Open boundary conditions with the advection-diffusion equation

Question

Following on from my previous equation I'm would like to apply open boundary condition to the advection-diffusion equation (with reaction term),

$$ \frac{\partial \phi}{\partial t} = \frac{\partial}{\partial x}\left( D\frac{\partial \phi}{\partial x} + \boldsymbol{v(x)} \phi \right) + S(x,t) .$$

I define "open" as meaning a boundary which allows unimpeded transport whether it be by diffusion or drift.

I'm unsure how to mathematically state this problem. Would I just impose that the open boundary take a Dirichlet boundary condition where the fixed by the initial conditions? Moreover, this would define a node where the value never changes. This doesn't seem right somehow.

Answer 1

That's not a simple question in my opinion. It is like if you were doing an experiment in a box and you just want to simulate the center of it. But obviously, the solution in the center of the box depends on what happens in the rest of the box. That is probably why it is a difficult problem to state.

I have no definite answer to this question, but I still have some clues, hopefully you will find a suitable solution. (I'm not an expert in any of these solutions)

• A first idea is to model the remaining part of the "box" with a reduced order model. For example, you could plug an ODE for the two extremities of your domain with are supposed to represent the behaviour of the rest of the box. This is a technique that is used sometimes to "end" vessels in blood flow simulations + You can easily imagine that the problem is well posed. You can also probably model incoming waves. It applies to FEM, probably to FD as well. - You have to find out the ODE if not found in the litterature.
• A second option is to used absorbing boundary conditions. These are tricky conditions that are supposed to absorb the information that is going out of the domain (also used sometimes for blood flows). If you google for "absorbing boundary conditions advection diffusion", you should find some papers about it + It (is supposed to) makes the effect you are looking for. - I think that it was developed for the FEM, so I don't know if it applies to FD.
• The third option I see is the one proposed by Bill Barth, which is a "no boundary" condition. It consists in prolongating the PDE up to the boundary and do not impose anything. + In a FEM context, it's easy to implement and works "surprisingly" well (I did try them for fluid flows, in a slightly different context). - I think that it's difficult to show that the problem is well posed (even if the discrete problem is). It applies to FEM, not sure for FD.

Hope it helps! (and good luck)

Answer 2

The best method I know of for this problem is by Griffiths.

Answer 3

The exact boundary condition for this case is not local. Rather, it is of the form $$ u(x,t)|_\Gamma = \int_\Gamma \int_{-\infty}^t Q(x,t,y,s) u(y,s) \; ds \; dy, $$ i.e., it is a (spatially and temporally) nonlocal operator with a kernel $Q$. In some interesting cases, you can write this operator as a pseudo-differential operator and only take the first few terms to get something local. However, these will only be approximations.