非截断玻尔兹曼方程的谱方法及数值掠射碰撞极限
Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit
AI总结：
本文研究从非截断玻尔兹曼方程到福克-普朗克-朗道方程在掠射碰撞极限下的数值过渡，提出了一种傅里叶谱方法，并证明了其一致性和均匀谱精度。
摘要：
本文研究了在所谓的掠射碰撞极限下，从空间均匀的非截断玻尔兹曼方程到福克-普朗克-朗道方程的数值过渡。为此，我们借鉴L. Pareschi, B. Perthame, TTSP 25, (1996)和L. Pareschi, G. Russo, SINUM 37, (2000)的思想，推导了非截断玻尔兹曼方程的傅里叶谱方法。我们证明了定义谱方法的核模式具有正确的掠射碰撞极限，从而为极限福克-普朗克-朗道方程提供了一致的谱方法。特别地，对于小散射角，我们推导了非截断玻尔兹曼方程核模式的近似公式，与福克-普朗克-朗道情况类似，该公式可以通过快速算法计算。此外，我们还证明了该方法关于掠射碰撞参数的均匀谱精度。
英文摘要：
In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cut-off to the Fokker-Planck-Landau equation in the so-called grazing collision limit. To this aim we derive a Fourier spectral method for the non cut-off Boltzmann equation in the spirit of L. Pareschi, B.Perthame, TTSP 25, (1996) and L.Pareschi, G.Russo, SINUM 37, (2000). We show that the kernel modes that define the spectral method have the correct grazing collision limit providing a consistent spectral method for the limiting Fokker-Planck-Landau equation. In particular, for small values of the scattering angle, we derive an approximate formula for the kernel modes of the non cut-off Boltzmann equation which, similarly to the Fokker-Planck-Landau case, can be computed with a fast algorithm. The uniform spectral accuracy of the method with respect to the grazing collision parameter is also proved.
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