On the order of accuracy of finite-volume schemes on unstructured meshes
<|reference_start|>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.<|reference_end|>
@article{bakhvalov2024on,
    title={On the order of accuracy of finite-volume schemes on unstructured meshes},
    author={Pavel Bakhvalov (1), Mikhail Surnachev (1) ((1) Keldysh Institute of
  Applied Mathematics)},
    journal={arXiv preprint arXiv:2404.04157},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.04157},
    primaryClass={math.NA cs.NA}
}