Analysis of a new implicit solver for a semiconductor model

TL;DR

This paper introduces a novel implicit solver for a simplified Boltzmann-Poisson system that simplifies previous methods by eliminating nesting, improves convergence with acceleration, and demonstrates efficiency through numerical comparisons.

Contribution

The paper presents a new fixed-point iterative solver for the Boltzmann-Poisson system that requires only one transport sweep per iteration, simplifying and potentially accelerating the solution process.

Findings

The new solver is contractive for a given electric potential.

Acceleration improves convergence rate in the drift-diffusion regime.

Numerical results show the new solver's efficiency compared to existing nested methods.

Abstract

We present and analyze a new iterative solver for implicit discretizations of a simplified Boltzmann-Poisson system. The algorithm builds on recent work that incorporated a sweeping algorithm for the Vlasov-Poisson equations as part of nested inner-outer iterative solvers for the Boltzmann-Poisson equations. The new method eliminates the need for nesting and requires only one transport sweep per iteration. It arises as a new fixed-point formulation of the discretized system which we prove to be contractive for a given electric potential. We also derive an accelerator to improve the convergence rate for systems in the drift-diffusion regime. We numerically compare the efficiency of the new solver, with and without acceleration, with a recently developed nested iterative solver.