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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
  • $\begingroup$ 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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