SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B792-B815, January 2020.
We propose a Hermite--Galerkin spectral method to numerically solve the spatially homogeneous Fokker--Planck--Landau equation with a singular quadratic collision model. To overcome the difficulty of the computationally expensive collision model, we adopt a novel approximation formulated by a combination of a simple linear term and a quadratic term very expensive to evaluate. Using the Hermite expansion, the quadratic term is evaluated exactly by calculating the spectral coefficients. To deal with singularities, we make use of Burnett polynomials so that even a very singular collision model can be handled smoothly. Numerical examples demonstrate that our method can capture low-order moments with satisfactory accuracy and performance.

中文翻译：

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Fokker-Planck-Landau方程中的奇异二次碰撞模型的逼近

SIAM科学计算杂志，第42卷，第3期，第B792-B815页，2020年1月。
我们提出了一种Hermite-Galerkin谱方法，用于用奇异二次碰撞模型对空间均匀的Fokker-Planck-Landau方程进行数值求解。为了克服计算量大的碰撞模型的困难，我们采用了一种新颖的近似方法，该方法是将一个简单的线性项和一个二次项组合在一起来制定的，因此评估起来非常昂贵。使用Hermite展开，可以通过计算频谱系数来精确评估二次项。为了处理奇点，我们使用了Burnett多项式，因此即使非常奇异的碰撞模型也可以平稳地处理。数值算例表明，我们的方法能够以令人满意的精度和性能捕获低阶矩。