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

Autori
Bukal, Mario ; Emmrich, Etienne ; Jüngel, Ansgar

Izvornik
Numerische Mathematik (0029-599X) 127 (2014), 2; 365-396

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Derrida-Lebowitz-Speer-Spohn equation; discrete entropy-dissipation inequality; Fisher information; BDF time discretization; numerical convergence; discrete variational derivative method
(Derrida-Lebowitz-Speer-Spohn equation; discrete entropy-dissipation inequality; Fisher information; BDF time discretization; numerical convergence; discrete variational derivative method.)

Sažetak
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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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