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M. Bukal, E. Emmrich, A. Jüngel:
"Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation";
Numerische Mathematik, 127 (2014), 365 - 396.

Structure-preserving numerical schemes for a nonlinear parabolic fourthorder
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.

Siehe englisches Abstract.

Quantum drift-diffusion model; entropy dissipation

http://dx.doi.org/10.1007/s00211-013-0588-7