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arXiv 1306.4625math.NAcs.NA
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一种用于玻尔兹曼方程各向异性散射的保守谱方法及近 grazing 碰撞极限

A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit

The University of Texas at Austin（德克萨斯大学奥斯汀分校）

Irene M. Gamba, Jeffrey R. Haack

备注：
arXiv admin note: text overlap with arXiv:1301.4195
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AI总结：

本文提出一种保守谱方法，用于处理具有各向异性散射机制的玻尔兹曼碰撞算子，以近似空间均匀情况下Landau方程的解。方法基于Gamba和Tharkabhushanam的保守谱方法，通过弱形式推导，并研究了傅里叶变换在近 grazing 碰撞极限下的收敛性。

摘要：

我们提出了一种用于Boltzmann碰撞算子的保守谱方案，以处理具有各向异性散射机制的碰撞极限情况，近似空间均匀设定下Landau方程的解。该方案基于Gamba和Tharkabhushanam[17, 18]的保守谱方法。该方案源于Boltzmann方程的弱形式，可以将碰撞项表示为傅里叶空间中的加权卷积。在此框架内，我们还研究了Boltzmann碰撞算子在近 grazing 碰撞极限下的傅里叶变换对Landau碰撞算子傅里叶变换的收敛率，对于一系列非可积的散射角截面。我们分析地证明了趋于平衡的衰减率依赖于与碰撞截面相关的参数，并具体研究了经典Rutherford散射角截面（在近似Landau时具有对数误差）与人工截面（具有线性误差）之间的差异。

英文摘要：

We present the formulation of a conservative spectral scheme for Boltzmann collision operators with anisotropic scattering mechanisms to model grazing collision limit regimes approximating the solution to the Landau equation in space homogeneous setting. The scheme is based on the conservative spectral method of Gamba and Tharkabhushanam [17, 18]. This formulation is derived from the weak form of the Boltzmann equation, which can represent the collisional term as a weighted convolution in Fourier space. Within this framework, we also study the rate of convergence of the Fourier transform for the Boltzmann collision operator in the grazing collisions limit to the Fourier transform for the Landau collision operator for a family of non-integrable angular scattering cross sections. We analytically show that the decay rate to equilibrium depends on the parameters associated with the collision cross section, and specifically study numerically the differences between the classical Rutherford scattering angular cross section, which has logarithmic error in approximating Landau, and an artificial cross section with a linear error.

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