Lattice points on circles and discrete velocity models for the Boltzmann equation
Lattice points on circles and discrete velocity models for the Boltzmann equation
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圆上的格点和玻尔兹曼方程的离散速度模型
DOI：
10.1137/040618916
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
B. Wennberg
中科院分区：
文献类型：
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作者：
Laura Fainsilber;P. Kurlberg;B. Wennberg
玻尔兹曼方程的离散速度模型或数值方法的构建，可能会导致计算碰撞算子的必要性，作为一个总和超过晶格点。碰撞算子涉及球面上的积分，这对应于能量和动量守恒。在第二个维度有困难，甚至在证明这种近似的收敛性，因为许多圆包含很少的格点，有些圆包含许多不良分布的格点。然而，通过表明大多数圆上的格点是等分布的，我们发现碰撞算子确实可以近似为二维情况下格点的和。证明使用了乘法函数的Halberstam-Richert不等式的一个弱形式（文中给出了一个证明），并估计了高斯素数的角分布。对于更高的维度，这个结果已经由Palczewski，Schneider和Bobylev [SIAM J. Numer.分析：第34（1997）号来文，第100页。1865-1883]。
The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties even in proving the convergence of such an approximation since many circles contain very few lattice points, and some circles contain many badly distributed lattice points. However, by showing that lattice points on most circles are equidistributed we find that the collision operator can indeed be approximated as a sum over lattice points in the two-dimensional case. The proof uses a weak form of the Halberstam-Richert inequality for multiplicative functions (a proof is given in the paper), and estimates for the angular distribution of Gaussian primes. For higher dimensions, this result has already been obtained by Palczewski, Schneider, and Bobylev [SIAM J. Numer. Anal., 34 (1997), pp. 1865-1883].