A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system (Q2719213)

The authors analyze a numerical method for capturing positive solutions of a nonlinear fourth order parabolic problem arising in macroscopic quantum semiconductor device modelling. Macroscopic models have the advantage of admitting a natural interpretation of the boundary conditions [\textit{R. Pinnau}, Appl. Math. Lett. 12, No. 5, 77-82 (1999; Zbl 0952.76100)]. After backward (implicit) Euler semidiscretization in the time variable, the authors prove solvability (being the solution such that the approximation of the electron density is strictly positive) first in one space dimension and then, under additional assumptions, in several space dimensions. The proof is based on an appropriate transformation of variables, variational formulations and compact Sobolev imbeddings in order to apply Leray-Schauder's fixed point theorem. A convergence theorem of the scheme is also proved in one space dimension. In the final section the authors present some numerical results for the simulation of the switching behavior of a resonant tunneling diode. The model studied in the paper has the especial interest of being evolutive.