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2014-01-13T20:39:05Z

Created page with "This page covers the solution of a pair of coupled ODEs using Fipy. =The Equation= This script solves the following differential equations: <math> \frac{dy}{dt} = - \frac..."

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This page covers the solution of a pair of coupled ODEs using [[Fipy]].

=The Equation=

This script solves the following differential equations:

<math>
\frac{dy}{dt} = -
\frac{dz}{dt} = \phi y - (1 - \phi) z
</math>

<math>
\phi = sin (2*pi*t) \left[ \exp^{- t/100} \right]
</math>

with initial condition

<math>
y(t=0) = 1.0
</math>

which has the exact solution:

<math>
y(t) = 1.5556 \exp( -1.8 t ) - 0.5556
</math>

More information about this problem is available via [http://wolfr.am/1eveai5 Wolfram Alpha].

=The Script=

==Import Statements==

We begin with our usual import statements, which allow us to use [[Fipy]] for solving the equation and [[Matplotlib]] for plotting results.

<source>
from numpy import *
import matplotlib.pylab as plt
from fipy import *

# Set up figure for plotting
fig = plt.figure()
ax = fig.add_subplot(111)
</source>

==Input Parameters==

The first thing we'll do is set some parameters, which we will then use to construct Fipy objects:

<source>
##############################
# Set Fipy parameters

# number of cells in mesh
nx = 1

# timesteps
deltat = 0.1
nt = 100

# initial value
iv = 1.0
</source>

==Fipy Objects==

Now assemble Fipy objects using user parameters

<source>
##############################
# Assemble Fipy objects

# create the mesh
mesh = Grid1D(nx=nx,dx=1.0)

# create the variable
y = CellVariable(mesh=mesh, value=iv, hasOld=1, name='y')

# create the equation
RHS = -1.8*y - 1.0
eq = TransientTerm() == RHS

# solver
from fipy.solvers import LinearPCGSolver
solver = LinearPCGSolver()
#from fipy.solvers import DefaultAsymmetricSolver
#solver = DefaultAsymmetricSolver()

# timesteps and steps
dt = Variable(deltat)
steps = nt
t = dt.value * steps
solutions = zeros([steps+1,nx])
solutions[0,:] = y.value
</source>

==Solution Loop==

Loop over each step and solve the differential equation over that step:

<source>
# loop over timesteps
for step in [i+1 for i in range(steps)]:
y.updateOld()
eq.solve(y,
solver = solver,
dt=dt)

# store solutions
solutions[step] = y.value
</source>

==Plot Computed and Exact Solution==

<source>
ts = linspace(0,t,steps)

# plot computed solution
ax.plot(ts,solutions[:step],'bo',label='Computed')#arange(len(y.value)),y.value)

# plot exact solution
y2 = 1.5556*exp(-1.8*ts) - 0.5556
plt.hold(True)
ax.plot(ts,y2,'r-',label='Exact')

ax.legend(); plt.show(); plt.draw()
</source>

=The Result=

This should result in the following plot:

[[Image:FipySimpleTransient.png]]

=The Py File=

http://files.charlesmartinreid.com/FipySimpleTransient.py

[[Category:Fipy]]
[[Category:Cantera]]

Fipy/Multiple Variable Transient Problem - Revision history

Admin: Created page with "This page covers the solution of a pair of coupled ODEs using Fipy. =The Equation= This script solves the following differential equations: \frac{dy}{dt} = - \frac..."

Admin: Created page with "This page covers the solution of a pair of coupled ODEs using Fipy. =The Equation= This script solves the following differential equations: $\frac{dy}{dt} = - \frac..."$