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.

翻译：我们考虑常系数线性双曲系统在非结构网格上的有限体积格式。在稳定性假设下，其收敛速度介于$p$和$p+1$之间，其中$p$为截断误差的阶数。我们的目标是解释这一现象。研究的核心在于，$(p+1)$阶多项式上的截断误差在网格周期内平均值为零。该条件已通过多项式重构、多斜率有限体积法、1-精确边基格式以及通量修正方法得到验证。我们证明该条件是必要的，且在附加假设下是$(p+1)$阶收敛的充分条件。此外，我们将多斜率方法应用于高雷诺数流动，并解释其精度特性。