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My question is about how do we discretize the equations in the conservative form using finite difference method.

I'm trying to solve Euler equations in conservative form. $$ \frac{\partial u}{\partial t}+\frac{\partial F(u)}{\partial x}=0 $$

I'm confused with the following two approaches. $$ \frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}+\frac{F_{i+\Delta x/2}^n-F_{i-\Delta x/2}^n}{\Delta x}=0 $$

and the other one is $$ \frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}+\frac{F_{i+1}^n-F_{i-1}^n}{2\Delta x}=0 $$

What is the right way of discretization and how are they different. It would be really helpful if you could point me out to some literature. I have been looking all over.

NOTE:(Answer) I dont have enough rights to add an answer, so I'm adding it here along with the question.

After a bit of research, I figured out that Case: 1 $$ \frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}+\frac{F_{i+\Delta x/2}^n-F_{i-\Delta x/2}^n}{\Delta x}=0 $$ can be used while incorporating TVD or WENO. Where as the second case is pretty straight forward for generic discretizations.

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    $\begingroup$ Could you give a little more details? What grids are used in the first case? How do these methods compare to the Lax-Wendroff schemes? $\endgroup$ – LutzL Jan 6 at 23:02

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