Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation

In this paper we present a new ultra efficient numerical method for solving kinetic equations. In this preliminary work, we present the scheme in the case of the BGK relaxation operator. The scheme, being based on a splitting technique between transport and collision, can be easily extended to other collisional operators as the Boltzmann collision integral or to other kinetic equations such as the Vlasov equation. The key idea, on which the method relies, is to solve the collision part on a grid and then to solve exactly the transport linear part by following the characteristics backward in time. The main difference between the method proposed and semi-Lagrangian methods is that here we do not need to reconstruct the distribution function at each time step. This allows to tremendously reduce the computational cost of the method and it permits for the first time, to the author's knowledge, to compute solutions of full six dimensional kinetic equations on a single processor laptop machine. Numerical examples, up to the full three dimensional case, are presented which validate the method and assess its efficiency in 1D, 2D and 3D

Solving the Boltzmann Equation on GPU

We show how to accelerate the direct solution of the Boltzmann equation using Graphics Processing Units (GPUs). In order to fully exploit the computational power of the GPU, we choose a method of solution which combines a finite difference discretization of the free-streaming term with a Monte Carlo evaluation of the collision integral. The efficiency of the code is demonstrated by solving the two-dimensional driven cavity flow. Computational results show that it is possible to cut down the computing time of the sequential code of two order of magnitudes. This makes the proposed method of solution a viable alternative to particle simulations for studying unsteady low Mach number flows.Comment: 18 pages, 3 pseudo-codes, 6 figures, 1 tabl

Efficient particle methods for solving the Boltzmann equation

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2007.Includes bibliographical references (leaves 85-86).A new particle simulation method for solving the Boltzmann equation is presented and tested. This method holds a significant computational efficiency advantage for low-signal flows compared to traditional particle methods such as the Direct Simulation Monte Carlo (DSMC). More specifically, the proposed algorithm can efficiently simulate arbitrarily small deviations from equilibrium (e.g. low speed flows) at a computational cost that does not scale with the deviation from equilibrium, while maintaining the basic algorithmic structure of DSIMC. This is achieved by incorporating the variance reduction ideas presented in [L. L. Baker and N. G. Hadjiconstantinou, Physics of Fluids, vol 17, art. no 051703, 2005] within a collision integral formulation; the latter ensures that the deviation from equilibrium remains finite and thus the calculation remains stable for collision dominated flows, in contrast to previous attempts. The formulation, developed within this thesis, is described in detail. The resulting scheme is validated for a wide range of Knudsen numbers (ratio of molecular mean free path to characteristic flow lengthscale) -- ranging from collision-dominated flow -- to collisionless flow- and a wide range of deviations from equilibrium. Excellent agreement is found with DSMC solutions for linear and weakly non-linear flows.by Thomas Homolle.S.M

The Moment Guided Monte Carlo method for the Boltzmann equation

In this work we propose a generalization of the Moment Guided Monte Carlo method developed in [11]. This approach permits to reduce the variance of the particle methods through a matching with a set of suitable macroscopic moment equations. In order to guarantee that the moment equations provide the correct solutions, they are coupled to the kinetic equation through a non equilibrium term. Here, at the contrary to the previous work in which we considered the simplified BGK operator, we deal with the full Boltzmann operator. Moreover, we introduce an hybrid setting which permits to entirely remove the resolution of the kinetic equation in the limit of infinite number of collisions and to consider only the solution of the compressible Euler equation. This modification additionally reduce the statistical error with respect to our previous work and permits to perform simulations of non equilibrium gases using only a few number of particles. We show at the end of the paper several numerical tests which prove the efficiency and the low level of numerical noise of the method.Comment: arXiv admin note: text overlap with arXiv:0908.026

Benchmark numérique pour intégrateurs explicites destinés à la dynamique d'impact

