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The Hermite Finite Volume Method with Global Conservation Law

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A Correction to this article was published on 28 February 2024

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Abstract

We construct a high-order (cubic) Hermite finite volume method (FVM-2L) with a two-layered dual strategy on triangular meshes, which possesses the conservation laws in both flux form and equation form. In particular, for problems with Dirichlet boundary conditions, the FVM-2L scheme preserves conservation laws on all triangles, whereas conservation properties may be lost on boundary dual elements by existing vertex-centered finite volume schemes. Theoretically, this is the first \(L^2\) result for the Hermite finite volume method on triangular meshes. Furthermore, the regularity requirement for the \(L^2\) theory of the FVM-2L scheme is reduced to \(u\in H^{k+1}\) (i.e. \(u\in H^{4}\)). While, as a comparison, all existing \(L^2\) results for high-order (\(k\ge 2\)) finite volume schemes require \(u\in H^{k+2}\) in the analysis. Finally, the conservation and convergence properties of the FVM-2L scheme are verified numerically for a selection of elliptic, linear elastic, Stokes, and heat conduction problems.

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  1. School of Mathematics, Jilin University, Changchun, 130012, China

    Xinyuan Zhang & Xiang Wang

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  2. Xiang Wang
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The original version of the article was revised: Table 1 was processed incorrectly. It has been corrected.

This work is supported in part by the National Natural Science Foundation of China (Nos. 12371396, 11701211) and the National Key Research and Development Program of China (Nos. 2020YFA0713602, 2022YFB3707301).

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Zhang, X., Wang, X. The Hermite Finite Volume Method with Global Conservation Law. J Sci Comput 98, 17 (2024). https://doi.org/10.1007/s10915-023-02407-y

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