用于动力学方程在扩散和异常扩散极限下的数值方案。第一部分：重尾平衡情况
Numerical schemes for kinetic equations in the diffusion and anomalous diffusion limits. Part I: the case of heavy-tailed equilibrium
AI总结：
本文提出用于线性动力学方程在扩散和异常扩散极限下的数值方案，针对重尾分布情况，发展了适用于不同 regime 的数值方法，具有渐进保持性质，不依赖标准显式时间积分器的稳定性条件。
摘要：
在本文中，我们为线性动力学方程在扩散和异常扩散极限下提出了一些数值方案。当平衡分布函数是麦克斯韦分布时，对于适当的时间尺度，小自由程极限导致扩散型方程。然而，当考虑重尾分布时，需要另一个时间尺度，小自由程极限导致分数异常扩散方程。我们的目标是开发适用于不同 regime 的原始动力学模型的数值方案，而不受标准显式时间积分器稳定性条件的限制。首先，我们为扩散渐进行为提出了一些数值方案；然后研究了它们向异常扩散极限的扩展。在这种情况下，捕捉重尾平衡的大速度效应至关重要，因此需要对为扩散渐进行为推导的方案进行一些重要变换。结果，我们获得了在异常扩散极限下具有渐进保持性质的数值方案，即：它们不受到时间步长的限制，并且当自由程趋于零时退化为分数扩散极限。我们还通过数值研究了统一精度，并构造了一类满足此性质的数值方案。最后，通过数值实验展示了不同数值方案的效率。
英文摘要：
In this work, we propose some numerical schemes for linear kinetic equations in the diffusion and anomalous diffusion limit. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion type equation. However, when a heavy-tailed distribution is considered, another time scale is required and the small mean free path limit leads to a fractional anomalous diffusion equation. Our aim is to develop numerical schemes for the original kinetic model which works for the different regimes, without being restricted by stability conditions of standard explicit time integrators. First, we propose some numerical schemes for the diffusion asymptotics; then, their extension to the anomalous diffusion limit is studied. In this case, it is crucial to capture the effect of the large velocities of the heavy-tailed equilibrium, so that some important transformations of the schemes derived for the diffusion asymptotics are needed. As a result, we obtain numerical schemes which enjoy the Asymptotic Preserving property in the anomalous diffusion limit, that is: they do not suffer from the restriction on the time step and they degenerate towards the fractional diffusion limit when the mean free path goes to zero. We also numerically investigate the uniform accuracy and construct a class of numerical schemes satisfying this property. Finally, the efficiency of the different numerical schemes is shown through numerical experiments.
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