Martin Deshaies-Jacques

deshaies.martin@sca.uqam.ca

www.science.martn.info

pyfKdV is a Fortran 95 pseudospectral propagator wrapped with python 2.7.

Program released in this bundle are licenced under the terms of the opensource GPL licence version 3 (http://www.gnu.org/licenses/gpl.html). Detailed can be found in ./doc/gpl.txt.

Use, share, remix! ...and criticize

See also:

• Data Assimilation Lab: dVar

• [Warn, T. and Brasnett, B. The amplification and capture of atmospheric solitons by topography:38, 1982.](http://journals.ametsoc.org/doi/abs/10.1175/1520-0469(1983\)040%3C0028%3ATAACOA%3E2.0.CO%3B2)
• Hodyss, D. and Nathan, T. R. Solitary rossby waves in zonally varying jet ows. Geophys. Astrophys., 262, 2002.

• FFTW3: fortran 77 fast-fourier transform librairy (http://www.fftw.org/)
• ARPACK is a collection of Fortran77 subroutines designed to solve large scale eigenvalue problems.

• Numpy
• Matplotlib
• Cython

• ./ refer to the root of pyfKdV installation;
• [something] means optional arguments;
• <something> means you must replace it with what is appropriate to your environment.

1. On Linux OS Debian/Ubuntu, you can install it running

   [sudo] apt-get install libfftw3* libarpack2*
   [sudo] apt-get install python python-matplotlib python-numpy
   

   The version of cython shipping with Debian at the moment (0.15.1) is not cutting edge enough (missing memoryviews component.) You should download the latest. (I used 0.19.1 personnaly). Installing is straightforward, instruction are included. You'll need python header files:

   [sudo] apt-get install python-dev
   

   (there was a ascII error and I had to include a --no-compile-something to the setup.py call but it work perfectly afterward.)

2. To compile, you'll have to modify ./src/Makefile, to specify your libraries directories, and fortran compiler:

   libDir=/usr/lib
   userLibDir=/home/<user>/lib
   compiler=gfortran
   

Be sure that ${userLibDir} is in your ${LD_LIBRARY_PATH}:

    echo $LD_LIBRARY_PATH

if not:

    export LD_LIBRARY_PATH=".:/home/<user>/lib/"

3. Build the ./src/ directory

   make pyKdV [clean]
   

4. Add the path to the python modules pyKdV and pseudoSpec1D to your PYTHONPATH environment variable, in Linux bash environment you can do it running.

   export PYTHONPATH=${PYTHONPATH}:<path to ./>
   

   Adding this export lines to your startup script (.bashrc or .profile) is a way to do it.

5. Have fun!

1. Write a launcher python script:

   import numpy as np
   import pyKdV as kdv
   import matplotlib.pyplot as plt 
   
   #----| Grid configuration |-------------------
   Ntrc=150
   L=300.
   grid=kdv.PeriodicGrid(Ntrc,L)
   
   tInt=40.
   dt=0.015
   #----| KdV parameters arguments |-------------
   def rhoProfile(x,t):
       x0=-10.
       sig=30.
       amp=0.01
       return -amp*kdv.gauss(x,x0,sig)+2.*amp*kdv.gauss(x,x0+30,sig/3.)
           
   def forcTime(x, t):
       return 0.1*np.sin(6.*x/L)*np.cos(t/10.)
   
   param=kdv.Param(grid, beta=1., gamma=-1, rho=rhoProfile,
           forcing=forcTime, tInt=tInt, dt=dt)
   print(param)
       
   #----| Initial condition |--------------------
   baseLF=kdv.rndSpecVec(grid, 15, amp=1.3)
   soliton=kdv.soliton(grid.x, 0., amp=3. , beta=1., gamma=-1)
   ic=baseLF+soliton
       
   #----| Integration |--------------------------
   launcher=kdv.kdvLauncher(param, dt=dt)
   print(launcher)
   traj=launcher.integrate(ic, tInt)
   
   
   #----| Perturbating the trajectory |----------
   pert=0.1*kdv.gauss(grid.x, 0., 20.)
   #-- linear perturbation
   tlmLauncher=kdv.kdvTLMLauncher(param)
   tlmLauncher.initialize(traj)
   print(tlmLauncher)
   fLinearPert=tlmLauncher.integrate(pert, tInt)
   #-- nonlinear perturbation
   fNLPert=launcher.integrate(ic+pert, tInt).final-traj.final
   
   
   #----| Plotting the result |------------------
   plt.figure(figsize=(12.,12.))
   subplt1=plt.subplot(311)
   subplt2=plt.subplot(312)
   subplt3=plt.subplot(313)
   
   traj.waterfall(axe=subplt1)
   subplt1.legend([r"$x(t)$"], loc="best")
   
   
   subplt2.plot(grid.x, pert, 'k:')
   subplt2.plot(grid.x, fLinearPert, 'g')
   subplt2.plot(grid.x, fNLPert, 'b')
   subplt2.legend(["$\delta x$", "$\mathbf{L}\delta x$", 
                   "$\mathcal{M}(x+\delta x)-\mathcal{M}(x)$"],
                   loc='best')
   
   
   traj.fftTraj().waterfall(axe=subplt3, color='r')
   subplt3.legend([ "$N_{trc}="+str(Ntrc)+"$", r"$\mathcal{F}[x(t)]$"], loc="best")
   
   plt.show()
   

2. Singular vector calculation work similarly, but are way longer to obtain:

   Nev=2
   svLauncher=kdv.kdvSVLauncher(param, traj)
   sVal=svLauncher.lanczos(Nev, tInt=1.)
   print(sVal)
   plt.figure()
   for sv in svLauncher.sVec:
       plt.plot(grid.x, sv) 
   

A special thank to Mauro Werder, through the blog of whom I could understand how to link python and fortran!