I'm trying to use finite difference to explicitly discretisize the following system:
$\frac{{\partial \rho }}{{\partial t}} = - \frac{{\partial \rho }}{{\partial x}}u - \rho \frac{{\partial u}}{{\partial x}}$
$\frac{{\partial u}}{{\partial t}} = - u\frac{{\partial u}}{{\partial x}} - \frac{\partial }{{\partial x}}\left[ {\frac{1}{{\sqrt \rho }}\frac{{{\partial ^2}(\sqrt \rho )}}{{\partial {x^2}}}} \right]$
where:
$\rho (x,t)\,\,\,,\,\,\,u(x,t)$
But i'm having difficulties since the $\rho$ time derivative is dependent on $u$ as well! I don't know which time step of $u$ to insert (or vice-versa for the $u$ time derivative).
Can it be done with finite differences ?
Thanks !