An Introduction to Computational Fluctuating Hydrodynamics

These notes are an introduction to fluctuating hydrodynamics (FHD) and theformulation of numerical schemes for the resulting stochastic partialdifferential equations (PDEs). Fluctuating hydrodynamics was originallyintroduced by Landau and Lifshitz as a way to put thermal fluctuations into acontinuum framework by including a stochastic forcing to each dissipativetransport process (e.g., heat flux). While FHD has been useful in modelingtransport and fluid dynamics at the mesoscopic scale, theoretical calculationshave been feasible only with simplifying assumptions. As such there is greatinterest 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 onGitHub.