I'm currently learning how to use FiPy and eventually want to use it to solve some biological problems. I have been trying to implement the following PDE system which describes hyphal growth of fungi:
I can implement the system without problems, as long as I ignore that the change of si over time depends on the absolute change of p over space abs(dp/dx) in the fourth equation dsi/dt. Here is my code without abs():
from fipy import *
##### produce mesh
nx= 100.
ny= nx
dx = 1/100
dy = dx
L = nx*dx
mesh = Grid2D(nx=nx,ny=ny,dx=dx,dy=dy)
x,y = mesh.cellCenters
##### parameters
b=1e+7 # branching rate (no. branches * cm^-1 * hyphae * day^-1 * (mol glucose)^-1)
f = 10. # fusion rate (no. fusions in cm * day^-1)
v = 1e+5 # cm * mol^-1 * day^-1
d = 0.5 # day^-1
r = 0. # degradation of hyphae
c1 = 9e+2 # cm * mol^-1 * day^-1
c2 = 1e-7 # mol * cm^-1
c3 = 1e+3 # cm * mol^-1 * day^-1, c1/c3 = 90% efficiency
c4 = 1e-8 # cm^-1
De = 1e-3 # diffusion of external substrate (cm² * s^-1)
Di = 1e-2 # diffusion of internal substrate (cm³ * day^-1)
Da = 0.
##### define state variables:
m = CellVariable(name="m",mesh=mesh,hasOld=True,value=0.)
mi = CellVariable(name="mi",mesh=mesh,hasOld=True,value=0.)
p = CellVariable(name="p",mesh=mesh,hasOld=True,value=0.)
si = CellVariable(name="si",mesh=mesh,hasOld=True,value=0.)
se = CellVariable(name="se",mesh=mesh,hasOld=True,value=0.)
##### differential equations
eqm = (TransientTerm(var=m)== si*v*p - ImplicitSourceTerm(var=m,coeff=d))
eqmi = (TransientTerm(var=mi)==m*d - ImplicitSourceTerm(var=mi,coeff=r))
eqp = (TransientTerm(var=p)== - ConvectionTerm(var=p,coeff=[[v]]*si) + b*si*m - ImplicitSourceTerm(var=p,coeff=f*m))
eqsi = (TransientTerm(var=si)== DiffusionTerm(var=si,coeff=Di*m)- DiffusionTerm(var=p,coeff=Da*m*si) + ImplicitSourceTerm(var=si,coeff=c1*m*se) - ImplicitSourceTerm(var=si,coeff=c2*v*p) - ConvectionTerm(var=p,coeff=[[c4*Da]]*(m*si)))
eqse = (TransientTerm(var=se) == DiffusionTerm(var=se,coeff=De) - ImplicitSourceTerm(var=se,coeff=c3*m*si))
# initial values
m0 = 100.
mi0 = 0.
p0 = 500.
si0 = 1e-5
se0 = 3e-5
r = 3. # radius of initial plug
m.setValue(m0,where=((x>(L/2-(r*dx)))&(x<(L/2+(r*dx)))&(y>(L/2-(r*dy)))&(y<(L/2+(r*dy)))))
mi.setValue(mi0,where=((x>(L/2-(r*dx)))&(x<(L/2+(r*dx)))&(y>(L/2-(r*dy)))&(y<(L/2+(r*dy)))))
p.setValue(p0,where=((x>(L/2-(r*dx)))&(x<(L/2+(r*dx)))&(y>(L/2-(r*dy)))&(y<(L/2+(r*dy)))))
si.setValue(si0,where=((x>(L/2-(r*dx)))&(x<(L/2+(r*dx)))&(y>(L/2-(r*dy)))&(y<(L/2+(r*dy)))))
se.setValue(se0)
# boundary conditions
#----------------
# simulation
eq = eqm & eqmi & eqp & eqsi & eqse
vi = Viewer((m))
from builtins import range
for t in range(100):
m.updateOld()
mi.updateOld()
p.updateOld()
si.updateOld()
se.updateOld()
eq.solve(dt=0.1)
print(t)
vi.plot()
Now, I tried to simply write
eqsi = (TransientTerm(var=si)== DiffusionTerm(var=si,coeff=Di*m)- DiffusionTerm(var=p,coeff=Da*m*si) + ImplicitSourceTerm(var=si,coeff=c1*m*se) - ImplicitSourceTerm(var=si,coeff=c2*v*p) - abs(ConvectionTerm(var=p,coeff=[[c4*Da]]*(m*si))))
which (expectedly) gives out an error: "bad operand type for abs(): 'PowerLawConvectionTerm'"
I tried to work my way around it by adding another CellVariable dpdx:
dpdx= CellVariable(name="dpdx",mesh=mesh,hasOld=True,value=0.)
eqdpdx= (dpdx == ConvectionTerm(var=p,coeff=[[1]]))
eqsi = (TransientTerm(var=si)== DiffusionTerm(var=si,coeff=Di*m)- DiffusionTerm(var=p,coeff=Da*m*si) + ImplicitSourceTerm(var=si,coeff=c1*m*se) - ImplicitSourceTerm(var=si,coeff=c2*v*p) - abs(dpdx)*c4*Da*m*si)
which then gives the error "ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()", which is probably caused by the fact that eqdpdx is not a derivative function?
My question is: Is it possible at all to perform mathematical operations with a ConvectionTerm? Can I somehow express the absolute change of p? And also, how can I express non derivative function that change over space like eqdpdx (or any dynamic parameters)? I looked through the FiPy manual and could not find a solution - I am sorry if my question is trivial or has been answered elsewhere.
