Based on the Navier-Stokes equations and a few parameterizations, the horizontal steady-state wind u(z)$u(z)$ within a forest of height H$H$ satisfies:
$a(\frac{du}{dz})^2 + b\frac{du}{dz} \frac{d^2u}{dz^2} + cu +d\frac{du}{dz} + eu^2 + f= 0$, for $0<z<H$. $$ a\Big(\frac{du}{dz}\Big)^{\!2} + b\frac{du}{dz} \frac{d^2u}{dz^2} + cu +d\frac{du}{dz} + eu^2 + f= 0\:\text{ for }0<z<H. $$
The coefficients a$a$ to f$f$ vary with the altitude z$z$ and are given initially (we can differentiable and integrate them as many times as needed).
At
At ground level: $u(z = 0) = 0$, $\frac{du}{dz}(z=0) = 0$.
At $$ u|_{z = 0} = 0, \quad\frac{du}{dz}\Big|_{z=0} = 0. $$ At canopy top: $u(z = H) = U_H$, $\frac{du}{dz}(z=H) = K$ (constant)
I $$ u|_{z = H} = U_H, \quad\frac{du}{dz}\Big|_{z=H} = K\text{ (constant)} $$ I am trying to solve this equation for u(z)$u(z)$ using a finite difference scheme, it would be great if someone could help me:
1/ Are Finite Differences even a good approach for this kind of problem ?
2/ If I rewrite the equation using the classical expressions $\frac{du}{dz} = \frac{u_{n+1}-u_{n-1}}{2h}, \frac{d^2u}{dz^2} =$ etc... I obtain square terms like $u_{i+1}u_{i-1}$ and I do not how what to do from there.
3/ I do not know how to use the Newton method or the Picard method correctly, is there a better way to rewrite the equation ? Using variables like $v = \frac{du}{dz}$ for example ?
Are Finite Differences even a good approach for this kind of problem ?
If I rewrite the equation using the classical expressions $\frac{du}{dz} = \frac{u_{n+1}-u_{n-1}}{2h}, \frac{d^2u}{dz^2} =$ etc... I obtain square terms like $u_{i+1}u_{i-1}$ and I do not how what to do from there.
I do not know how to use the Newton method or the Picard method correctly, is there a better way to rewrite the equation ? Using variables like $v = \frac{du}{dz}$ for example ?
At that point, I am not even sure if I am missing something obvious or if this is a really hard problem, any help would be greatly appreciated.
Thanks
