In this paper, we introduce a novel Gauss–Jacobi quadrature rule designed for infinite intervals, which is specifically applied to the velocity discretization in multi-scale Boltzmann solvers. Our method utilizes a newly formulated bell-shaped weight function for numerical integration. We establish the relationship between this new quadrature and the classical Gauss–Jacobi, as well as the Gauss–Hermite quadrature rules, and we compare the resulting discrete velocity distributions with several commonly used methods. Additionally, we validate the performance of our method through numerical simulations of flows with various Knudsen numbers. The proposed quadrature provides fresh insights into velocity space discretization.

中文翻译：

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用于多尺度玻尔兹曼求解器的新型高斯-雅可比求积


在本文中，我们介绍了一种为无限区间设计的新颖的高斯-雅可比求积规则，该规则专门应用于多尺度玻尔兹曼求解器中的速度离散化。我们的方法利用新制定的钟形权重函数进行数值积分。我们建立了这种新求积法与经典高斯-雅可比求积法以及高斯-埃尔米特求积规则之间的关系，并将所得的离散速度分布与几种常用的方法进行了比较。此外，我们还通过各种努森数的流动数值模拟来验证我们方法的性能。所提出的求积法为速度空间离散化提供了新的见解。