A nonlinear correction finite volume scheme preserving maximum principle for diffusion equations with anisotropic and discontinuous coefficient

Yao Yu; Guanyu Xue; Yao Yu; Guanyu Xue

doi:10.3934/era.2025075

Electronic Research Archive

2025, Volume 33, Issue 3: 1589-1609. doi: 10.3934/era.2025075

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Research article

A nonlinear correction finite volume scheme preserving maximum principle for diffusion equations with anisotropic and discontinuous coefficient

Yao Yu ^{1
,
,},
Guanyu Xue ²

1.
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China
2.
School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China

Received: 20 January 2025 Revised: 09 March 2025 Accepted: 14 March 2025 Published: 19 March 2025

A nonlinear correction finite volume scheme preserving the discrete maximum principle (DMP) was presented to solve diffusion equations with anisotropic and discontinuous coefficients. It is well-known that existing cell-centered finite volume schemes for the diffusion problem with the general discontinuous coefficient often impose severe restrictions on the mesh-cell geometry to maintain the DMP. We proposed a nonlinear method for modifying the flux to obtain a new scheme which eliminated the requirement for nonnegative interpolation coefficients at the midpoint of cell-edge unknowns while still preserving the DMP. That is, our new scheme satisfied the DMP unconditionally and can be applied to the diffusion problem with the discontinuous coefficient on arbitrary distorted meshes. We then provided a priori estimation under a coercivity assumption and proved that the scheme satisfied the DMP. Numerical results were presented to demonstrate that our scheme can handle diffusion equations with anisotropic and discontinuous coefficients, satisfied the DMP, and, in some cases, outperformed existing schemes which preserved the DMP in terms of accuracy.
- maximum principle,
- nonlinear correction,
- finite volume scheme,
- discontinuous coefficient,
- diffusion equations
Citation: Yao Yu, Guanyu Xue. A nonlinear correction finite volume scheme preserving maximum principle for diffusion equations with anisotropic and discontinuous coefficient[J]. Electronic Research Archive, 2025, 33(3): 1589-1609. doi: 10.3934/era.2025075

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Abstract

A nonlinear correction finite volume scheme preserving the discrete maximum principle (DMP) was presented to solve diffusion equations with anisotropic and discontinuous coefficients. It is well-known that existing cell-centered finite volume schemes for the diffusion problem with the general discontinuous coefficient often impose severe restrictions on the mesh-cell geometry to maintain the DMP. We proposed a nonlinear method for modifying the flux to obtain a new scheme which eliminated the requirement for nonnegative interpolation coefficients at the midpoint of cell-edge unknowns while still preserving the DMP. That is, our new scheme satisfied the DMP unconditionally and can be applied to the diffusion problem with the discontinuous coefficient on arbitrary distorted meshes. We then provided a priori estimation under a coercivity assumption and proved that the scheme satisfied the DMP. Numerical results were presented to demonstrate that our scheme can handle diffusion equations with anisotropic and discontinuous coefficients, satisfied the DMP, and, in some cases, outperformed existing schemes which preserved the DMP in terms of accuracy.

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