I am working on a MATLAB assingment, where I try to work out water waves. The height of the waves is given by $u$. Now $x \in [-L,L]$, which to me is the position of the wave. The PDE is:
\begin{equation} \frac{\partial u}{\partial t} = - \frac{\partial}{\partial x} \left( \frac{\partial^{2}u}{\partial x^{2}} +3u^{2} \right). \end{equation}
The first thing I need to do is to discretize the spatial derivatives using centered differences. Now I really have no clue what they mean by this. I looked online, but no luck there either. As far as my English goes, I expect that we 'cut up' the time, thereby making time discreet, and than see how the function reacts to this. I guess I am trying to find the $u$ given an initial value for each position $x$ at each discrete time point $t_{n}$.
Does this make sense at all?