Hi,
I would like to solve "infinite" diffusion PDE (3d):
1/(a\nabla\phi) a \otimes a \frac{\partial^2}{\partial x_i \partial
x_j} \phi -\alpha \phi + \beta = \partial \phi /\partial t
where "a" is a vector and a \otimes a is outer product of this vector
(symmetric matrix a_ij). This equation reduces to:
\frac{\partial^2 \phi / \partial x^2 }{\partial \phi / \partial x}
-\alpha \phi + \beta = \partial \phi /\partial t
for one dimensional case and a scalar.
Equilibrium solution for 1d case is reducing to Burgers equation which
can be solved analytically(Hopf-Cole transformation). The solution is
-tanh(2\alpha/a(x-\delta)) where \delta is position of the interface
(boundary condition are +1 for x=-\inf and -1 for x=+\inf) . This
represents
a traveling wave solution with certain velocity \delta=velocity * time.
I tried to solve 1d equation with Fipy something like:
from fipy import *
nx = 40
Lx = 2.5 * nx / 100.
dx = Lx / nx
mesh = Grid1D(dx=dx, nx=nx)
timeStepDuration = 0.0002
phaseTransientCoeff = 1
thetaSmallValue = 1e-6
beta = 1e5
mu = 1e3
gamma = 1000
s = 1
phase = CellVariable(name='phase field',mesh=mesh, value=1.)
x = mesh.getCellCenters()[0]
phase.setValue(x*(1-x),where=mesh.getCellCenters()[0] < Lx )
#phase.setValue(1./3,where=mesh.getCellCenters()[0] > Lx /3.)
#phase.setValue(1-x,where=mesh.getCellCenters()[0] > 2* Lx / 3.)
print phase
gradMag = phase.getFaceGrad().getMag()
eps = 1./gamma/10.
gradMag+=(gradMag < eps) * eps
IGamma = (gradMag > 1./gamma) * (1/gradMag-gamma) + gamma
#print IGamma
diffusionCoeff = s*IGamma
phaseEq = TransientTerm(phaseTransientCoeff) ==
ImplicitDiffusionTerm(diffusionCoeff)
#print diffusionCoeff
if __name__ == '__main__':
phaseViewer = Viewer(vars=phase, datamin=0., datamax=2)
phaseViewer.plot()
steps = 100
for i in range(steps):
phase.updateOld();
phaseEq.solve(phase,dt=timeStepDuration)
if __name__ =='__main__':
phaseViewer.plot()
If I fix the boundary condition in fipy to 1 an -1 at the domain
boundaries, the numerical solution converges to step like function.
It seems very difficult to control the terms involving the 1/grad(Phi).
Is it risky to handle this term as a very high diffusion coefficient ???
The 1d solution of the previous equation has exact equilibrium solution
as tangent hyperbolic and not step function.
I would be grateful if somebody can help me with this problem.
Thank you,
Toni Ivas
