针对弗拉索夫-泊松（Vlasov-Poisson, VP）系统的、基于非对称加权（AW）埃尔米特函数的伽辽金近似的收敛性
A stable and efficient Galerkin spectral method with asymmetrically-weighted Hermite functions for the Vlasov-Poisson system
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中文总结 AI 辅助
本工作分析了基于非对称加权埃尔米特函数的伽辽金近似在弗拉索夫-泊松系统中的收敛性，推导了误差估计并证实其在索伯列夫空间具有谱精度。
中文摘要 AI 辅助
本工作分析了应用于弗拉索夫-泊松（VP）系统的、基于非对称加权（AW）埃尔米特函数的伽辽金近似的收敛性。该VP系统通过速度空间中的埃尔米特函数被表述为双曲型系统。为获得数值方法的稳定性性质，我们考虑了无加权内积，并自然建立了关于无加权范数的稳定性，进而证明了该方法的收敛性。此外，我们推导了数值解与VP系统光滑解之间的误差估计。我们的结果证实，基于AW埃尔米特函数的伽辽金近似在索伯列夫空间中具有谱精度。
英文摘要
We analyze a Galerkin spectral method applied to the Vlasov-Poisson (VP) system based on time-independent asymmetrically-weighted (AW) Hermite functions in velocity. The VP system is written as an hyperbolic system using AW Hermite functions in velocity. Unlike the classical Petrov-Galerkin spectral method, which is not stable, the proposed Galerkin spectral method admits a natural stability in the unweighted L2 norm. This stability property enables a rigorous convergence analysis of the method. For sufficiently regular solutions with exponential decay in velocity, we establish error estimates between the exact and numerical solutions and prove convergence of the Galerkin spectral method. In particular, the method achieves spectral convergence in Sobolev spaces, with convergence rates determined by the regularity of the exact solution. Since the Gram matrix arising from the proposed Galerkin method is dense, we derive an equivalent form that retains the same approximation properties while preserving the sparsity structure similar to the classical Petrov-Galerkin method.
