Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling

Liu Liu

doi:10.3934/krm.2018044

Article Contents

2018, Volume 11, Issue 5: 1139-1156. Doi: 10.3934/krm.2018044

This issue Previous Article A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation Next Article Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit

Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling

Liu Liu

Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin, 53706, USA

Received: June 2017

Revised: October 2017

Published: May 2018

The author was supported in part by Prof. Shi Jin’s NSF grants DMS-1522184 and DMS-1107291: RNMS KI-Net.

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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 [15], 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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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