====== continuesoln ======

''continuesoln'' continues an equilibrium, traveling wave, or periodic orbit in a parameter μ. The parameter μ is one of Reynolds number, //Lx, Lz, Lx/Lz, sqrt(Lx^2 + Lz^2)//, or //dPdx//. Continuation is done by a predictor-corrector method, using quadratic extrapolation in pseudo-arclength //s// along curves in the //(μ,D)// plane for prediction, and Newton-Krylov-hookstep search for correction. 

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

<code>
main options : 
  -r         --restart                                             start from three previously computed solutions
  -eqb       --equilibrium                                         search for equilibrium or relative equilibrium (trav wave)
  -orb       --periodicorbit                                       search for periodic orbit or relative periodic orbit
  -poinc     --poincare                                            (relative) periodic orbit search constrained to I-D=0 Poincare section
  -xrel      --xrelative                                           search over x phase shift for relative orbit or eqb
  -zrel      --zrelative                                           search over z phase shift for relative orbit or eqb
  -contRe    --continueRe                                          continue solution in Reynolds number
  -contdPdx  --continuedPdx                                        continue solution in imposed pressure gradient
  -contLx    --continueLx                                          continue solution in streamwise width Lx
  -contLz    --continueLz                                          continue solution in spanwise width Lz
  -contAsp   --continueAspect                                      continue solution in aspect ratio Lx/Lz
  -contDiag  --continueDiagonal                                    continue solution along diagonal with const aspect ratio Lx/Lz
  -contLtarg --continueLtarget                                     continue solution towards a target Lx,Lz
  -up        --upwards                                             for non-restart searches, search in dir of increasing free parameter
  -Lxtarg    --LxTarget          <real>      default == 6.28319    aim for this value of Lx in target Lx,Lz continuation
  -Lztarg    --LzTarget          <real>      default == 3.14159    aim for this value of Lz in target Lx,Lz continuation
  -T         --maptime           <real>      default == 15         initial guess for orbit period or time of eqb/reqb map f^T(u)
  -R         --Reynolds          <real>      default == 400        Reynolds number
  -dPdx      --dPdx              <real>      default == 0          imposed mean pressure gradient
  -s0        --s0                <real>      default == 0          start value for arclength (arbitrary)
  -ds        --ds                <real>      default == 0.001      initial arclength increment for quadratic extrapolation
  -dsmin     --dsmin             <real>      default == 0.0001     minimum arclength increment (in normalized D,Re space)
  -dsmax     --dsmax             <real>      default == 0.05       maximum arclength increment (in normalized D,Re space)
  -errmin    --errmin            <real>      default == 1e-05      minimum error for extrapolated guesses
  -errmax    --errmax            <real>      default == 0.001      maximum error for extrapolated guesses
  -sigma     --sigma             <string>                          file containing sigma of sigma f^T(u) - u = 0 (default == identity)
  -symms     --symmetries        <string>                          file containing generators of isotropy group for symmetry-constrained search

</code>

===== output =====

''continuesoln'' produces a lot of output data. For each step in the continuation it produces a new subdirectory of the working directory and populates it with data files and log files. The output directories are named ''startup-2'', ''startup-1'', ''startup-0'', and ''search-0'', ''search-1'', ''search-1'', .... The three ''startup'' directories contain solutions computed either for very small perturbations of a single initial solutions, or solutions from previous computation (see the ''--restart'' option below). The ''search'' directories are indefinite in number, they just keep going as the continuation proceeds. Each of these directories contains data files for the solution's velocity field, phase shift σ, etc., as well as a logfile ''findsoln.log'' with detailed information about the Newton-Krylov search. The ''convergence.asc'' file shows the convergence of the residual with each Newton-Krylov step.

===== initial data and the --restart option  =====

The quadratic extrapolation method requires three consecutive data points. These can either be generated by very making small perturbations in the continuation parameter from a single existing solution, or by reading in three consecutive solutions from a previous continuation calculation. 

This example starts a continuation of a periodic orbit in Reynolds number, by specifying an initial solution //(u,T,σ)// on the command-line. 

<code>
   continuesoln -orb -contRe -T 87.25853994 -sigma sigma.asc P87p25.h5
</code>

This example starts from three consecutive solutions from a previous computation. The three arguments following the ''--restart'' option are directories containing solutions generated by a previous continuation. Those directories 
contain values of //(u,T,σ)// at different values of μ.

<code>
   continuesoln -orb -contRe --restart ../previous/search-22 ../previous/search-23 ../previous/search-24
</code>


===== solution type options =====

Choose one of ''-eqb'' or ''-orb'' to specify whether the solution type is (relative) equilibrium or
(relative) periodic orbit. 

For periodic orbit searches, ''-poinc'' specifies that the orbit starting and end points are confined to the //I-D=0//Poincare section. If you use this, make sure your initial data lie on the section (use ''fieldprops -e'').

To specify a search for a relative eqb or orbit, use the ''-xrel'' and/or ''-zrel'' options.