{"title":"Spectral Convergence of a Semi-discretized Numerical System for the Spatially Homogeneous Boltzmann Equation with Uncertainties","authors":"Liu Liu, Kunlun Qi","doi":"10.1137/24m1638483","DOIUrl":null,"url":null,"abstract":"SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 3, Page 812-841, September 2024. <br/> Abstract.In this paper, we study the Boltzmann equation with uncertainties and prove that the spectral convergence of the semi-discretized numerical system holds in a combined velocity and random space, where the Fourier spectral method is applied for approximation in the velocity space, whereas the generalized polynomial chaos (gPC)-based stochastic Galerkin (SG) method is employed to discretize the random variable. Our proof is based on a delicate energy estimate for showing the well-posedness of the numerical solution as well as a rigorous control of its negative part in our well-designed functional space that involves high-order derivatives of both the velocity and random variables. This paper rigorously justifies the statement proposed in Remark 4.4 of [J. Hu and S. Jin, J. Comput. Phys., 315 (2016), pp. 150–168].","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1137/24m1638483","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 3, Page 812-841, September 2024. Abstract.In this paper, we study the Boltzmann equation with uncertainties and prove that the spectral convergence of the semi-discretized numerical system holds in a combined velocity and random space, where the Fourier spectral method is applied for approximation in the velocity space, whereas the generalized polynomial chaos (gPC)-based stochastic Galerkin (SG) method is employed to discretize the random variable. Our proof is based on a delicate energy estimate for showing the well-posedness of the numerical solution as well as a rigorous control of its negative part in our well-designed functional space that involves high-order derivatives of both the velocity and random variables. This paper rigorously justifies the statement proposed in Remark 4.4 of [J. Hu and S. Jin, J. Comput. Phys., 315 (2016), pp. 150–168].
SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 3, Page 812-841, September 2024. 摘要.本文研究了具有不确定性的玻尔兹曼方程，并证明了半离散化数值系统的谱收敛性在速度空间和随机空间的组合中成立，其中傅立叶谱方法用于速度空间的逼近，而基于广义多项式混沌（gPC）的随机伽勒金（SG）方法用于随机变量的离散化。我们的证明基于一个微妙的能量估算，以显示数值解的好求解性，以及在我们精心设计的函数空间中对其负部分的严格控制，该函数空间涉及速度和随机变量的高阶导数。本文严格证明了[J. Hu and S. Jin, J. Comput. Phys., 315 (2016), pp.］
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