The gold-standard definition of the Direct Simulation Monte Carlo (DSMC) method is given in the 1994 book by Bird [Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Clarendon Press, Oxford, UK, 1994)], which refined his pioneering earlier papers in which he first formulated the method. In the intervening 25 years, DSMC has become the method of choice for modeling rarefied gas dynamics in a variety of scenarios. The chief barrier to applying DSMC to more dense or even continuum flows is its computational expense compared to continuum computational fluid dynamics methods. The dramatic (nearly billion-fold) increase in speed of the largest supercomputers over the last 30 years has thus been a key enabling factor in using DSMC to model a richer variety of flows, due to the method’s inherent parallelism. We have developed the open-source SPARTA DSMC code with the goal of running DSMC efficiently on the largest machines, both current and future. It is largely an implementation of Bird’s 1994 formulation. Here, we describe algorithms used in SPARTA to enable DSMC to operate in parallel at the scale of many billions of particles or grid cells, or with billions of surface elements. We give a few examples of the kinds of fundamental physics questions and engineering applications that DSMC can address at these scales.

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Problems in hydrology frequently have moving fronts and dynamic driving mechanisms such as wells. Since the location of important features changes during a simulation, accurate modeling requires uniformly fine resolution or the ability to change resolution during the simulation. We will describe an algorithm for refinement and unrefinement of tetrahedral/triangular meshes that has been implemented in the adaptive hydrology (ADH) code. The codes including the refinement/unrefinement algorithms are implemented in parallel to accommodate problems with large run time and memory requirements. In this paper, we describe the parallel, adaptive grid algorithm used in ADH and show the resulting grids from some example problems.

A new second-order accurate, hybrid, incremental projection method for time-dependent incompressible viscous flow is introduced in this paper. The hybrid finite-element/finite-volume discretization circumvents the well-known Ladyzhenskaya-Babuška-Brezzi conditions for stability, and does not require special treatment to filter pressure modes by either Rhie-Chow interpolation or by using a Petrov-Galerkin finite element formulation. The use of a covelocity with a high-resolution advection method and a linearly consistent edge-based treatment of viscous/diffusive terms yields a robust algorithm for a broad spectrum of incompressible flows. The high-resolution advec-* Corresponding author.

Abstract. In this paper we discuss the design and implementation of a Newton-Krylov-Schwarz solver for the implicit temporal integration on an unstructured three-dimensional spatial mesh of Richards' equation for groundwater flow in unsaturated porous media. We use aggregation techniques from the algebraic multigrid literature to construct a coarse mesh for two-level Schwarz methods. Our coarse mesh differs from other constructions in that no coarse mesh geometry need be created and we do not need geometric information about the subdomains. We report on a computational example to illustrate the performance of the preconditioner.