A correction finite volume scheme preserving the discrete extremum principle for convection-diffusion equations on tetrahedral meshes

Fei Zhao; Chan Bi; Yao Yu; Fei Zhao; Chan Bi; Yao Yu

doi:10.3934/era.2026027

Electronic Research Archive

2026, Volume 34, Issue 1: 583-605. doi: 10.3934/era.2026027

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Research article

A correction finite volume scheme preserving the discrete extremum principle for convection-diffusion equations on tetrahedral meshes

Fei Zhao ^{1
,
,},
Chan Bi ¹,
Yao Yu ²

1.
College of Science, North China University of Technology, Beijing 100144, China
2.
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China

Received: 18 November 2025 Revised: 31 December 2025 Accepted: 08 January 2026 Published: 19 January 2026

In this paper, we present a nonlinear correction finite volume scheme that preserves the discrete extremum principle for convection-diffusion equations on tetrahedral meshes. The approximation of the diffusive flux is based on a nonlinear correction of the second-order linear flux, while the convection flux is approximated by the upwind scheme with second-order accuracy given by a Taylor expansion. In the construction of the new scheme, the requirement to represent auxiliary unknowns as convex combinations of primary unknowns can be removed, which greatly reduces the restrictions on diffusion coefficients and mesh geometry. Moreover, the obtained new scheme is cell-centered and satisfies local conservation. In addition, we theoretically testify that the numerical solution of the new scheme keeps the discrete extremum principle and verify it numerically. Numerical results also show that our scheme has almost second-order accuracy.
- convection-diffusion equation,
- extremum principle,
- tetrahedral meshes,
- second-order accuracy
Citation: Fei Zhao, Chan Bi, Yao Yu. A correction finite volume scheme preserving the discrete extremum principle for convection-diffusion equations on tetrahedral meshes[J]. Electronic Research Archive, 2026, 34(1): 583-605. doi: 10.3934/era.2026027

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Abstract

In this paper, we present a nonlinear correction finite volume scheme that preserves the discrete extremum principle for convection-diffusion equations on tetrahedral meshes. The approximation of the diffusive flux is based on a nonlinear correction of the second-order linear flux, while the convection flux is approximated by the upwind scheme with second-order accuracy given by a Taylor expansion. In the construction of the new scheme, the requirement to represent auxiliary unknowns as convex combinations of primary unknowns can be removed, which greatly reduces the restrictions on diffusion coefficients and mesh geometry. Moreover, the obtained new scheme is cell-centered and satisfies local conservation. In addition, we theoretically testify that the numerical solution of the new scheme keeps the discrete extremum principle and verify it numerically. Numerical results also show that our scheme has almost second-order accuracy.

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