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Spectral convergence of a semi-discretized numerical system for the spatially homogeneous Boltzmann equation with uncertainties

Liu Liu, Kunlun Qi

TL;DR

This document documents the SIAM LaTeX online-only class and its associated BibTeX style, detailing class options, front matter, cross-referencing, mathematics, theorem-like environments, tables, figures, algorithms, and sections. It also covers supplementary materials, templates, and metadata fields (DOI, URL, eprint, etc.), plus guidelines for customizing appearance and bookmarks. The key contributions are a comprehensive set of templates (ex_article, ex_shared, ex_supplement), automatic linking features, and updated practices for modern SIAM publishing, including persistent identifiers and software citations. Its practical impact lies in streamlining the preparation of SIAM manuscripts with consistent formatting, accessible referencing, and cohesive integration of supplementary materials.

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, J. Hu and S. Jin, J. Comput. Phys., 315 (2016), pp. 150-168].

Spectral convergence of a semi-discretized numerical system for the spatially homogeneous Boltzmann equation with uncertainties

TL;DR

This document documents the SIAM LaTeX online-only class and its associated BibTeX style, detailing class options, front matter, cross-referencing, mathematics, theorem-like environments, tables, figures, algorithms, and sections. It also covers supplementary materials, templates, and metadata fields (DOI, URL, eprint, etc.), plus guidelines for customizing appearance and bookmarks. The key contributions are a comprehensive set of templates (ex_article, ex_shared, ex_supplement), automatic linking features, and updated practices for modern SIAM publishing, including persistent identifiers and software citations. Its practical impact lies in streamlining the preparation of SIAM manuscripts with consistent formatting, accessible referencing, and cohesive integration of supplementary materials.

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, J. Hu and S. Jin, J. Comput. Phys., 315 (2016), pp. 150-168].
Paper Structure (29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 14 more sections

Key Result

Theorem 6.1

\newlabelthm:mvt0 Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure 1: Example figure using external image files.
  • Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

  • Theorem 6.1: Mean Value Theorem
  • Corollary 6.2
  • Proof 1
  • Claim 6.3
  • Proof 2: Proof of main theorem