On the order of accuracy of finite-volume schemes on unstructured meshes

We consider finite-volume schemes for linear hyperbolic systems with constant coefficients on unstructured meshes. Under the stability assumption, they exhibit the convergence rate between p and p+1 where p is the order of the truncation error. Our goal is to explain this effect. The central point of our study is that the truncation error on (p+1) -th order polynomials has zero average over the mesh period. This condition is verified for schemes with a polynomial reconstruction, multislope finite-volume methods, 1-exact edge-based schemes, and the flux correction method. We prove that this condition is necessary and, under additional assumptions, sufficient for the (p+1) -th order convergence. Furthermore, we apply the multislope method to a high-Reynolds number flow and explain its accuracy.