基于Maxwell多项式的伪谱插值在具有能量扩散的动力学方程中的应用
Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion
AI总结：
本文研究了基于Maxwell多项式的伪谱插值在动力学方程中用于速度变量离散化的方法，发现其在大多数情况下优于其他常用方案，但需注意非模态时间步长不稳定性问题。
摘要：
我们研究了一种近期流行的用于动力学方程中速度变量离散化策略的近似和稳定性性质，该策略基于伪谱插值在由非标准正交多项式族——Maxwell多项式的零点定义的网格上进行离散化。以描述由于Fokker-Planck碰撞导致能量扩散的一维方程作为性能测试平台，我们发现基于Maxwell的离散化在大多数情况下优于其他常用方案，通常高出多个数量级。这为在高维gyrokinetic模拟中使用它们提供了强大动力。然而，我们还显示基于Maxwell的方案在其最直接的实现中受到非模态时间步长不稳定性的影响，因此必须特别注意离散线性算子的表示，以充分利用Maxwell多项式提供的优势。
英文摘要：
We study the approximation and stability properties of a recently popularized discretization strategy for the speed variable in kinetic equations, based on pseudo spectral collocation on a grid defined by the zeros of a non-standard family of orthogonal polynomials called Maxwell polynomials. Taking a one-dimensional equation describing energy diffusion due to Fokker-Planck collisions with a Maxwell-Boltzmann background distribution as the test bench for the performance of the scheme, we find that Maxwell based discretizations outperform other commonly used schemes in most situations, often by orders of magnitude. This provides a strong motivation for their use in high-dimensional gyrokinetic simulations. However, we also show that Maxwell based schemes are subject to a non-modal time stepping instability in their most straightforward implementation, so that special care must be given to the discrete representation of the linear operators in order to benefit from the advantages provided by Maxwell polynomials.
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