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

doi:10.1007/s42967-023-00292-8

Abstract

Abstract: Numerical schemes for the transport equation on unstructured meshes usually exhibit the convergence rate p ∈ [k, k + 1], where k is the order of the truncation error. For the discontinuous Galerk in 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≥1, there are no theoretical results for general meshes. In this paper, we consider three edge-based finitevolume 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 Petersontype example showing that the values 3∕2 and 5∕4 are optimal.

Key words: Finite-volume method, Edge-based scheme, Superconvergence

CLC Number:

Pavel Bakhvalov, Mikhail Surnachev. On the Order of Accuracy of Edge-Based Schemes: a Peterson-Type Counter-Example[J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 372-391.

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References

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