Hi Christian

Thank you very much for your suggestion. Please correct me if I am wrong.

The insulated BC is ∫_ΓN grad(u)*n*ds where grad(u)*n=0; and it is what I have done. As you can see in the second picture, the concentration of u (the solution) is increasing since there is no flux in or out, and in addition, u does not have any space to diffuse.

What I am looking for is a BC that allows u to keep the same values as it is far from the BC, like u the first picture.

Please let me know if you have any idea.

Best regards

Sorry to say that your question is properly posed. First, you have to think about the physical problem you want to solve. That is the first step that must be translate in the correct set of equation and boundary condition that fulfill the physics. Then you can think about the translation of the PDE in the numerical model.

Now you wrote only about a generic Poisson equation but I don't understand what is the problem you are solving.

As example, prescribing homogeneous Neumann conditions (adiabatic walls for example in the heat problem) leave the solution determined up to an arbitrary constant.

On the other hand, Dirichlet conditions require to know the physical value at the boundaries.

Sometimes it is possible to suppose the a second normal derivative is disregardable but that depends on the problem.

Thus, please give the details of your problem.

Dear Filippo

Thank you for your suggestion. My question above is related to the following PDE shown in the picture; where f is Gaussian function representing a source that moves from the left to the right.

My real problem requires to simulate a huge domain where f scans totally the domain in lines. To save time and sources, I want to look at small portion of that domain, which leads to, I should have appropriate BC that does not change u when the source f enters and leaves the small domain. My first idea is Neumann BC saying grad(u)*n=0 or grad(u)*n = -grad(u)*n.

Besides, my real problem is simulate drift-diffusion equation used in semiconductor science, shown in picture 2.

If you have any idea, please let me know.

Best regards

Well, from a look at the original problem in figure 2 you see a coupled system of PDE, the time dependent equations are not of elliptic type.

Therefore, your simplified model of a single elliptic equation could model the steady-state with a known source term. But if you need to model the moving of the source term into the domain you can not do with this simple elliptic model. When f enters or leave a boundary it must have effect on u.

I found an interesting discussion about this problem.

Link: https://scicomp.stackexchange.com/questions/5471/open-boundary-conditions-with-the-advection-diffusion-equation

The paper mentioned in the discussion: Griffiths 1997 - THE ‘NO BOUNDARY CONDITION’ OUTFLOW BOUNDARY CONDITION

This kind of BC is classical for the transport equations, that is for hyperbolic equations where you have a clear direction of propagation of the characteristics.

But if you want to solve an elliptic problem you have no privileged directions. But, as I wrote before, I cannot see how you can simulate the movement of the source if you do not solve the original system of equations

Well, it seem that I can not have "open BC" for the Poisson equation. About the movement of the source, did you mean I should add the derivative of u over time to the Poisson equation?

Yes, this is a parabolic equation with a source term that changes position in time. It is a simple problem to solve numerically either implicitly or explicitly in time

I see. Thank you very much.

Dear Nguyen,

Would periodic BC help? Basically this would not change at all the behaviour of the solution close to the boundary.

On the other hand, the BC are defined by the practical context. E.g. if you want conservation of some quantity, you should have for the corresponding equation.

Good luck,

Sorin

Dea Sorin,

unfortunately the periodic BC.s imply that the source is periodically distributed everywhere into the domain adn that is a very different representation of the original physics.

Dear Filippo,

To be honest I don't agree. The BC's have nothing to do with the source.

OK, if you go for periodic BC you assume that the entire domain R^n is divided into hypercubes repeating periodically and that the whole model is periodic. In consequence, the behaviour of the solution is periodic. But, on the other hand, you are interested in what happens in one of the hypercube, so what happens with the source outside the hypercube of interest must not be of relevance.

Anyway, you know better what the precise context is.

Good luck,

Sorin

The original problem has a source moving into the domain that cannot be mimicked by a set of periodical sources. One should consider a moving box into the domain having a single source

Still, I was not saying that the source should behave periodically, or that a window/moving box should be moved. In fact, there is absolutely no need to do anything like this. The simplified problem is actually a standard parabolic one, with nothing moving in the domain. One may view f(x(t)) as a standard source term g(x, t) = f(x(t)). This is well understood in the literature.

I was suggesting that one can consider periodic solutions, Anyway, Nguyen knows better which BC's are more suited for his mathematical model.