These notes are an introduction to fluctuating hydrodynamics (FHD) and the formulation of numerical schemes for the resulting stochastic partial differential equations (PDEs). Fluctuating hydrodynamics was originally introduced by Landau and Lifshitz as a way to put thermal fluctuations into a continuum framework by including a stochastic forcing to each dissipative transport process (e.g., heat flux). While FHD has been useful in modeling transport and fluid dynamics at the mesoscopic scale, theoretical calculations have been feasible only with simplifying assumptions. As such there is great interest in numerical schemes for Computational Fluctuating Hydrodynamics (CFHD). There are a variety of algorithms (e.g., spectral, finite element, lattice Boltzmann) but in this introduction we focus on finite volume schemes. Accompanying these notes is a demonstration program in Python available on GitHub.

翻译：本讲义旨在介绍涨落流体力学（FHD）及其对应随机偏微分方程（PDEs）的数值格式构建。涨落流体力学最初由朗道和栗弗席兹提出，其核心思想是通过在每个耗散性输运过程（例如热通量）中引入随机强迫项，将热涨落纳入连续介质框架。尽管FHD在介观尺度的输运和流体动力学建模中具有重要价值，但理论计算仅在简化假设下才可行。因此，计算涨落流体力学（CFHD）的数值方法备受关注。现有多种算法（如谱方法、有限元法、格子玻尔兹曼法），但本导论将重点阐述有限体积格式。随附讲义提供基于Python的演示程序，可在GitHub获取。