Tuesday, November 24, 2015

2D Diffusion Equation using Python, Scipy, and VPython

 I got it from here, but modify it here and there.

 I also add animation using vpython but can't find 3d or surface version, so I planned to go to matplotlib surface plot route, :)

 (update: here it is, :) )

#!/usr/bin/env python
A program which uses an explicit finite difference
scheme to solve the diffusion equation with fixed
boundary values and a given initial value for the

Two steps of the solution are stored: the current
solution, u, and the previous step, ui. At each time-
step, u is calculated from ui. u is moved to ui at the
end of each time-step to move forward in time.


he uses matplotlib

I use visual python

import scipy as sp
import time
from visual import *
from visual.graph import *

graph1 = gdisplay(x=0, y=0, width=600, height=400, 
          title='x vs. T', xtitle='x', ytitle='T', 
          foreground=color.black, background=color.white)

# Declare some variables:

dx=0.01        # Interval size in x-direction.
dy=0.01        # Interval size in y-direction.
a=0.5          # Diffusion constant.
timesteps=500  # Number of time-steps to evolve system.

nx = int(1/dx)
ny = int(1/dy)

dx2=dx**2 # To save CPU cycles, we'll compute Delta x^2
dy2=dy**2 # and Delta y^2 only once and store them.

# For stability, this is the largest interval possible
# for the size of the time-step:
dt = dx2*dy2/( 2*a*(dx2+dy2) )

# Start u and ui off as zero matrices:
ui = sp.zeros([nx,ny])
u = sp.zeros([nx,ny])

# Now, set the initial conditions (ui).
for i in range(nx):
 for j in range(ny):
  if ( ( (i*dx-0.5)**2+(j*dy-0.5)**2 <= 0.1)
   & ((i*dx-0.5)**2+(j*dy-0.5)**2>=.05) ):
    ui[i,j] = 1
def evolve_ts(u, ui):
 global nx, ny
 This function uses two plain Python loops to
 evaluate the derivatives in the Laplacian, and
 calculates u[i,j] based on ui[i,j].
 for i in range(1,nx-1):
  for j in range(1,ny-1):
   uxx = ( ui[i+1,j] - 2*ui[i,j] + ui[i-1, j] )/dx2
   uyy = ( ui[i,j+1] - 2*ui[i,j] + ui[i, j-1] )/dy2
   u[i,j] = ui[i,j]+dt*a*(uxx+uyy)
def evolve_ts(u, ui):
 This function uses a numpy expression to
 evaluate the derivatives in the Laplacian, and
 calculates u[i,j] based on ui[i,j].
 u[1:-1, 1:-1] = ui[1:-1, 1:-1] + a*dt*( 
                (ui[2:, 1:-1] - 2*ui[1:-1, 1:-1] + ui[:-2, 1:-1])/dx2 + 
                (ui[1:-1, 2:] - 2*ui[1:-1, 1:-1] + ui[1:-1, :-2])/dy2 )
# Now, start the time evolution calculation...
#tstart = time.time()

f1 = gcurve(color=color.blue)

while True:
    #for m in range(1, timesteps+1):
    if t<timesteps:
 evolve_ts(u, ui)
        ui[:] = u[:] # I add this line to update ui value (not present in original code)
 #print "Computing u for m =", m
    f1.gcurve.pos   =   []
    for i in arange(nx):
#tfinish = time.time()

#print "Done."
#print "Total time: ", tfinish-tstart, "s"
#print "Average time per time-step using numpy: ", ( tfinish - tstart )/timesteps, "s."