Implicit-Explicit Runge-Kutta schemes for the Boltzmann-Poisson system for semiconductors

TL;DR

This paper introduces Implicit-Explicit Runge-Kutta schemes tailored for the multi-scale Boltzmann-Poisson system in semiconductors, effectively handling kinetic, intermediate, and diffusive regimes with high accuracy and stability.

Contribution

The paper develops high-order IMEX Runge-Kutta schemes that remain stable and accurate across multiple regimes of the semiconductor Boltzmann equation, overcoming stiffness issues.

Findings

Schemes accurately capture different physical regimes

Methods avoid parabolic stiffness in diffusive limit

Numerical results demonstrate high performance

Abstract

In this paper we develop a class of Implicit-Explicit Runge-Kutta schemes for solving the multi-scale semiconductor Boltzmann equation. The relevant scale which characterizes this kind of problems is the diffusive scaling. This means that, in the limit of zero mean free path, the system is governed by a drift-diffusion equation. Our aim is to develop a method which accurately works for the different regimes encountered in general semiconductor simulations: the kinetic, the intermediate and the diffusive one. Moreover, we want to overcome the restrictive time step conditions of standard time integration techniques when applied to the solution of this kind of phenomena without any deterioration in the accuracy. As a result, we obtain high order time and space discretization schemes which do not suffer from the usual parabolic stiffness in the diffusive limit. We show different numerical results which permit to appreciate the performances of the proposed schemes.