# What is the rationale of second-order finite volume discretization?

When it comes to a second-order accurate finite volume discretization of Navier-Stokes equations, which one of the two following rationales is adopted?
1- Second-order accuracy is a direct consequence of how we eliminate higher order terms in the Taylor expansion of all the terms;
2- From the beginning, we assume that the dependent variable varies linearly between a pair of neighboring cell centers; with such an assumption, high-order terms won't appear at all.

• With #2, why would it be second-order accurate, not third-order or higher? Jun 24, 2021 at 17:07

1. Usually you speak of a $$n$$'th order (accurate) method if your Taylor truncation error is of order $$n+1$$. This means your approximation is accurate up to order $$n$$ terms, and your errors are of order $$n+1$$. However, in FVM methods you often have no easy way of obtaining the truncation error of your formulation, since you reconstruct the numerical fluxes $$F$$ based on some procedure, which are in turn based on the your reconstructed trace/edge values $$u_L, u_R$$.
2. I guess you are referring here to the linear reconstruction of the face values. This reconstruction is second order accurate since you have a first order approximation with truncation error $$\mathcal{O}(\Delta x^2)$$ on an interval of size $$\mathcal{O}(\Delta x)$$, resulting in an overall truncation order of $$\mathcal{O}(\Delta x^3)$$.