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Given a standard advection equation, we write the update as $$ q_i^{n+1}=q_i^n+\frac{\Delta t}{\Delta x}\left(F_i^{n+1/2}-F_{i+1}^{n+1/2}\right) $$ with $F_i^{n+1/2}=F\left(q_i^n,\,q_i^*\right)$ and $q^*_i$ a predicted value, other variables take their normal meaning.

It is possible to reconstruct what numeric value $F_i^{n+1/2}$ possessed if I know $q^n_i$ and $q^{n+1}_i$? Or is it once I deallocate the array F, that information is lost to me?

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    $\begingroup$ This question too little information. You don't say for what the equation provides an update, what $F(\cdot)$ is, and in particular you don't say anything about any array F and how it relates to the formulas. You need to be more specific in your question. $\endgroup$ – Wolfgang Bangerth Jan 31 '14 at 4:02
  • $\begingroup$ @KyleKanos: I agree with Paul here. I don't think a personal attack was intended, nor any retribution on your part called for. Most members of the community want to see questions clarified so that they can be better answered. I am going to delete your comment because it is not constructive. Although I think I know what you mean by an advection equation (I'd assume a system of advecting passive scalars), it doesn't hurt to list briefly the equation and nomenclature. (Even to say that you're using the nomenclature from LeVeque is probably fine.) $\endgroup$ – Geoff Oxberry Feb 1 '14 at 0:44
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    $\begingroup$ @GeoffOxberry: The question is clear: I have an advection equation (of which there is only one equation dubbed advection, so there cannot possibly be any confusion as to what it is I'm talking about) and want to know if I can reconstruct the flux values $F_i$ from the new, $q^{n+1}$, and old, $q^n$, values of the variable I am advecting. There is zero need and zero capability to clarify any further. $\endgroup$ – Kyle Kanos Feb 1 '14 at 1:54
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    $\begingroup$ I'm curious... How is your $q_i^*$ predicted? Is $q_i^*=q_i^{n+1}$? The way your notation is written, it's unclear if knowledge of $q_i^{n+1}$ helps determine the value of $F(q_i^n,q_i^*)$ at all. For this reason, it might help to write out what you understand as the formula for $F(q_i^n,q_i^*)$. $\endgroup$ – Paul Feb 1 '14 at 7:18
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You could solve a linear system in terms of the $F_{i}^{n+1/2}$.

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I am somewhat confused by this question. Why are you evaluating the fluxes at a midpoint of the time step instead of evaluating them at $t = t_{n}$ or $t = t_{n+1}?

From what I have seen, fluxes are generally using data at the same time step to produce a value $q^{*}$ at the interface of two elements, where $q^{*}$ is the solution to the Riemann problem generated at the interface. This $q^{*}$ is then used as the variable to evaluate the flux function at, at this interface.

For nonlinear problems, finding $q^{*}$ is difficult and many people opt to use approximate solutions to $F(q^{*})$ via linearization about $\hat{q}$ which is usually some function that generates some sort of average of the right and left values of $q$ at the interface, or other fancier approaches to approximating the correct flux value at the interface.

Whether you are aiming for an explicit or implicit scheme, the approach is the same (at least in my experience), even if the value $q^{*}$ ends up helping create a coupled system in the implicit scheme case. I will recognize that I still have plenty to learn, so I definitely don't know everything about how to approach these problems, just the ways I have been taught.

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