====== poincare ======

Compute crossings of a Poincare section in a pre-computed trajectory. The Poincare condition is %%(u(t) - ueqb, e) == 0%%, i.e. the displacement from a given point %%ueqb%% is orthogonal to a direction %%e%%. The program checks for changes in sign in the inner product in the stored trajectory (with large %%dT%%, and when it finds one, goes back and reintegrates with small %%dt%%, and then does quadratic interpolation when the fine-scale trajectory recrosses the section.

===== usage =====

<code>
  poincare -T0 0 -T1 400 -d datadir ueqb.ff e.ff
</code>
reads a trajectory u0, u1, ..., u400 stored in %%datadir%%, computes the crossings of the Poincare section %%(u(t)-ueqb, e) == 0%%, and saves them in a %%poincare/%% directory. 

===== options =====

<code>
options :
  -T0       --T0                <real>      default == 0           start time
  -T1       --T1                <real>      default == 100         end time
  -dT       --dT                <real>      default == 1           save interval
  -d        --datadir           <string>    default == data/       flowfield series directory
  -o        --poindir           <string>    default == poincare/   output dir for Poincare crossings
  -vdt      --variabledt                                           adjust dt to keep CFLmin<=CFL<CFLmax
  -dt       --dt                <real>      default == 0.03125     timestep
  -dtmin    --dtmin             <real>      default == 0.001       minimum time step
  -dtmax    --dtmax             <real>      default == 0.05        maximum time step
  -CFLmin   --CFLmin            <real>      default == 0.4         minimum CFL number
  -CFLmax   --CFLmax            <real>      default == 0.6         maximum CFL number
  -ts       --timestepping      <string>    default == sbdf3       timestepping algorithm
  -nl       --nonlinearity      <string>    default == rot         method of calculating nonlinearity
  -R        --Reynolds          <real>      default == 400         Reynolds number
  -c        --channel                                              channelflow instead of plane Couette
  -b        --bulkvelocity                                         hold bulk velocity fixed, not pressure gradient
  -P        --dPdx              <real>      default == 0           value for fixed pressure gradient
  -U        --Ubulk             <real>      default == 0           value for fixed bulk velocity
  <flowfield>      (trailing arg 2)                                fixed pt of poincare map
  <flowfield>      (trailing arg 1)                                e, for poincare condition (u(t)-u*, e) = 0
</code>

===== Usage example =====


Here is the basic idea of usage. Suppose you have an equilibrium ueqb with an unstable complex eigenvalue pair whose eigenfunctions are ef1 and ef2. Integrate a trajectory with initial condition %%u(0) = ueqb + 0.001 ef1%% as follows
<code>
   addfields 1 ueqb 0.001 ef1 ueqb_001ef1
   couette -T0 0 -T1 1000 -o data-ueqb_0001ef1 ueqb_001ef1
</code>
Then you can define a Poincare section as, say, %%(u-ueqb, ef2) = 0%%, that is, the displacement from the equilibrium is orthogonal to the ef2 eigenfunction. To compute crossings of the Poincare section within the
previously integrated trajectory, run
<code>
   poincare -T0 0 -T1 1000 -d data-ueqb_0001ef1 ueqb ef2
</code>

===== Hiccups =====

:-) Hey John, I was wondering if there was any particular reason you put the cfpause() line in your code other than it was helpful in writing the code. --- //[[dustin.spieker@gatech.edu|Dustin Spieker]] 2009-03-02 10:24//