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


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