Adaptive Mesh and Algorithm Refinement Using Direct Simulation Monte Carlo

Abstract

Adaptive mesh and algorithm refinement (AMAR) embeds a particle method within a continuum method at the finest level of an adaptive mesh refinement (AMR) hierarchy. The coupling between the particle region and the overlaying continuum grid is algorithmically equivalent to that between the fine and coarse levels of AMR. Direct simulation Monte Carlo (DSMC) is used as the particle algorithm embedded within a Godunov-type compressible Navier-Stokes solver. Several examples are presented and compared with purely continuum calculations.

Algorithm Refinement for Fluctuating Hydrodynamics

“…As in that paper, particles that enter the particle patch have velocities drawn from the either the Maxwell-Boltzmann distribution or the Chapman-Enskog distribution. While the Chapman-Enskog distribution is preferred in deterministic hybrids (see [4]) we find that in the stochastic hybrid the Maxwell-Boltzmann distribution sometimes yields better results for the second moment statistics (see Sections 4.1 and 4.3). While Chapman-Enskog yields slightly more accurate results for time-dependent problems, where we focus on the mean behavior of the system (see Sections 4.4 and 4.5), one must recall that the derivation of the LLNS equations is based on the assumption of local equilibrium (e.g., gradients do not appear in the amplitudes of the stochastic fluxes).…”

“…The implementation of reservoir boundary conditions for DSMC is described in [4]. As in that paper, particles that enter the particle patch have velocities drawn from the either the Maxwell-Boltzmann distribution or the Chapman-Enskog distribution.…”

“…It is well-known that correlations such as those appearing in the deterministic hybrid are present when a fluid is out of thermodynamic equilibrium (see Sec. 4…”

“…An improved simulation does not result from continued refinement of the mesh, but rather "refinement" of the algorithm, i.e., switching from the continuum model to a particle simulation. Introduced in [4], AMAR has proved to be a useful paradigm for multiscale fluid modeling. In this paper, we describe AMAR for fluctuating hydrodynamics.…”

“…A hybrid coupling Navier-Stokes and DSMC was developed in [4], with several of the technical issues necessary for implementation extended in [9]. Stochastic hybrid methods were developed in [10] (mass diffusion), [11] (the "train model" for momentum diffusion), and [12] (Burgers' equation).…”