //T// (and integer multiples)

You specify which kind of solution to find by telling the search algorithm which variables to treat as unknowns, with the ''-eqb, -orb, -xrel, -zrel'' options. The ''-eqb'' option specifies that //T// is held fixed; the ''-orb'' option specifies that is unknown. The ''-xrel'' and ''-zrel'' options specifying that the //x// and //z// phase shifts in σ are unknown. The appropriate combinations for each solution type are best shown by example.

^command^search type^
|''findsoln -eqb uguess.h5'' | equilibrium solution |
|''findsoln -eqb -xrel uguess.h5'' | traveling wave in //x// |
|''findsoln -eqb -xrel -zrel uguess.h5'' | traveling wave in //x,z// |
|''findsoln -eqb -zrel -sigma sigma.asc -T 20 uguess.h5'' | traveling wave in //z// with an initial guess for wavespeed |
|''findsoln -orb -T 64.3 uguess.asc'' | periodic orbit with initial guess for period //T// |
|''findsoln -orb -T 34.8 -sigma sigma.asc uguess.h5'' | pre-periodic orbit |
|''findsoln -orb -xrel -T 49.1 -sigma sigma.asc uguess.h5'' | relative periodic orbit with phase shift in //x// |


([[docs:math:newton_krylov_hookstep|mathematical details of algorithm]]
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