Numerische Mathematik

, Volume 127, Issue 2, pp 365–396 | Cite as

Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation

  • Mario Bukal
  • Etienne Emmrich
  • Ansgar Jüngel


Structure-preserving numerical schemes for a nonlinear parabolic fourth-order equation, modeling the electron transport in quantum semiconductors, with periodic boundary conditions are analyzed. First, a two-step backward differentiation formula (BDF) semi-discretization in time is investigated. The scheme preserves the nonnegativity of the solution, is entropy stable and dissipates a modified entropy functional. The existence of a weak semi-discrete solution and, in a particular case, its temporal second-order convergence to the continuous solution is proved. The proofs employ an algebraic relation which implies the G-stability of the two-step BDF. Second, an implicit Euler and \(q\)-step BDF discrete variational derivative method are considered. This scheme, which exploits the variational structure of the equation, dissipates the discrete Fisher information (or energy). Numerical experiments show that the discrete (relative) entropies and Fisher information decay even exponentially fast to zero.

Mathematics Subject Classification (2000)

65M06 65M12 65M15 35Q40 82D37 


  1. 1.
    Barrett, J., Blowey, J., Garcke, H.: Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer. Math. 80, 525–556 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bleher, P., Lebowitz, J., Speer, E.: Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations. Commun. Pure Appl. Math. 47, 923–942 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bukal, M., Jüngel, A., Matthes, D.: A multidimensional nonlinear sixth-order quantum diffusion equation. Ann. Inst. H. Poincaré Anal. non linéaire 30, 337–365 (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Carrillo, J.A., Jüngel, A., Tang, S.: Positive entropic schemes for a nonlinear fourth-order equation. Discrete Contin. Dyn. Syst. B 3, 1–20 (2003)zbMATHGoogle Scholar
  5. 5.
    Chainais-Hillairet, C., Gisclon, M., Jüngel, A.: A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors. Numer. Methods Part. Differ. Eqs. 27, 1483–1510 (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chainais-Hillairet, C., Jüngel, A., Schuchnigg, S.: Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities (2013, Preprint). arXiv:1303.3791Google Scholar
  7. 7.
    Dahlquist, G.: \(G\)-stability is equivalent to \(A\)-stability. BIT 18, 384–401 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Degond, P., Méhats, F., Ringhofer, C.: Quantum energy-transport and drift-diffusion models. J. Stat. Phys. 118, 625–665 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Derrida, B., Lebowitz, J., Speer, E., Spohn, H.: Fluctuations of a stationary nonequilibrium interface. Phys. Rev. Lett. 67, 165–168 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Düring, B., Matthes, D., Milišić, J.-P.: A gradient flow scheme for nonlinear fourth order equations. Discrete Contin. Dyn. Syst. B 14, 935–959 (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Emmrich, E.: Error of the two-step BDF for the incompressible Navier–Stokes problem. M2AN Math. Model. Numer. Anal. 38, 757–764 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Emmrich, E.: Stability and error of the variable two-step BDF for semilinear parabolic problems. J. Appl. Math. Comput. 19, 33–55 (2005)Google Scholar
  13. 13.
    Emmrich, E.: Two-step BDF time discretization of nonlinear evolution problems governed by monotone operators with strongly continuous perturbations. Comput. Methods Appl. Math. 9, 37–62 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Furihata, D., Matsuo, T.: Discrete Variational Derivative Method. Chapman and Hall/CRC Press, Boca Raton (2010)CrossRefGoogle Scholar
  15. 15.
    Gianazza, U., Savaré, G., Toscani, G.: The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation. Arch. Ration. Mech. Anal. 194, 133–220 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Girault, V., Raviart, P.-A.: Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Mathematics. Springer, Berlin (1981)Google Scholar
  17. 17.
    Glitzky, A., Gärtner, K.: Energy estimates of continuous and discretized electro-reaction-diffusion systems. Nonlinear Anal. Theory Methods Appl. 70, 788–805 (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Grün, G.: On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions. Math. Comput. 72, 1251–1279 (2003)CrossRefzbMATHGoogle Scholar
  19. 19.
    Grün, G., Rumpf, M.: Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87, 113–152 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hill, A., Süli, E.: Approximation of the global attractor for the incompressible Navier–Stokes equations. IMA J. Numer. Anal. 20, 633–667 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Jüngel, A.: Transport Equations for Semiconductors. Lecture Notes in Physics. Springer, Berlin (2009)CrossRefGoogle Scholar
  22. 22.
    Jüngel, A., Matthes, D.: The Derrida–Lebowitz–Speer–Spohn equation: existence, non-uniqueness, and decay rates of the solutions. SIAM J. Math. Anal. 39, 1996–2015 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Jüngel, A., Milišić, J.-P.: A sixth-order nonlinear parabolic equation for quantum systems. SIAM J. Math. Anal. 41, 1472–1490 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Jüngel, A., Pinnau, R.: Global non-negative solutions of a nonlinear fourth-order parabolic equation for quantum systems. SIAM J. Math. Anal. 32, 760–777 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Jüngel, A., Pinnau, R.: A positivity preserving numerical scheme for a nonlinear fourth-order parabolic equation. SIAM J. Numer. Anal. 39, 385–406 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Jüngel, A., Pinnau, R.: Convergent semidiscretization of a nonlinear fourth order parabolic system. M2AN Math. Model. Numer. Anal. 37, 277–289 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Jüngel, A., Violet, I.: First-order entropies for the Derrida–Lebowitz–Speer–Spohn equation. Discrete Contin. Dyn. Syst. B 8, 861–877 (2007)CrossRefzbMATHGoogle Scholar
  28. 28.
    Kreth, H.: Time-discretisations for nonlinear evolution equations. In: Numerical Treatment of Differential Equations in Applications (Proc. Meeting, Math. Res. Center, Oberwolfach, 1977), pp. 57–63. Lecture Notes in Mathematics, vol. 679. Springer, Berlin (1978)Google Scholar
  29. 29.
    Moore, P.: A posteriori error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension. SIAM J. Numer. Anal. 31, 149–169 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Palais, R.S., Palais, R.A.: A Web Companion for Differential Equations, Mechanics, and Computation.
  31. 31.
    Westdickenberg, M., Wilkening, J.: Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations. M2AN Math. Model. Numer. Anal. 44, 133–166 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Willett, D., Wong, J.: On the discrete analogues of some generalizations of Gronwall’s inequality. Monatsh. Math. 69, 362–367 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Zhornitskaya, L., Bertozzi, A.: Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37, 523–555 (2000)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria
  2. 2.Department of Control and Computer Engineering, Faculty of Electrical Engineering and ComputingUniversity of ZagrebZagrebCroatia
  3. 3.Institute for MathematicsTechnical University of BerlinBerlinGermany

Personalised recommendations