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I often hear that cell centred finite volume is second order accurate but at the same time I come across notions of high order FVM flux schemes. Is there a distinction between the two? If I were to use something like a first order upwind referencing vs something like a third order MUSCL would that make my entire FVM scheme first or third order accurate?

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    $\begingroup$ You can’t really know until you’ve done the math or measured the convergence rate. You really need/ought to do both. $\endgroup$
    – Bill Barth
    Oct 17, 2021 at 17:43
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    $\begingroup$ Typically, the order of accuracy of your scheme is the lowest order of accuracy of discretisation schemes you use for different terms and stabilisation. For example, if you use an nth-order scheme (n>1) for viscous fluxes but use the first-order stabilisation scheme, then the overall accuracy is only first-order. $\endgroup$
    – Chenna K
    Oct 18, 2021 at 19:37

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For methods that discretize simultaneously space and time, such as Lax-Wendroff or Upwind with Forward/Backward Euler, you can manually compute the truncation errors (for a sufficiently regular $u$). For instance, the former (LW) is both second order accurate in space and time, while the latter (Euler-Upwind) is only first order accurate.

To obtain overall higher schemes, it is common to split spatial and temporal discretization (resulting in the REA algorithm; see for instance this paper, or chapter 5 of my favorite FVM notes). For the well-known reconstruction of the trace(cell-face) values by means of piece-wise linear functions and slope limiters you are getting in principle a second order reconstruction. However, around local maxima and minima, the limiters cause the reconstruction to fall back to the $\mathcal{O}(\Delta x)$ case. Popular even higher reconstruction methods are ENO and WENO.

Then, however you need a time integrator. In general, standard RK solvers do not posess the desired TVD property, so you need some specially designed time integrators.

By choosing reconstruction and time integrator of certain order, you obtain in principle a FVM scheme of that particular order. In practice, however, you may observe a worse rate of convergence since the true solution is often discontinuous, rendering the Taylor-Expansion based way of computing truncation errors inapplicable. Also, as already mentioned above, the limiting step (to get the TVD property) might cause your method to approximate maxima and minima not as accurate as desired.

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