I am trying to solve three equations which are of the form shown below numerically:
$\frac{\partial v}{\partial t}=f(v,y)$
$\frac{\partial}{\partial t}\left(vy\right)=f(v,y,z)$
$\frac{\partial}{\partial t}\left(vy^{2}\right)+\frac{\partial}{\partial t}\left(z\right)=f(v,y,z)$
where $v$, $y$ and $z$ are all functions of time ($t$) and space ($x$).
I know the initial values of the variables. I am desperate to know which numerical method to use to solve them. I was thinking of explicit Euler but I think that would not be possible due to the non-linearity cause by the $y^2$ term. I will be coding in Matlab. Would I need to linearize the equation and how?
Thanks very much.