Basically, for solving 1D flow equations, one uses the mass, momentum and energy equations (Also known as Euler equations). Say now I have $N$ elements in series, will I be using $3N$ equations?
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Not exactly. You still have your three Euler equations: $$ \partial_t\rho + \partial_x\rho v_x=0 $$ $$ \partial_t\left(\rho v_x\right)+\partial_x\left(\rho v_x^2+P\right)=0 $$ $$ \partial_te +\partial_x\left[\left(e+P\right)v_x\right]=0 $$ but what you are doing is solving the system of 3 equations $N$ times (once for each cell).
It will be worth your while to read these lecture notes on partial differential equations from Arizona State University.