> I'm particularly interested in using FiPy to study weakly-nonlinear
>> equations associated with pattern formation in multiple dimensions. A
>> classical problem there is the Kuramoto-Sivashinsky equation:
>>
>> h_t = - \nabla^2 h - \nabla^4 h - (1/2) | \nabla h |**2
>>
>> where h is the dependent variable.
>
>
> One possible way to do this (with the coupled equation capability) would be
>
> h_t = - \nabla^2 h - \nabla^2 f - (1/2) \nabla \cdot \left(h \nabla
> h \right) + h f
>
> f = \nabla^2 h
>
> I can't say how well this will work, but it seems logical. The second to
> last term will be a convection term with the velocity "\nabla h" and the
> last term would be an implicit source term on f (h will be the
> coefficient), since it is based on the derivatives of h.
>
Thanks so much, and for the links to documentation on coupled equations.
So far, this single equation seems to work:
mesh = PeriodicGrid1D(nx = 100, dx = 0.5)
u = CellVariable(name="u", mesh=mesh)
v = CellVariable(name="v=lapl u", mesh=mesh)
eqn0 = TransientTerm(var=u) == - DiffusionTerm(coeff=1., var=u) -
DiffusionTerm(coeff=(1., 1.), var=u) \
+ PowerLawConvectionTerm(coeff=.5*u.faceGrad, var=u) -
ImplicitSourceTerm(coeff=.5*u.faceGrad.divergence, var=u)
But replacing the last line with this coupled equation fails:
eqn1 = TransientTerm(var=u) == - DiffusionTerm(coeff=1., var=u) -
DiffusionTerm(coeff=1., var=v) \
+ PowerLawConvectionTerm(coeff=.5*u.faceGrad, var=u) -
ImplicitSourceTerm(coeff=.5*v, var=u)
eqn2 = ImplicitSourceTerm(var=v) == DiffusionTerm(1, var=u)
eqn = eqn1 & eqn2
It seems to run, but quickly fails with message alluding to:
Warning: overflow encountered in square
Warning: overflow encountered in square
Warning: overflow encountered in square
Warning: overflow encountered in square
Warning: overflow encountered in square
(many more of this, then:)
...
Traceback (most recent call last):
File "dKS-1D.py", line 76, in <module>
eqn.solve(dt=dt)
File
"/usr/local/lib/python2.7/dist-packages/FiPy-3.0_dev5303-py2.7.egg/fipy/terms/term.py",
line 211, in solve
solver._solve()
File
"/usr/local/lib/python2.7/dist-packages/FiPy-3.0_dev5303-py2.7.egg/fipy/solvers/pysparse/pysparseSolver.py",
line 103, in _solve
self._solve_(self.matrix, array, self.RHSvector)
File
"/usr/local/lib/python2.7/dist-packages/FiPy-3.0_dev5303-py2.7.egg/fipy/solvers/pysparse/linearLUSolver.py",
line 87, in _solve_
b = b * (1 / maxdiag)
ValueError: cannot convert float NaN to integer
I note either way I try to do it, I still seem to have to use faceGrad() in
the ConvectionTerm for $ \nabla (u \nabla u) $. Is this destroying the
implicit nature of the method? Is there a way around it?
--
Dr. Scott A. Norris
Assistant Professor of Mathematics
Southern Methodist University, Dallas, TX
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