Fokker-Planck-Landau方程守恒谱方法的收敛性和误差估计
Convergence and Error Estimates for the Conservative Spectral Method for Fokker-Planck-Landau Equations
摘要：
严格推导出半离散形式的保守谱方法的误差估计，用于近似与硬势相关的空间齐次Fokker-Planck-Landau（FPL）方程。分析表明，半离散问题有一个唯一的解决方案，有界的时刻。此外，在一定条件下，这种解的任意阶导数在$L^2$空间和整体时间中也是有界的。这些估计与光谱投影的控制相结合，足以获得解析解的误差估计并收敛到平衡态。应该注意的是，这是第一次，误差估计已产生的任何数值方法，近似FPL方程与任何范围的潜力。
英文摘要：
Error estimates are rigorously derived for a semi-discrete version of a conservative spectral method for approximating the space-homogeneous Fokker-Planck-Landau (FPL) equation associated to hard potentials. The analysis included shows that the semi-discrete problem has a unique solution with bounded moments. In addition, the derivatives of such a solution up to any order also remain bounded in $L^2$ spaces globally time, under certain conditions. These estimates, combined with control of the spectral projection, are enough to obtain error estimates to the analytical solution and convergence to equilibrium states. It should be noted that this is the first time that an error estimate has been produced for any numerical method which approximates FPL equations associated to any range of potentials.
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