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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
Article

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.

Mathematics Subject Classification (2000)

65M06 65M12 65M15 35Q40 82D37 

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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

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