# Uniform Spectral Convergence of the Stochastic Galerkin Method for the Linear Semiconductor Boltzmann Equation with Random Inputs and Diffusive Scalings

## TL;DR

This paper proves uniform spectral convergence of a stochastic Galerkin method applied to the linear semiconductor Boltzmann equation with random inputs, improving understanding of solution regularity and numerical accuracy in diffusive regimes.

## Contribution

The paper provides a sharper regularity estimate of the solution, leading to uniform spectral convergence results for the stochastic Galerkin method under diffusive scaling.

## Key findings

- Exponential decay of the solution towards local equilibrium.
- Uniform spectral convergence of the stochastic Galerkin method.
- Sharper regularity estimates in the random space.

## Abstract

In this paper, we study the generalized polynomial chaos (gPC) based stochastic Galerkin method for the linear semiconductor Boltzmann equation under diffusive scaling and with random inputs from an anisotropic collision kernel and the random initial condition. While the numerical scheme and the proof of uniform-in-Knudsen-number regularity of the distribution function in the random space has been introduced in [Jin-Liu-16'], the main goal of this paper is to first obtain a sharper estimate on the regularity of the solution-an exponential decay towards its local equilibrium, which then lead to the uniform spectral convergence of the stochastic Galerkin method for the problem under study.