boundary conditions with pdepe when m>0

Hello,

I'm trying to simulate a 2d diffusion problem where the source at the boundary is time-dependent. I have no problem imposing this boundary conditions when m=0 (i.e., problem is in 1 dimension), but as soon as I try to run the simulation in 2 dimensions (m=1), it doesn't work.

In the pdepe documentation I found the following that pertains to my problem:

When m > 0 and a = 0, boundedness of the solution near x = 0 requires that the flux f vanish at a = 0. pdepe imposes this boundary condition automatically and it ignores values returned in pl and ql.

thus, my time-dependent source is ignored entirely when m=1. Is there any work around to this? Is there anyway to use the 1d solver to simulate the 2d problem?

Thanks

Nick

Answer 1

[Torsten]

Setting m=1 does not make your computation 2-dimensional, but calculations are done over the radius of a cylinder. You can't prescribe a source at the axis of a cylinder (x=0) because the area over which this source would be added is zero.

Best wishes

Torsten.

[Nicholas Mikolajewicz]

Wait so do you mean that the source must have a spatial component to it? (i.e., I can't have a point source for m=1?). I noticed is that if I increase the source radius, then my choice of spacing of xmesh influences the outcome of the simulation (this is also true for a single point source). How can I ensure that the concentrations I introduce at the source are correctly dffused and independent of my xmesh settings? Sorry these questions are extremely basic but I want to make sure I get this right.

If I want to model 2-dimension diffusion, I would use m=1 over m=0 correct? (I'm assuming the particle diffuse radially).

Thanks again,

Nick

[Torsten]

You can have a source for m=1, but not at the left boundary point (r=0).

As you described your problem in your first post, the source must be added at the boundary of your computational domain. Thus there is no such thing as a source radius.

If you want to model 2d-diffusion, you must use a 2-dimensional solver (e.g. the PDE Toolbox) - "pdepe" is 1-dimensional. Again: The variable "m" has nothing to do with the dimensionality of your problem.

Best wishes

Torsten.

[Nicholas Mikolajewicz]

So if pdepe is 1-dimensional, there's no way to model experimental observations? Or is there someway to relate the two?

[Torsten]

I don't know your experimental set-up. So I can't say whether it is legitimate to use a 1d-model to reproduce your experimental results.

E.g. if you have a cylinder which is heated at r=R with a constant heat flux over the height, it is legitimate to model the temperature over the radius as an 1d-problem using "pdepe".

If the heat flux is not constant over the height, you will have to set up the problem 2-dimensional (radius and height as independent variables).

Best wishes

Torsten.

[Nicholas Mikolajewicz]

the source height is approximately 20 micrometers while the height of medium is 3 millimeters...so I suppose 2d is more appropriate... This can all be achieved using the pde toolbox?