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====== findsoln ====== find an equilibrium, traveling wave, periodic orbit, or relative periodic orbit of plane Couette or channel flow ====== 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]]
