Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

https://www.ki-net.umd.edu/pubs/files/FRG-2009-Jin-Shi.Filbet-Jin.pdf

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

https://hal.archives-ouvertes.fr/hal-00986714/document

Numerical methods for kinetic equations

A unified gas-kinetic scheme for continuum and rarefied flows IV

This paper presents a review and critical analysis on the modeling of the dynamics of vehicular traffic, human crowds and swarms seen as living and, hence, complex systems. It contains a survey of ...

/pdf/vehicular-traffic-crowds-and-swarms-from-kinetic-theory-and-2e3l5tqiwc.pdf

Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives

The computation of compressibleflows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incom- pressible flow simulation. In this paper, we develop an all-speed asymptotic preserv- ing (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyper- bolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved im- plicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solu- tion techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes. AMS subject classifications: 35Q35, 65M08, 65M99, 76M12, 76N99

/pdf/an-all-speed-asymptotic-preserving-method-for-the-isentropic-64o6rxruoa.pdf

An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

We propose a simple fast spectral method for the Boltzmann collision operator with general collision kernels. In contrast to the direct spectral method [L. Pareschi and G. Russo, SIAM J. Numer. Anal., 37 (2000), pp. 1217--1245; I. M. Gamba and S. H. Tharkabhushanam, J. Comput. Phys., 228 (2009), pp. 2012--2036], which requires $O(N^6)$ memory to store precomputed weights and has $O(N^6)$ numerical complexity, the new method has complexity $O(MN^4\log N)$, where $N$ is the number of discretization points in each of the three velocity dimensions and $M$ is the total number of discretization points on the sphere and $M\ll N^2$. Furthermore, it requires no precomputation for the variable hard sphere model and only $O(MN^4)$ memory to store precomputed functions for more general collision kernels. Although a faster spectral method is available [C. Mouhot and L. Pareschi, Math. Comp., 75 (2006), pp. 1833--1852] (with complexity $O(MN^3\log N)$), it works only for hard sphere molecules, thus limiting its use for...

A Fast Spectral Method for the Boltzmann Collision Operator with General Collision Kernels

We propose a simple fast spectral method for the Boltzmann collision operator with general collision kernels. In contrast to the direct spectral method \cite{PR00, GT09} which requires $O(N^6)$ memory to store precomputed weights and has $O(N^6)$ numerical complexity, the new method has complexity $O(MN^4\log N)$, where $N$ is the number of discretization points in each of the three velocity dimensions and $M$ is the total number of discretization points on the sphere and $M\ll N^2$. Furthermore, it requires no precomputation for the variable hard sphere (VHS) model and only $O(MN^4)$ memory to store precomputed functions for more general collision kernels. Although a faster spectral method is available \cite{MP06} (with complexity $O(MN^3\log N)$), it works only for hard sphere molecules, thus limiting its use for practical problems. Our new method, on the other hand, can apply to arbitrary collision kernels. A series of numerical tests is performed to illustrate the efficiency and accuracy of the proposed method.

A fast spectral method for the Boltzmann collision operator with general collision kernels

We derive a conservative multispecies BGK model that follows the spirit of the original, single species BGK model by making the specific choice to conserve species masses, total momentum, and total kinetic energy and to satisfy Boltzmann’s $$\mathcal {H}$$
 -Theorem. The derivation emphasizes the connection to the Boltzmann operator which allows for direct inclusion of information from higher-fidelity collision physics models. We also develop a complete hydrodynamic closure via the Chapman-Enskog expansion, including a general procedure to generate symmetric diffusion coefficients based on this model. We numerically investigate velocity and temperature relaxation in dense plasmas and compare the model with previous multispecies BGK models and discuss the trade-offs that are made in defining and using them. In particular, we demonstrate that the BGK model in the NRL plasma formulary does not conserve momentum or energy in general.

A Conservative, Entropic Multispecies BGK Model

https://www.ki-net.umd.edu/pubs/files/AnisotropicBoltzSubmission_july2.pdf

Jeffrey R. Haack

Papers

An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

A Fast Spectral Method for the Boltzmann Collision Operator with General Collision Kernels

A fast spectral method for the Boltzmann collision operator with general collision kernels

A Conservative, Entropic Multispecies BGK Model

A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit