I am trying to write a finite difference scheme to solve numerically a 2-nd order non-linear equation :
$$\boxed{\frac{\partial h}{\partial t} = A\frac{\partial}{\partial x}\left(h^{3}\frac{\partial h}{\partial x}\right)}$$
With following boundary conditions :
- $h(x,t=0) = 0$
- $\frac{\partial h}{\partial x} = 0 \space \mbox{for} \space \lvert {x} \rvert = \infty $
I am attempting to write an implicit scheme like this :
$$\left[\frac{h_{i}^{n+1}-h_{i}^{n}}{\delta t}\right] = A \left[\left(h_{i}^{n}\right)^{3}\left(\frac{h_{i+1}^{n+1}-2h_{i}^{n+1}+h_{i-1}^{n+1}}{(\delta x)^{2}}\right)+3\left(h_{i}^{n}\right)^{2}\left(\frac{h_{i+1}^{n+1}-h_{i}^{n+1}}{\delta x}\right)^{2}\right]$$
By noting that : $$\frac{\partial}{\partial x}\left(h^{3}\frac{\partial h}{\partial x}\right) = \left(h^{3} \frac{\partial^{2}h}{\partial x^{2}} + 3h^{2}\left(\frac{\partial h}{\partial x}\right)^{2}\right)$$
How do you think I can write the matrix calculation in order to implement the problem numerically ? I am somehow disturbed by those $h_{i}^{n}$ terms...
Thank you in advance for your help.