在N log N复杂度下求解玻尔兹曼方程
Solving the Boltzmann equation in N log N
AI总结：
基于谱方法提出快速确定性算法，实现玻尔兹曼碰撞算子的O(N log N)计算，并应用于均匀和非均匀解，与蒙特卡洛方法比较精度和效率，同时展示熵泛函趋向平衡时的振荡现象。
摘要：
在[C. Mouhot and L. Pareschi, "Fast algorithms for computing the Boltzmann collision operator," Math. Comp., to appear; C. Mouhot and L. Pareschi, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 71-76]中，针对一类相互作用（包括三维硬球模型）的玻尔兹曼碰撞算子，推导了基于谱方法的快速确定性算法。这些算法被用于求解二维和三维玻尔兹曼方程，首先用于均匀解，然后用于一般非均匀解。结果与显式解（如果可用）和蒙特卡洛方法进行了比较。特别地，计算成本和精度与蒙特卡洛方法以及先前的谱方法进行了比较。最后，对于非均匀解，我们利用该方法的高计算效率展示了熵泛函在趋向平衡过程中的振荡现象，这一现象在[L. Desvillettes and C. Villani, Invent. Math., 159 (2005), pp. 245-316]中曾被提出。
英文摘要：
In [C. Mouhot and L. Pareschi, "Fast algorithms for computing the Boltzmann collision operator," Math. Comp., to appear; C. Mouhot and L. Pareschi, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 71-76], fast deterministic algorithms based on spectral methods were derived for the Boltzmann collision operator for a class of interactions including the hard spheres model in dimension three. These algorithms are implemented for the solution of the Boltzmann equation in two and three dimension, first for homogeneous solutions, then for general non homogeneous solutions. The results are compared to explicit solutions, when available, and to Monte-Carlo methods. In particular, the computational cost and accuracy are compared to those of Monte-Carlo methods as well as to those of previous spectral methods. Finally, for inhomogeneous solutions, we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium, which was suggested in the work [L. Desvillettes and C. Villani, Invent. Math., 159 (2005), pp. 245-316].
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