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Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states. (English)
J. Comput. Phys. 228, No. 6, 2012-2036 (2009).
Summary: We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann transport equation (BTE) in $d$-dimensions for variable hard sphere (VHS) collision kernels with conservative or non-conservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computation is reduced to a separate integral over the unit sphere $S^{d-1}$. The conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space, is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro-reversibility (inelastic interactions) or elastic models of slowing down process. Our simulations are benchmarked with available exact self-similar solutions, exact moment equations and analytical estimates for the homogeneous Boltzmann equation, both for elastic and inelastic VHS interactions. Benchmarking of the simulations involves the selection of a time self-similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in {\it A. V. Bobylev}, {\it C. Cercignani} and {\it G. Toscani} [J. Stat. Phys. 111, No. 1‒2, 403‒417 (2003; Zbl 1119.82318)] and generalized to a wide range of related models in {\it A.V. Bobylev, C. Cercignani} and {\it I.M. Gamba} [On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, Commun. Math. Phys., in press; cf. \url{arXiv:math-ph/0608035}]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods [{\it T. V. Noije, M. Ernst}, Velocity distributions in homogeneously cooling and heated granular fluids, Granular Matter 57, No. 1, (1998); {\it M.. H. Ernst} and {\it R. Brito} [J. Stat. Phys. 109, No. 3‒4, 407‒432 (2002; Zbl 1015.82030); {\it S. J. Moon, M. D. Shattuck, J. Swift}, Velocity distributions and correlations in homogeneously heated granular media, Phys. Rev. E 64, 031303 (2001); {\it I. M. Gamba, S. Rjasanow, W. Wagner}, Math. Comput. Modelling 42, No. 5‒6, 683‒700 (2005; Zbl 1088.35049)] and rigorously proven by {\it I. M. Gamba, V. Panferov} and {\it C. Villani} [Commun. Math. Phys. 246, No. 3, 503‒541 (2004; Zbl 1106.82031)] and {\it A. V. Bobylev, I. M. Gamba} and {\it V. Panferov} [J. Stat. Phys. 116, No. 5‒6, 1651‒1682 (2004; Zbl 1097.82021)].