Multi-species Boltzmann Equation Research Articles

Rigorous derivation of the Fick cross-diffusion system from the multi-species Boltzmann equation in the diffusive scaling

We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann equation in a rigorous manner in Sobolev spaces. To this end, we formally show that, in a diffusive scaling, the hydrodynamical limit of the kinetic system is the Fick model supplemented with a closure relation and we give explicit formulae for the macroscopic diffusion coefficients from the Boltzmann collision operator. Then, we provide a perturbative Cauchy theory in Sobolev spaces for the constructed Fick system, which turns out to be a dilated parabolic equation. We finally prove the stability of the system in the Boltzmann equation, ensuring a rigorous derivation between the two models.

Hermite spectral method for multi-species Boltzmann equation

We introduce a numerical scheme for the full multi-species Boltzmann equation based on Hermite spectral method. With the proper choice of expansion centers for different species, a practical algorithm is derived to evaluate the complicated multi-species binary collision operator. New collision models are built by combining the quadratic collision model and the simple BGK collision model under the framework of the Hermite spectral method, which enables us to balance the computational cost and accuracy. Several numerical experiments are implemented to validate the dramatic efficiency of this new Hermite spectral method. Moreover, we can handle the problems with as many as 100 species, which is far beyond the capability of the state-of-art algorithms.

Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system

<p style='text-indent:20px;'>Recently, the authors proved [<xref ref-type="bibr" rid="b2">2</xref>] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely investigated, but the particular case of a perturbative incompressible setting around a non constant equilibrium state of the mixture (needed in [<xref ref-type="bibr" rid="b2">2</xref>]) seems absent of the literature. We thus establish a quantitative perturbative Cauchy theory in Sobolev spaces for it. More precisely, by reducing the analysis of the Maxwell-Stefan system to the study of a quasilinear parabolic equation on the sole concentrations and with the use of a suitable anisotropic norm, we prove global existence and uniqueness of strong solutions and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture. As a by-product, we show that the equimolar diffusion condition naturally appears from this perturbative incompressible setting.</p>

On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior – exponential decay to the global equilibrium – of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E.S. Daus,Arch. Ration. Mech. Anal.3(2016) 1367–1443] for the deterministic problem in the perturbative regime, and in [E.S. Daus, S. Jin and L. Liu,Kinet. Relat. Models12(2019) 909–922] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far neither in the single species case nor in the multi-species case.

A discontinuous Galerkin fast spectral method for the multi-species Boltzmann equation

We introduce a fast Fourier spectral method for the multi-species Boltzmann collision operator. The method retains the advantages of the single-species fast spectral method Gamba et al. (2017) including: (a) spectral accuracy, (b) reduced computational complexity compared to direct spectral method, (c) reduced memory requirement in the precomputation, and (d) applicability to general collision kernels. The fast collision algorithm is then coupled with discontinuous Galerkin discretization in the physical space Jaiswal et al. (2019) to result in a highly accurate deterministic method (DGFS) for the full Boltzmann equation of gas mixtures. A series of numerical tests is performed to illustrate the efficiency and accuracy of the proposed method. Various benchmarks highlighting different collision kernels, different mass ratios, momentum transfer, heat transfer, and in particular the diffusive transport have been studied. The results are directly compared with the direct simulation Monte Carlo (DSMC) method.

Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings

We prove the stability of global equilibrium in a multi-species mixture, where the different species can have different masses, on the $3$-dimensional torus. We establish stability estimates in $L^\infty_{x,v}(w)$ where $w=w(v)$ is either polynomial or exponential, with explicit threshold. Along the way we extend recent estimates and stability results for the mono-species Boltzmann operator not only to the multi-species case but also to more general hard potential and Maxwellian kernels.

The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium

We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in $${L^1_vL^\infty_x(m)}$$ , where $${m\sim (1+ |v|^k)}$$ is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an $${L^2-L^\infty}$$ theory à la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the $${L^1_vL^\infty_x}$$ framework is dealt with for any $${k > k_0}$$ , recovering the optimal physical threshold of finite energy $${k_0=2}$$ in the particular case of a multi-species hard spheres mixture with the same masses.

