This paper introduces an adaptive mesh and algorithm refinement method for fluctuating hydrodynamics. This particle-continuum hybrid simulates the dynamics of a compressible fluid with thermal fluctuations. The particle algorithm is direct simulation Monte Carlo (DSMC), a molecular-level scheme based on the Boltzmann equation. The continuum algorithm is based on the Landau–Lifshitz Navier–Stokes (LLNS) equations, which incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. It uses a recently developed solver for the LLNS equations based on third-order Runge–Kutta. We present numerical tests of systems in and out of equilibrium, including time-dependent systems, and demonstrate dynamic adaptive refinement by the computation of a moving shock wave. Mean system behavior and second moment statistics of our simulations match theoretical values and benchmarks well. We find that particular attention should be paid to the spectrum of the flux at the interface between the particle and continuum methods, specifically for the nonhydrodynamic (kinetic) time scales.

MSC codes

  1. 65C30
  2. 60H35
  3. 76M12
  4. 76M28


  1. stochastic Navier–Stokes equations
  2. multiscale hybrid algorithm
  3. computational fluid dynamics
  4. direct simulation Monte Carlo
  5. adaptive mesh refinement

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Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 1256 - 1280
ISSN (online): 1540-3467


Submitted: 3 July 2007
Accepted: 24 October 2007
Published online: 1 February 2008

MSC codes

  1. 65C30
  2. 60H35
  3. 76M12
  4. 76M28


  1. stochastic Navier–Stokes equations
  2. multiscale hybrid algorithm
  3. computational fluid dynamics
  4. direct simulation Monte Carlo
  5. adaptive mesh refinement



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