  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.
