# A fast implicit solver for semiconductor models in one space dimension

## TL;DR

This paper introduces a fast implicit solver for one-dimensional semiconductor models based on a simplified Boltzmann-Poisson system, utilizing a sweeping algorithm and acceleration techniques to improve computational efficiency.

## Contribution

It develops and compares multiple iterative solvers with acceleration schemes for efficient implicit discretization of semiconductor models.

## Key findings

- Accelerated iterative solvers significantly reduce convergence time.
- The sweeping algorithm effectively solves fixed-point problems in this context.
- Performance varies with electron mean-free-path, influencing solver choice.

## Abstract

Several different approaches are proposed for solving fully implicit discretizations of a simplified Boltzmann-Poisson system with a linear relaxation-type collision kernel. This system models the evolution of free electrons in semiconductor devices under a low-density assumption. At each implicit time step, the discretized system is formulated as a fixed-point problem, which can then be solved with a variety of methods. A key algorithmic component in all the approaches considered here is a recently developed sweeping algorithm for Vlasov-Poisson systems. A synthetic acceleration scheme has been implemented to accelerate the convergence of iterative solvers by using the solution to a drift-diffusion equation as a preconditioner. The performance of four iterative solvers and their accelerated variants has been compared on problems modeling semiconductor devices with various electron mean-free-path.