Numerical problems in semiconductor simulation using the hydrodynamic model: a second-order finite difference scheme

In this paper, a second-order Total Variation Diminishing (TVD) finite difference scheme of upwind type is employed for the numerical approximation of the classical hydrodynamic model for semiconductors proposed by Bløtekjær and Baccarani–Wordeman. In particular, the high-order hyperbolic fluxes are evaluated by a suitable extrapolation on adjacent cells of the first-order fluxes of Roe, while total variation diminishing is achieved by limiting the slopes of the discrete Riemann invariants using the so-called Flux Corrected Transport approach. Extensive numerical simulations are performed on a submicron nþ n nþ ballistic diode. The numerical experiments show that the spurious oscillations arising in the electron current are not completely suppressed by the TVD scheme, and can lead to serious numerical instabilities when the solution of the hydrodynamic model is non-smooth and the computational mesh is coarse. The accuracy of the numerical method is investigated in terms of conservation of the steady electron current. The obtained results show that the second-order scheme does not behave much better than a corresponding first-order one due to a poor performance of the slope limiters caused by the presence of local extrema of the Riemann invariant associated with the hyperbolic system.