On the Order of Accuracy of Edge-Based Schemes: a Peterson-Type Counter-Example

Abstract

Numerical schemes for the transport equation on unstructured meshes usually exhibit the convergence rate $p \in [k, k+1]$, where k is the order of the truncation error. For the discontinuous Galerkin method, the result $p = k+1/2$ is known, and the example where the convergence rate is exactly $k+1/2$ was constructed by Peterson (SIAM J. Numer. Anal. 28: 133–140, 1991) for $k=0$ and $k=1$. For finite-volume methods with $k \geqslant 1$, there are no theoretical results for general meshes. In this paper, we consider three edge-based finite-volume schemes with $k=1$, namely the Barth scheme, the Luo scheme, and the EBR3. For a special family of meshes, under stability assumption we prove the convergence rate $p=3/2$ for the Barth scheme and $p=5/4$ for the other ones. We also present a Peterson-type example showing that the values $3/2$ and $5/4$ are optimal.

Keywords

Finite-volume method / Edge-based scheme / Superconvergence / 65M08 / 65M12 / 65M15

References

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Funding

Russian Science Foundation(22-11-00199)

RIGHTS & PERMISSIONS

Shanghai University