International audienceCe travail vise à établir un benchmark numérique pour les schémas d'intégration temporels explicites dédiés aux problèmes dynamiques non-linéaire avec impact. Ce benchmark est constitué de plusieurs cas tests, simples à implémenter et à analyser, dont quatre sont présentés ici. Chaque cas vise à tester une propriété numérique du schéma importante en dynamique non-linéaire avec impact : conservation de l'énergie à l'impact et en temps long, conservation du moment angulaire, capacité à surmonter des accumulations d'impacts... Certains schémas référents en dynamique d'impact seront testés à travers ce benchmark

A fast spectral method for the Boltzmann equation for monatomic gas mixtures

Although the fast spectral method has been established for solving the Boltzmann equation for single-species monatomic gases, its extension to gas mixtures is not easy because of the non-unitary mass ratio between the di↵erent molecular species. The conventional spectral method can solve the Boltzmann collision operator for binary gas mixtures but with a computational cost of the order m3rN6, where mr is the mass ratio of the heavier to the lighter species, and N is the number of frequency nodes in each frequency direction. In this paper, we propose a fast spectral method for binary mixtures of monatomic gases that has a computational cost O(pmrM2N4 logN), where M2 is the number of discrete solid angles. The algorithm is validated by comparing numerical results with analytical Bobylev- Krook-Wu solutions for the spatially-homogeneous relaxation problem, for mr up to 36. In spatially-inhomogeneous problems, such as normal shock waves and planar Fourier/Couette flows, our results compare well with those of both the numerical kernel and the direct simulation Monte Carlo methods. As an application, a two-dimensional temperature-driven flow is investigated, for which other numerical methods find it difficult to resolve the flow field at large Knudsen numbers. The fast spectral method is accurate and elective in simulating highly rarefied gas flows, i.e. it captures the discontinuities and fine structures in the velocity distribution functions

Asynchronous coupling of hybrid models for efficient simulation of multiscale systems

We present a new coupling approach for the time advancement of multi-physics models of multiscale systems. This extends the method of E et al. (2009) [5] to deal with an arbitrary number of models. Coupling is performed asynchronously, with each model being assigned its own timestep size. This enables accurate long timescale predictions to be made at the computational cost of the short timescale simulation. We propose a method for selecting appropriate timestep sizes based on the degree of scale separation that exists between models. A number of example applications are used for testing and benchmarking, including a comparison with experimental data of a thermally driven rarefied gas flow in a micro capillary. The multiscale simulation results are in very close agreement with the experimental data, but are produced almost 50,000 times faster than from a conventionally-coupled simulation

Discrete unified gas kinetic scheme on unstructured meshes

The recently proposed discrete unified gas kinetic scheme (DUGKS) is a finite volume method for multiscale flow computations with asymptotic preserving property. The solution of the Boltzmann model equation is directly used for the construction of numerical flux and makes the scheme applicable in all flow regimes. In previous applications of the DUGKS, structured meshes have been mostly employed, which may have difficulties for problems with complex geometries. In this paper we will extend the DUGKS to unstructured meshes, with the implementation of computational fluid dynamics techniques to the DUGKS. Several test cases, i.e., the cavity flow ranging from continuum to free molecular regimes, a multiscale flow problem between two connected cavities with large pressure and density variations, high speed flows past multiple cylinders in slip and transitional regimes, and an impulsive start problem are performed. The results are compared with the well-defined Direct Simulation Monte Carlo (DSMC) or Navier-Stokes (NS) solutions in their applicable regimes. The numerical results demonstrate the effectiveness of the proposed DUGKS for the study of multiscale flow problems

Nouvelles de M. Léon Rey, directeur du Service archéologique de l'armée d'Orient.

Homolle Théophile. Nouvelles de M. Léon Rey, directeur du Service archéologique de l'armée d'Orient. In: Comptes rendus des séances de l'Académie des Inscriptions et Belles-Lettres, 62ᵉ année, N. 4, 1918. p. 307