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

Fast spectral solution of the generalised Enskog equation for dense gases

We propose a fast spectral method for solving the generalized Enskog equation for dense gases. For elastic collisions, the method solves the Enskog collision operator with a computational cost of O(Md-1Nd logN), where d is the dimension of the velocity space, and Md-1 and Nd are the number of solid angle and velocity space discretizations, respectively. For inelastic collisions, the cost is N times higher. The accuracy of this fast spectral method is assessed by comparing our numerical results with analytical solutions of the spatially homogeneous relaxation of heated granular gases. We also compare our results for force driven Poiseuille flow and Fourier flow with those from molecular dynamics and Monte Carlo simulations. Although it is phenomenological, the generalized Enskog equation is capable of capturing the flow dynamics of dense granular gases, and the fast spectral method is accurate and efficient. As example applications, Fourier and Couette flows of a dense granular gas are investigated. In additional to the temperature profile, both the density and the high-energy tails in the velocity distribution functions are found to be strongly influenced by the restitution coefficient

Fast spectral solution of the generalized Enskog equation for dense gases

AbstractWe propose a fast spectral method for solving the generalized Enskog equation for dense gases. For elastic collisions, the method solves the Enskog collision operator with a computational cost of O(Md−1Ndlog⁡N), where d is the dimension of the velocity space, and Md−1 and Nd are the number of solid angle and velocity space discretizations, respectively. For inelastic collisions, the cost is N times higher. The accuracy of this fast spectral method is assessed by comparing our numerical results with analytical solutions of the spatially-homogeneous relaxation of heated granular gases. We also compare our results for force-driven Poiseuille flow and Fourier flow with those from molecular dynamics and Monte Carlo simulations. Although it is phenomenological, the generalized Enskog equation is capable of capturing the flow dynamics of dense granular gases, and the fast spectral method is accurate and efficient. As example applications, Fourier and Couette flows of a dense granular gas are investigated. In addition to the temperature profile, both the density and the high-energy tails in the velocity distribution functions are found to be strongly influenced by the restitution coefficient

Derivation and numerical comparison of Shakhov and Ellipsoidal Statistical kinetic models for a monoatomic gas mixture

Gas mixtures are important for many practical applications. Extending kinetic model equations of the Bhatnagar–Gross–Krook (BGK) type from a single-species gas to a multi-species gas mixture presents a number of significant challenges. First, obtaining the correct species diffusions, viscous stresses as well as heat conduction in the continuum limit requires a careful design of the collision terms in the kinetic equations. Secondly, the derived model collision terms need to guarantee positivity of the macroscopic fields. In the present work, two new kinetic models are introduced and compared: an approach based on the Shakhov kinetic model and an approach involving an anisotropic Gaussian equilibrium function. The two new models are capable of modelling a binary mixture of monoatomic gases, with updated definitions for the relaxation parameters and target species velocities and temperature. Both methods account for separate species-mean velocity such that the species diffusion and velocity drift are accurately represented. The key contribution of the models is the exact recovery of the Fick, Newton and Fourier laws in the continuum limit, while preserving positive temperature fields and crucial properties of the Boltzmann equation. The profile of a normal shock wave is inspected under various flow conditions to numerically validate the two models. The results show improvement upon comparison with a model, which has two correct transport coefficients, and demonstrate the ability to reliably model inert gas mixtures

Projective and Telescopic Projective Integration for Non-Linear Kinetic Mixtures

We propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and \acf{BGK} equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the efficient solution of equations in the hyperbolic scaling with very small Knudsen numbers. We validate the schemes on a range of scenarios, demonstrating its prowess in dealing with extreme mass ratios, fluid instabilities, and other complex phenomena

An efficient numerical method for solving the Boltzmann equation in multidimensions

International audienceIn this paper we deal with the extension of the Fast Kinetic Scheme (FKS) [J. Comput. Phys., Vol. 255, 2013, pp 680-698] originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3D×3D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations