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.