Central differencing accuracy in FVM

Question

I've read that central differencing is second order accurate to interpolate face center values when using the finite volume method. I'm not sure why that's the case when, in general, the center of two neighboring polyhedron and the center of their common face do not lie on a straightline. Is it just assumed that the volumes are sufficiently close to forming a lattice that they tend to be aligned anyway, or is there something deeper going on?

Answer 1

The Central Differencing Scheme is always second-order by the truncation error in the Taylor series. Technical books recommend it when diffusion plays a big role, Pèclet number below 2 in steady problems; but it useable if it rearranges as

$$\frac{\phi_{i+1} - \phi_{i-1}}{2\Delta x} = \frac{\phi_{i} - \phi_{i-1}}{\Delta x} + \frac{\Delta x}{2}\frac{\phi_{i+1} - 2\phi_i + \phi_{i-1}}{\Delta x^2},$$

where the second term, named anti-diffusion, is computed in a previous level iteration. This correction technique can be a class of deferred method.

On the other hand, the scheme can still be applied to a polyhedron as follows:

$$\phi_{i+\frac{1}{2}} = \frac{\phi_{i}+\phi_{i+1}}{2} + \frac{\nabla\phi_i\cdot\mathbf{r}_i + \nabla\phi_{i+1}\cdot\mathbf{r}_{i+1}}{2},$$

where the $\nabla\phi$'s are the gradients in each cell; the $\mathbf{r}$'s, the vector from the cell centroid toward the face centroid.