For an unconfined aquifer we have this PDE for the water table position( of course after somehow making the original Boussinesq equation linearized ):
$$ \frac{\partial^2(h^2)}{\partial x^2} + \frac{\partial^2(h^2)}{\partial y^2} = \frac{2s}{k} \frac{\partial h}{\partial t} - \frac{2N}{k}$$
where $N$, $k$ and $s$ are constants .
Would you tell me what type mathematically it is? In respect to $h^2$ it seems elliptic, but physically it is not. Can we treat it and solve it like 2D heat equation? I need to solve it numerically via finite difference. Which FD scheme would you recommend?
I tried it with the simple explicit and it works OK and I'm aware of instability restrictions. but with methods like , ADE (alternating-direction explicit), because the second derivatives are in power 2 , it will lead to quadratic equation for each node in each time.