How to represent source terms depending on position and time in Fipy python

Question

I have the following partial differential equation

I would like to know how I can represent the source term in Fipy python. I have tried the following

from fipy import *
nx = 50
ny = 1
dx = dy = 0.025                     # grid spacing
L = dx * nx
mesh = Grid2D(dx=dx, dy=dy, nx=nx, ny=ny)
phi = CellVariable(name="solution variable", mesh=mesh, value=0.)
convCoeff = ((10.,), (10.,))
Gamma = 1.
eqX = TransientTerm() == DiffusionTerm(coeff=Gamma) - ConvectionTerm(coeff=convCoeff) + t*numerix(exp(x*y))
valueTopLeft = 0
valueBottomRight = 1
X, Y = mesh.faceCenters
facesTopLeft = ((mesh.facesLeft & (Y > L / 2)) | (mesh.facesTop & (X < L / 2)))
facesBottomRight = ((mesh.facesRight & (Y < L / 2)) |
                    (mesh.facesBottom & (X > L / 2)))
#
phi.constrain(valueTopLeft, facesTopLeft)
phi.constrain(valueBottomRight, facesBottomRight)
timeStepDuration = 10 * 0.9 * dx ** 2 / (2 * D)
steps = 10
results = []
for step in range(steps):
    eqX.solve(var=phi, dt=timeStepDuration)
    results.append(phi.value)

and this is not working. According to fipy manual I saw they said source terms are represnted the way they appear and its recommendated using numerix module over other modules like numpy. I dont know that am missing in this code. Thanks

Answer 1

I found a number of problems with the code

• t was undefined
• the use of numerix and exp in the source was nonsensical and gave an error
• D was undefined

To make the source time dependent, t needs to be a Variable so that the source will be reevaluated whenever time changes. The time variable also needs to be updated at each step.

Here is a corrected version of your code that actually runs.

from fipy import *
nx = 50
ny = 1
dx = dy = 0.025                     # grid spacing
L = dx * nx
mesh = Grid2D(dx=dx, dy=dy, nx=nx, ny=ny)
phi = CellVariable(name="solution variable", mesh=mesh, value=0.)
t = Variable(0.)
x = mesh.x
y = mesh.y
convCoeff = ((10.,), (10.,))
Gamma = 1.
D = Gamma
eqX = TransientTerm() == DiffusionTerm(coeff=Gamma) - ConvectionTerm(coeff=convCoeff) + t*numerix.exp(x*y)
valueTopLeft = 0
valueBottomRight = 1
X, Y = mesh.faceCenters
facesTopLeft = ((mesh.facesLeft & (Y > L / 2)) | (mesh.facesTop & (X < L / 2)))
facesBottomRight = ((mesh.facesRight & (Y < L / 2)) |
                    (mesh.facesBottom & (X > L / 2)))
#
phi.constrain(valueTopLeft, facesTopLeft)
phi.constrain(valueBottomRight, facesBottomRight)
timeStepDuration = 10 * 0.9 * dx ** 2 / (2 * D)
steps = 10
results = []
for step in range(steps):
    eqX.solve(var=phi, dt=timeStepDuration)
    results.append(phi.value)
    t.setValue(t.value + timeStepDuration)
    print('step:', step)