An asymptotic-preserving scheme for the semiconductor Boltzmann equation toward the energy-transport limit

Abstract

We design an asymptotic-preserving scheme for the semiconductor Boltzmann equation which leads to an energy-transport system for electron mass and energy as mean free path goes to zero. As opposed to the classical drift-diffusion limit where the stiff collisions are all in one scale, new difficulties arise in the two-scale stiff collision terms because the simple BGK penalization [15] fails to drive the solution to the correct limit. We propose to set up a spatially dependent threshold on the penalization of the stiffer collision operator such that the evolution of the solution resembles a Hilbert expansion at the continuous level. Formal asymptotic analysis and numerical results confirm the efficiency and accuracy of our scheme.

Plasmonic HEMT Terahertz Transmitter based on the Dyakonov-Shur Instability: Performance Analysis and Impact of Nonideal Boundaries

“…In our analysis, we follow the methodology introduced in [27] to simulate the 2DEG as a 1D system and utilize the up-wind approach to stabilize the discretization of the system. Then, we use a homogeneous and uniform mesh to generalize to two dimensions for the EM solver [28][29][30]. In the Appendix, we provide the derivation of the finite-element discretization of the governing equations and the conditions for the system to be numerically stable.…”

On Deterministic Numerical Methods for the Quantum Boltzmann-Nordheim Equation. I. Spectrally Accurate Approximations, Bose-Einstein Condensation, Fermi-Dirac Saturation

“…The full extent of this convolution idea was used in [11], allowing to write a fast (spectral) method able to compute the full collision operator with the "reasonably low" numerical cost of O(N 5 log(N )) operations, where N is the number of unknowns in each velocity dimension, and with spectral accuracy. Extension of this method to the full inhomogeneous case, and to the asymptotic preserving setting was then done in the series of works [21,17,19] for the fermionic (electrons) case only. Nevertheless, these works only deal with the simplified 1-dimensional (1d) in x, 2-dimensional in v setting and use an approximation of the collision operator.…”

A stochastic asymptotic-preserving scheme for the bipolar semiconductor Boltzmann-Poisson system with random inputs and diffusive scalings

“…An efficient AP scheme in the high field regime was developed in [25]. The authors in [16] further study the semiconductor Boltzmann equation with a two-scale stiff collision operators, by taking into account different effects including the interactions between electrons and the lattice defects caused by ionized impurities [3]; they design and demonstrate the efficiency and accuracy of an asymptoticpreserving scheme that leads to an energy-transport system as mean free path goes to zero at a discretized level.…”