# need help in discretizing the fluid equations

I have the two fluid equations:

$$\frac{\partial \rho}{\partial t}$$ = - $$\frac{\partial (\rho v)}{\partial x}$$

$$\frac{\partial (\rho v)}{\partial t}$$ = - $$\frac{\partial (\rho v^2 + p)}{\partial x} + \rho g$$

I need to find a discretized solution for $$\rho, v$$ so that it could be solved numerically, meaning I need to express $$\rho_m^{n+1}$$ and $$v_m^{n+1}$$ where m is the space index and n the time index. The solution needs to be accurate to second order, equivalent to the Lax-Wendroff method.

I managed to find $$\rho_m^{n+1}$$ but I am stuck with $$v_m^{n+1}$$. I wrote:

$$(\rho v)(x,t+\Delta t)$$ = $$(\rho v)(x,t)$$+$$\Delta t \frac{\partial (\rho v)}{\partial t}$$ + $$0.5\Delta t ^2 \frac{\partial^2 (\rho v)}{\partial t^2}$$

The problem is with the third term, I am stuck with a time derivative even after using the given equations. What can I do?

thanks in advance