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Positive Scharfetter-Gummel finite volume method for convection-diffusion equations on polygonal meshes

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  • Quenjel, El Houssaine

Abstract

In this paper, we develop and study a fully implicit positive finite volume scheme that allows an accurate approximation of the nonlinear highly anisotropic convection-diffusion equations on almost arbitrary girds. The key idea is to relate the oscillatory fluxes, including the convective ones, to the normal monotonic diffusive flux thanks to a technique used in the Scharfetter-Gummel discretizations. Then, we obtain a nonlinear two-point-like scheme with positive coefficients on primal and dual meshes. We check that the structure of the scheme naturally ensures the nonnegativity of the approximate solutions. We also establish energy estimates, which leads to a proof of existence of the numerical solutions. This analytical study is accompanied with a series of numerical results and simulations. They highlight the fulfillment of the discrete maximum principle, the optimal accuracy of our scheme, and its robustness with respect to the mesh and to high ratios of anisotropy.

Suggested Citation

  • Quenjel, El Houssaine, 2022. "Positive Scharfetter-Gummel finite volume method for convection-diffusion equations on polygonal meshes," Applied Mathematics and Computation, Elsevier, vol. 425(C).
    • Handle: RePEc:eee:apmaco:v:425:y:2022:i:c:s0096300322001552
    DOI: 10.1016/j.amc.2022.127071
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    Cited by:

    1. Quenjel, El Houssaine & Beljadid, Abdelaziz, 2023. "Node-Diamond approximation of heterogeneous and anisotropic diffusion systems on arbitrary two-dimensional grids," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 450-472.

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