We have a system of equations, wherein the BC of one PDE is coupled with the source term of another PDE.
We have a regular 2D unit grid in x and y.
There are two PDEs to be solved
- The first PDE (elliptic diffusion problem) is defined only at $y = 1$, acting along the x-axis (i.e. it acts in the x-direction and only along the top of the cartesian grid). This x-axis is discretized with a fixed grid-spacing, generating a finite number of nodes. Let this set of nodes/co-ordinates be represented by $‘X’$.
- The second PDE is a time-varying diffusion problem. This is defined only along the y-axis, but for all x-nodes (i.e. for all $‘X’$), where the 1st PDE is being solved.
PDE1:
$$\nabla.(S \nabla a) = f(\omega,a)$$ BC1: $\frac{\partial a}{\partial x} = 1$, at $x=1$ (Neumann)
BC2: $a = 0$, at $x=0$ (Dirichelet)
PDE2:
$$\frac{\partial B}{\partial t} = \nabla.( \left( \begin{array}{cc} 0 & 0 \\ 0 & D \end{array} \right) \nabla B)$$
BC1: $B = 0$, at $y=0$ for all $'X'$, i.e. along the bottom face
BC2: $\frac{\partial B}{\partial y} = g(\omega, a)$, at $y=1$ (Neumann)
$f$ and $g$ are linear functions in $\omega(x,y,t)$ and $a(x,y,t)$. $B$ is defined in the 2-D grid as $B(x,y,t)$.
Importantly, $$\omega=B_{y=1} \text{ for all } 'X'$$
i.e., the BC2 of the 2nd PDE couples with the Implicit Source Term of the 1st PDE along the top face of the Cartesian mesh.
How would one approach this problem in order to obtain a numerical solution via a FD/FV implementation?