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.

  poincare -T0 0 -T1 400 -d datadir ueqb.ff e.ff

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 :
  -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

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

   addfields 1 ueqb 0.001 ef1 ueqb_001ef1
   couette -T0 0 -T1 1000 -o data-ueqb_0001ef1 ueqb_001ef1

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

   poincare -T0 0 -T1 1000 -d data-ueqb_0001ef1 ueqb ef2

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 2009-03-02 10:24