1. Introduction

1.1. What is fluctuating hydrodynamics?

At mesoscopic scales (from nanometres to micrometres) thermal fluctuations change how fluids behave. They drive diffusion enhancement, giant concentration fluctuations, the onset of instabilities and spontaneous pattern formation. Fluctuating hydrodynamics models these effects by adding stochastic fluxes to the usual conservation laws of continuum fluid dynamics. The noise amplitude is fixed by the fluctuation–dissipation theorem, so that the equations reproduce the correct equilibrium fluctuations.

The simplest example is the incompressible fluctuating Stokes / Navier–Stokes system for the velocity \(\mathbf{v}\) and pressure \(\pi\),

(1.1)\[\rho\,\partial_t \mathbf{v} + \nabla\pi = -\rho\,\nabla\cdot(\mathbf{v}\mathbf{v}^T) + \eta\nabla^2\mathbf{v} + \nabla\cdot\left(\sqrt{2\eta k_B T}\;\boldsymbol{\mathcal{W}}\right), \qquad \nabla\cdot\mathbf{v} = 0,\]

where \(\boldsymbol{\mathcal{W}}\) is a white-noise random tensor field. FHDeX solves this system and extends it to multispecies mixtures, compressible flows, chemical reactions, charged species, and coupling to particles and immersed boundaries.

The equations and numerical methods of all solver families are described in Theory and Numerical Methods.

1.2. Solver families

FHDeX contains several groups of solvers. Each group is a set of source libraries (src_*) together with one or more applications (exec/*):

Family	Description
Incompressible hydrodynamics	Staggered-grid (MAC) fluctuating Navier–Stokes with a GMRES Stokes solver and multigrid preconditioner (`src_hydro`, `src_gmres`; application `exec/hydro`).
Low-Mach multispecies	Multispecies mixtures with stochastic mass fluxes, optional electrodiffusion for charged species, inertial and Boussinesq time integrators (`src_multispec`; application `exec/multispec`).
Compressible	Compressible fluctuating Navier–Stokes for multispecies gas mixtures, in a collocated (`src_compressible`) and a staggered (`src_compressible_stag`) formulation, with optional chemistry and surface chemistry.
Reaction–diffusion	Stochastic reaction–diffusion of species number densities, with deterministic, chemical-Langevin, SSA or tau-leaping reactions and explicit, semi-implicit or multinomial diffusion (`src_reactDiff`; application `exec/reactDiff`).
Particle methods	DSMC, Brownian / immersed ions and immersed-boundary markers built on AMReX particle containers (`src_particles`, `src_immersed-boundary`).
Multiscale coupling	Coupling of continuum solvers to external particle codes through the MUI library (`src_mui`, `src_surfchem_mui`).

1.3. Relationship to AMReX

FHDeX applications are AMReX applications. All grid data are stored in AMReX MultiFab objects, staggered velocities are face-centred (nodal in one direction) MultiFab objects, parallelism uses AMReX’s MPI domain decomposition and GPU offloading, and input parameters are read with ParmParse. Readers new to AMReX should first read:

Box, IntVect and IndexType, BoxArray and DistributionMapping – how the domain is split into boxes and distributed over MPI ranks;
FabArray, MultiFab and iMultiFab and MFIter and Tiling – the data containers and loops used in every FHDeX kernel;
ParmParse – the input-file format;
Building with GNU Make – the GNU Make build system used by every exec/* directory.

1.4. Repository layout

FHDeX/
  src_*/          source libraries, one per physics component (see Code Structure)
  exec/           applications: one directory per executable, each with a
                  GNUmakefile and a set of inputs_* files (see Applications)
  python/         post-processing scripts (AMReX particle output -> ParaView)
  tools/          template for a local Path.local (sets AMREX_HOME)
  unmaintained/   old applications kept for reference; not documented here
  Docs/           this guide