# What determines the order of a finite volume scheme?

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?

• You can’t really know until you’ve done the math or measured the convergence rate. You really need/ought to do both. Oct 17, 2021 at 17:43
• 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. Oct 18, 2021 at 19:37

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.