An asymptotic preserving Monte Carlo method for the multispecies Boltzmann equation

An asymptotic preserving (AP) scheme is efficient in solving multiscale kinetic equations with a wide range of the Knudsen number. In this paper, we generalize the asymptotic preserving Monte Carlo method (AP-DSMC) developed in 25 to the multispecies Boltzmann equation. This method is based on the successive penalty method 26 originated from the BGK-penalization-based AP scheme developed in 7. For the multispecies Boltzmann equation, the penalizing Maxwellian should use the unified Maxwellian as suggested in 12. We give the details of AP-DSMC for multispecies Boltzmann equation, show its AP property, and verify through several numerical examples that the scheme can allow time step much larger than the mean free time, thus making it much more efficient for flows with possibly small Knudsen numbers than the classical DSMC.

Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation

AbstractThis paper generalizes the exponential Runge-Kutta asymptotic preserving (AP) method developed in [G. Dimarco and L. Pareschi,SIAM Numer. Anal., 49 (2011), pp. 2057-2077] to compute the multi-species Boltzmann equation. Compared to the single species Boltzmann equation that the method was originally applied on, this set of equation presents a new difficulty that comes from the lack of local conservation laws due to the interaction between different species. Hence extra stiff nonlinear source terms need to be treated properly to maintain the accuracy and the AP property. The method we propose does not contain any nonlinear nonlocal implicit solver, and can capture the hydrodynamic limit with time step and mesh size independent of the Knudsen number. We prove the positivity and strong AP properties of the scheme, which are verified by two numerical examples.

A BGK‐penalization‐based asymptotic‐preserving scheme for the multispecies Boltzmann equation

AbstractAn asymptotic‐preserving (AP) scheme is efficient in solving multiscale problems where kinetic and hydrodynamic regimes coexist. In this article, we extend the BGK‐penalization‐based AP scheme, originally introduced by Filbet and Jin for the single species Boltzmann equation (Filbet and Jin, J Comput Phys 229 (2010) 7625–7648), to its multispecies counterpart. For the multispecies Boltzmann equation, the new difficulties arise due to: (1) the breaking down of the conservation laws for each species and (2) different convergence rates to equilibria for different species in disparate masses systems. To resolve these issues, we find a suitable penalty function—the local Maxwellian that is based on the mean velocity and mean temperature and justify various asymptotic properties of this method. This AP scheme does not contain any nonlinear nonlocal implicit solver, yet it can capture the fluid dynamic limit with time step and mesh size independent of the Knudsen number. Numerical examples demonstrate the correct asymptotic‐behavior of the scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Kinetic solution of the structure of a shock wave in a nonreactive gas mixture

The multispecies Boltzmann equation is numerically integrated to characterize the internal structure of a Mach 3 shock wave in a hard sphere gas. The collision integral is evaluated by the conservative discrete ordinate method [F. G. Tcheremissine, Comput. Math. Math. Phys. 46, 315 (2006)]. There was excellent agreement of macroscopic variables [Kosuge et al.., Eur. J. Mech. B/Fluids 20, 87 (2001)]. The effect of species concentration and mass ratio on the behavior of macroscopic variables and distribution functions in the structure of the shock wave is considered for both two- and three-species gas mixtures. In a binary mixture of gases with different masses and varying concentrations, the temperature overshoot of the parallel component of temperature near the center of the shock wave is highest for the heavy component when the concentration of the heavy component is the smallest. The rise in the parallel component of temperature is revealed by the behavior of the distribution function.

A Micro-Macro Decomposition-Based Asymptotic-Preserving Scheme for the Multispecies Boltzmann Equation

In this paper we extend the micro-macro decomposition-based asymptotic-preserving scheme developed by Bennoune, Lemou, and Mieussens [J. Comput. Phys., 227 (2008), pp. 3781-3803] for the single species Boltzmann equation to the multispecies problems. An asymptotic-preserving scheme for the kinetic equation is very efficient in the fluid regime where the Knudsen number is small and the collision term becomes stiff. It allows a coarse (independent of the Knudsen number) mesh size and a large time step in the fluid regime. The difficulty associated with multispecies problems is that there are no local conservation laws for each species, resulting in extra stiff nonlinear source terms that need to be discretized properly in order to (1) avoid Newton-type solvers for nonlinear algebraic systems, and (2) to be asymptotic-preserving. We show that these extra nonlinear source terms can be solved using only linear system solvers, and the scheme preserves the correct Euler and Navier-Stokes limits. Numerical examples are used to demonstrate the efficiency and applicability of the schemes for both Euler and Navier-Stokes regimes.