A Positivity-Preserving Numerical Scheme for a Nonlinear Fourth Order Parabolic System

A positivity-preserving numerical scheme for a fourth order nonlinear parabolic system arising in quantum semiconductor modeling is studied. The system is numerically treated by introducing an additional nonlinear potential and a subsequent semidiscretization in time. The resulting sequence of nonlinear second order elliptic systems admits at each time level strictly positive solutions, which is proved by an exponential transformation of variables. The stability of the scheme is shown and convergence is proved in one space dimension. Numerical results concerning the switching behavior of a resonant tunneling diode are presented.