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首页> 外文期刊>Journal of Computational Physics >Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states
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Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states

机译：非平衡统计状态碰撞模型的谱-拉格朗日方法

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摘要

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 Sd-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 Bobylev et al. [A.V. Bobylev, C. Cercignani, G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, journal of Statistical Physics 111 (2003) 403-417] and generalized to a wide range of related models in Bobylev et al. [A.V. Bobylev, C. Cercignani, I.M. Gamba, On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, Communication in Mathematical Physics, in press. URL: ]. 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 [T.V. Noije, M. Ernst, Velocity distributions in homogeneously cooling and heated granular fluids, Granular Matter 1(57) (1998); M.H. Ernst, R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, journal of Statistical Physics 109 (2002) 407-432; SJ. Moon, M.D. Shattuck, J. Swift, Velocity distributions and correlations in homogeneously heated granular media, Physical Review E 64 (2001) 031303; I.M. Gamba, S. Rjasanow, W. Wagner, Direct simulation of the uniformly heated granular Boltzmann equation, Mathematical and Computer Modelling 42 (2005) 683-700] and rigorously proven in Gamba et al. [I.M. Gamba, V. Panferov, C. Villani, On the Boltzmann equation for diffusively excited granular media, Communications in Mathematical Physics 246 (2004) 503-541(39)] and JAN. Bobylev, I.M. Gamba, V. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, Journal of Statistical Physics 116 (2004) 1651-1682]. (C) 2008 Elsevier Inc. All rights reserved.
机译：针对具有保守或非保守二元相互作用的可变硬球体（VHS）碰撞核的d维非线性非线性Boltzmann输运方程（BTE），我们提出了一种基于拉格朗日光谱的确定性求解器。该方法基于碰撞积分的傅立叶变换的对称性，其中其计算复杂度降低到单位球面Sd-1上的单独积分。矩守恒由拉格朗日约束约束。在自由空间中实施的结果方案非常通用，可以通过几种非常简单的方式进行调整，以适应由于局部微可逆性（非弹性相互作用）或减慢过程的弹性模型而导致能量耗散的情况。我们的模拟使用可用的精确自相似解，精确矩方程和齐次Boltzmann方程的解析估计作为基准，以用于弹性和非弹性VHS相互作用。模拟基准测试包括选择数值分布函数的时间自相似重标度，这是使用麦克斯韦分子方程的连续谱进行的，如Bobylev等人首先研究的那样。 [A.V. Bobylev，C. Cercignani，G. Toscani，玻尔兹曼方程的颗粒材料自相似解的渐近性质的证明，《统计物理学》 111（2003）403-417]，并推广到了Bobylev等。 [A.V. Bobylev，C.Cercignani，I.M.Gamba，关于广义非线性动力学麦克斯韦模型的自相似渐近性，《数学物理通讯》，印刷中。网址：]。在用于硬球体的非弹性扩散Boltzmann方程（热浴下的非弹性碰撞）的情况下，该方法也可得出准确的结果，其中通过随机方法在计算中推测出了人口过多的非高斯指数尾巴。 Noije，M. Ernst，均匀冷却和加热的颗粒状流体中的速度分布，Granular Matter 1（57）（1998）;硕士Ernst，R.Brito，具有过度填充的高能尾部的非弹性Boltzmann方程的比例解，《统计物理学》 109（2002）407-432; SJ。 Moon，M.D.Shattuck，J.Swift，均匀加热的粒状介质中的速度分布和相关性，Physical Review E 64（2001）031303; I.M. Gamba，S。Rjasanow，W。Wagner，均匀加热的颗粒Boltzmann方程的直接模拟，《数学和计算机建模》 42（2005）683-700]，并在Gamba等人中得到了严格的证明。 [I.M. Gamba，V. Panferov，C. Villani，关于扩散激发颗粒介质的Boltzmann方程，《数学物理学通讯》 246（2004）503-541（39）]和JAN。 Bobylev，I.M。Gamba，V。Panferov，具有非弹性相互作用的Boltzmann方程的矩不等式和高能尾，统计物理学杂志116（2004）1651-1682]。 （C）2008 Elsevier Inc.保留所有权利。

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