# Finite difference scheme for unconfined aquifer equation

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.

• Assuming by $K$ you mean $k$, that's a forced nonlinear heat equation, and thus parabolic rather than elliptic. eqworld.ipmnet.ru/en/solutions/npde/npde1201.pdf – origimbo Jul 11 '17 at 20:25
• Yes, if $h>0$, then the equation has a Laplacian on the left hand side and a single time derivative on the right (if you apply the product rule to the derivatives of $h^2$). That makes it parabolic. – Wolfgang Bangerth Jul 11 '17 at 23:25