Adding more boundary conditions to pdepe

so bcfun has the following signiture

function [pl,ql,pr,qr] = bcfun(xl,ul,xr,ur,t)

So it lets me set BCs at the left and right BCs.

My spacial domain is divided into 3 parts 0 to a, a to b, and then b to L. a, b, L are just arbitrary lengths. This means I need to have a BC at 0, a, b, and L.

So my question is that is it possible to set BCs at a and b. It is straight forward setting BCs at 0 and L but I cannot find a way to set them for the boundarys in the middle (if it is even possible)

Answer 1

[Torsten]

So my question is that is it possible to set BCs at a and b.

It depends what boundary conditions you want to set. If it is simply continuity of function and first derivative, you don't need to divide the region in parts. It's enough to include a and b in the mesh you supply to "pdepe". If you want to set jump conditions, it's not possible with "pdepe".

[Mohamad Keblawi]

It's the former. I just need to specify values if the first deriviative at a and b. You're saying include a and b in the mesh and then change the bcfun to somthing like this

[pl,ql,pr,qr,pa,qa,qb,pb] = bcfun(xl,ul,xr,ur,xa,ua,xb,ub,t)

would that be sufficient?

obviously xa and xb will be included in the mesh vector as well

[Torsten]

Please write down what conditions you want to impose at a and b.

[Mohamad Keblawi]

sure thing

where r and s are just some constants

[Torsten]

At interfaces a and b, you set conditions that couple the function values and possibly the derivatives to the left and to the right of the interface.

As noted above, a condition that is often used is the continuity of the function and its derivative over the interface. This condition can be handled by pdepe if you use a and b as grid points in your mesh.

Conditions like the one you state above make no sense at interfaces because they do not include a coupling of function values and/or derivatives on both sides of the interface. They are usually end conditions in the interval of integration.

[Mohamad Keblawi]

Gotcha, So if I can couple the my BCs at a and b to my left and right BCs then I'll be to put it in the pdepe bcfun. I appreiciate your help.

[Torsten]

So if I can couple the my BCs at a and b to my left and right BCs then I'll be to put it in the pdepe bcfun. I appreiciate your help.

No.

For pdepe, you can only prescribe boundary conditions at both ends of the integration interval. You cannot put additional boundary conditions in the function bcfun.

If you want to solve problems which are somehow differently defined in two or three regions, you can nonetheless use pdepe if over the separation points, you want to prescribe continuity of the solution and the flux. In this case, you only have to include the points separating the regions in your mesh vector. Changes to bcfun don't need to be made.

If you want to prescribe more difficult transmission conditions at the points that separate two regions (e.g. jump conditions), pdepe cannot be used.