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A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme

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Abstract

We propose a finite volume scheme for convection–diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter–Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.

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References

  1. Alt H.W., Luckhaus S., Visintin A.: On nonstationary flow through porous media. Annali di Matematica Pura ed Applicata 136(1), 303–316 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arimburgo, F., Baiocchi, C., Marini, L.D.: Numerical approximation of the 1-D nonlinear drift-diffusion model in semiconductors. In: Nonlinear kinetic theory and mathematical aspects of hyperbolic system (Rapallo, 1992). Ser. Adv. Math. Appl. Sci., vol. 9, pp. 1–10. World Sci. Publ., River Edge, NJ (1992)

  3. Brezzi F., Marini L.D., Pietra P.: Méthodes d’éléments finis mixtes et schéma de Scharfetter–Gummel. C. R. Acad. Sci. Paris Sér. I Math. 305(13), 599–604 (1987)

    MathSciNet  MATH  Google Scholar 

  4. Brezzi F., Marini L.D., Pietra P.: Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal. 26(6), 1342–1355 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brézis H.: Analyse fonctionnelle: théorie et applications. Masson, Paris (1983)

    MATH  Google Scholar 

  6. Carrillo J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147(4), 269–361 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carrillo J.A., Toscani G.: Asymptotic L 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49(1), 113–142 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chainais-Hillairet C., Filbet F.: Asymptotic behavior of a finite volume scheme for the transient drift-diffusion model. IMA J. Numer. Anal. 27(4), 689–716 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chainais-Hillairet C., Liu J.G., Peng Y.J.: Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis. M2AN 37(2), 319–338 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chainais-Hillairet C., Peng Y.J.: Convergence of a finite volume scheme for the drift-diffusion equations in 1-D. IMA J. Numer. Anal. 23, 81–108 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chainais-Hillairet C., Peng Y.J.: Finite volume approximation for degenerate drift-diffusion system in several space dimnesions. M3AS 14(3), 461–481 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Courant R., Isaacson E., Rees M.: On the solution of nonlinear hyperbolic differential equations by finite differences. Commun. Pure. Appl. Math. 5, 243–255 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eymard R., Fuhrmann J., Gärtner K.: A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems. Numer. Math. 102, 463–495 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eymard R., Gallouët T.: H-convergence and numerical schemes for elliptic problems. SIAM J. Numer. Anal. 41(2), 539–562 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of numerical analysis, vol. VII. Handb. Numer. Anal., vol. VII, pp. 713–1020. North-Holland, Amsterdam (2000)

  16. Eymard R., Gallouët T., Herbin R., Michel A.: Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92, 41–82 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eymard R., Hilhorst D., Vohralík M.: A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105(1), 73–131 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Il’in A.M.: A difference scheme for a differential equation with a small parameter multiplying the highest derivative. Math. Zametki 6, 237–248 (1969)

    MATH  Google Scholar 

  19. Jüngel A.: Numerical approximation of a drift-diffusion model for semiconductors with nonlinear diffusion. ZAMM 75(10), 783–799 (1995)

    Article  MATH  Google Scholar 

  20. Jüngel A.: Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors. Math. Models Methods Appl. Sci. 5(4), 497–518 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jüngel A., Pietra P.: A discretization scheme for a quasi-hydrodynamic semiconductor model. Math. Models Methods Appl. Sci. 7(7), 935–955 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lazarov R.D., Mishev I.D., Vassilevski P.S.: Finite volume methods for convection–diffusion problems. SIAM J. Numer. Anal. 33(1), 31–55 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Markowich P.A., Ringhofer C.A., Schmeiser C.: Semiconductor equations. Springer-Verlag, Vienna (1990)

    Book  MATH  Google Scholar 

  24. Markowich P.A.: The stationary semiconductor device equations, Springer edn. Computational Microelectronics, Vienna (1986)

    Google Scholar 

  25. Markowich P.A., Unterreiter A.: Vacuum solutions of the stationary drift-diffusion model. Ann. Scuola Norm. Sup. Pisa 20, 371–386 (1993)

    MathSciNet  MATH  Google Scholar 

  26. Scharfetter D.L., Gummel H.K.: Large signal analysis of a silicon Read diode. IEEE Trans. Elec. Dev. 16, 64–77 (1969)

    Article  Google Scholar 

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  1. Laboratoire de Mathématiques UMR 6620, CNRS, Université Blaise Pascal, Campus des Cézeaux, B. P. 80026, 63177, Aubière Cedex, France

    Marianne Bessemoulin-Chatard

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  1. Marianne Bessemoulin-Chatard
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Correspondence to Marianne Bessemoulin-Chatard.

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Bessemoulin-Chatard, M. A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme. Numer. Math. 121, 637–670 (2012). https://doi.org/10.1007/s00211-012-0448-x

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