A generalized Fourier–Hermite semi-discretization for the Vlasov–Poisson equation is introduced. The formulation of the method includes as special cases the symmetrically-weighted and asymmetrically-weighted Fourier–Hermite methods from the literature. The numerical scheme is formulated as a weighted Galerkin method with two separate scaling parameters for the Hermite polynomial and the exponential part of the new basis functions. Exact formulas for the error in mass, momentum, and energy conservation of the method depending on the parameters are devised and \(L^2\) stability is discussed. The numerical experiments show that an optimal choice of the additional parameter in the generalized method can yield improved accuracy compared to the existing methods, but also reveal the distinct stability properties of the symmetrically-weighted method.

引入了Vlasov-Poisson方程的广义Fourier-Hermite半离散化。该方法的制定包括特殊文献中的对称加权和非对称加权傅里叶-赫尔米特方法。数值方案被公式化为加权Galerkin方法，具有两个独立的缩放参数用于Hermite多项式和新基础函数的指数部分。根据参数，设计了该方法的质量，动量和能量守恒误差的精确公式，并且\（L ^ 2 \）讨论稳定性。数值实验表明，与现有方法相比，广义方法中附加参数的最优选择可以提高精度，同时也揭示了对称加权方法的独特稳定性。