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

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Abstract

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.

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Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 , Vienna, Austria

    Mario Bukal & Ansgar Jüngel

  2. Department of Control and Computer Engineering, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 , Zagreb, Croatia

    Mario Bukal

  3. Institute for Mathematics, Technical University of Berlin, Straße des 17. Juni 136, 10623 , Berlin, Germany

    Etienne Emmrich

Authors
  1. Mario Bukal
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  2. Etienne Emmrich
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  3. Ansgar Jüngel
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Correspondence to Ansgar Jüngel.

Additional information

M. Bukal and A. Jüngel acknowledge partial support from the Austrian Science Fund (FWF), Grants P20214, P22108, and I395, and the Austrian-French Project of the Austrian Exchange Service (ÖAD). M. Bukal is currently supported by the European Communities Seventh Framework Programme under Grant 285939 (ACROSS).

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Bukal, M., Emmrich, E. & Jüngel, A. Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation. Numer. Math. 127, 365–396 (2014). https://doi.org/10.1007/s00211-013-0588-7

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  • DOI: https://doi.org/10.1007/s00211-013-0588-7

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