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I was reading through this document about FVM. I understood all up to the point where we have on page 15 the following

$$(\bar u^{n+1}_i - \bar u^n_i) \Delta x + \int^{t^{n+1}}_{t^n} f(u(x_{i+1/2},t))-f(u(x_{i-1/2},t)) dt = 0$$

How am I suppose to compute the integral from $t_n$ to $t_{n+1}$ of the flux?

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You can't. You need to approximate the integral via quadrature and or other approximations. See the following section in the document you cite.

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  • $\begingroup$ I see, but then, the approximation of the flux in the interval $[t_n, t_{n+1}]$ is done with values of $u$ at time $t_n$ only as in equation (2.49), how can this be right? $\endgroup$ – BRabbit27 Nov 12 '14 at 9:20
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    $\begingroup$ Well, it's an approximation. You can of course also use values of $u$ at time $t_{n+1}$, in which case you get an implicit time stepping scheme. $\endgroup$ – Wolfgang Bangerth Nov 13 '14 at 0:17
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Solving the Riemann problem at the interface gives you the values of the solution at this interface just after $t_n$. Then, the basic assumption is that this interface state remains constant during the whole time step, which is true for piece-wise constant reconstruction (i.e. the solution is assumed uniform in each cell), and as long as the CFL condition is met everywhere, i.e. characteristic waves coming from the other interfaces do not cross this interface in one time step. This property is "pleasant" from a physical point of view, at least for explicit time schemes.

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