We develop in this paper a Petrov-Galerkin spectral method for the inelastic Boltzmann equation in one dimension. Solutions to such equations typically exhibit heavy tails in the velocity space so that domain truncation or Fourier approximation would suffer from large truncation errors. Our method is based on the mapped Chebyshev functions on unbounded domains, hence requires no domain truncation. Furthermore, the test and trial function spaces are carefully chosen to obtain desired convergence and conservation properties. Through a series of examples, we demonstrate that the proposed method performs better than the Fourier spectral method and yields highly accurate results.

中文翻译：

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基于映射的Chebyshev函数的非弹性Boltzmann方程的Petrov-Galerkin谱方法

我们在本文中为一维非弹性玻尔兹曼方程开发了一种彼得罗夫-加勒金谱方法。此类方程式的解通常在速度空间中显示出较重的尾巴，因此域截断或傅立叶近似将遭受较大的截断误差。我们的方法基于无边界域上映射的Chebyshev函数，因此不需要域截断。此外，仔细选择测试和试验函数空间以获得所需的收敛性和守恒性。通过一系列的例子，我们证明了所提出的方法比傅立叶光谱法性能更好，并且产生了高度准确的结果。