On Sep 19, 2012, at 1:46 PM, Allawala, Altan wrote:
> I'm trying to find the steady state solution to the following linear PDE:
>
> \frac{\partial \varphi}{\partial t} = - x * \frac{\partial \varphi}{\partial
> x} - x^2 \varphi
>
> Below is my code to solve this. However, the numerical solution is not
> matching the analytical one. Any idea where the problem is?
No solution, but a couple of observations:
PowerLawConvectionTerm(coeff = x*[[1]], var=numerical)
expresses
\frac{\partial}{\partial x} \left( x \varphi\right)
or
x \frac{\partial \varphi}{\partial x} + \varphi
not
x * \frac{\partial \varphi}{\partial x}
Changing the equation declaration to
X = mesh.getFaceCenters()[0]
eqn = (TransientTerm(coeff=1., var=numerical) ==
- PowerLawConvectionTerm(coeff = X*[[1]], var=numerical)
+ ImplicitSourceTerm(coeff = 1. - x*x, var=numerical))
seems to help, but the solution still seems to dissipate to zero with time,
irrespective of boundary conditions. Wheeler may know why the constraints are
being ignored, noting that...
> It seems that neither of the left hand side boundary conditions are being
> obeyed.
At least part of the issue is that a 1st order PDE only admits one boundary
condition.
> In addition, I've observed that adding a minus sign in front of every term on
> both sides of the equality entirely changes the solution, even though
> mathematically it shouldn't.
Rather than change solution, changing the sign renders the problem unstable. I
don't know why.
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