We develop a spectral method for the spatially homogeneous Boltzmann equation using Burnett polynomials in the basis functions. Using the sparsity of the coefficients in the expansion of the collision term, the computational cost is reduced by one order of magnitude for general collision kernels and by two orders of magnitude for Maxwell molecules. The proposed method can couple seamlessly with the BGK-type modelling techniques to make future applications affordable. The implementation of the algorithm is discussed in detail, including a numerical scheme to compute all the coefficients accurately, and the design of the data structure to achieve high cache hit ratio. Numerical examples are provided to demonstrate the accuracy and efficiency of our method

中文翻译：

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空间齐次 Boltzmann 方程的 Burnett 谱方法

我们在基函数中使用伯内特多项式为空间齐次玻尔兹曼方程开发了一种谱方法。利用碰撞项展开中系数的稀疏性，对于一般碰撞核，计算成本降低了一个数量级，对于麦克斯韦分子则降低了两个数量级。所提出的方法可以与 BGK 类型的建模技术无缝耦合，使未来的应用程序负担得起。详细讨论了算法的实现，包括精确计算所有系数的数值方案，以及实现高缓存命中率的数据结构设计。提供了数值例子来证明我们方法的准确性和效率