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A structure-preserving scheme for computing effective diffusivity and anomalous diffusion phenomena of random flows

Tan Zhang, Zhongjian Wang, Jack Xin, Zhiwen Zhang

TL;DR

This document provides a comprehensive guide to the SIAM LaTeX style, detailing class options, front matter formatting, cross-referencing, math and theorem environments, tables, figures, algorithms, sections, supplemental materials, and bibliographic features. It describes automated hyperlinking via cleveref and hyperef, as well as support for PDF bookmarks, DOIs, arXiv preprints, article IDs, and software citations. The contribution lies in offering a cohesive template and updated conventions (including template files and change logs) to ensure consistent, publication-ready formatting for SIAM submissions and supplements. The practical impact is improved formatting fidelity, easier manuscript preparation, and reproducible referencing across main articles and supplementary materials.

Abstract

This paper aims to investigate the diffusion behavior of particles moving in stochastic flows under a structure-preserving scheme. We compute the effective diffusivity for normal diffusive random flows and establish the power law between spatial and temporal variables for cases with anomalous diffusion phenomena. From a Lagrangian approach, we separate the corresponding stochastic differential equations (SDEs) into sub-problems and construct a one-step structure-preserving method to solve them. Then by modified equation systems, the convergence analysis in calculating the effective diffusivity is provided and compared between the structure-preserving scheme and the Euler-Maruyama scheme. Also, we provide the error estimate for the structure-preserving scheme in calculating the power law for a series of super-diffusive random flows. Finally, we calculate the effective diffusivity and anomalous diffusion phenomena for a series of 2D and 3D random fields.

A structure-preserving scheme for computing effective diffusivity and anomalous diffusion phenomena of random flows

TL;DR

This document provides a comprehensive guide to the SIAM LaTeX style, detailing class options, front matter formatting, cross-referencing, math and theorem environments, tables, figures, algorithms, sections, supplemental materials, and bibliographic features. It describes automated hyperlinking via cleveref and hyperef, as well as support for PDF bookmarks, DOIs, arXiv preprints, article IDs, and software citations. The contribution lies in offering a cohesive template and updated conventions (including template files and change logs) to ensure consistent, publication-ready formatting for SIAM submissions and supplements. The practical impact is improved formatting fidelity, easier manuscript preparation, and reproducible referencing across main articles and supplementary materials.

Abstract

This paper aims to investigate the diffusion behavior of particles moving in stochastic flows under a structure-preserving scheme. We compute the effective diffusivity for normal diffusive random flows and establish the power law between spatial and temporal variables for cases with anomalous diffusion phenomena. From a Lagrangian approach, we separate the corresponding stochastic differential equations (SDEs) into sub-problems and construct a one-step structure-preserving method to solve them. Then by modified equation systems, the convergence analysis in calculating the effective diffusivity is provided and compared between the structure-preserving scheme and the Euler-Maruyama scheme. Also, we provide the error estimate for the structure-preserving scheme in calculating the power law for a series of super-diffusive random flows. Finally, we calculate the effective diffusivity and anomalous diffusion phenomena for a series of 2D and 3D random fields.
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