We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

翻译：本文分析了一维Vlasov-Poisson方程谱离散格式的反对称特性。该离散化方案采用速度方向的对称加权Hermite基函数谱展开、空间中心有限差分及时间隐式Runge Kutta积分器。所提出的离散格式保持了Vlasov方程中平流算子的反对称结构，从而形成稳定的数值方法。我们将此离散化应用于两种表述形式：标准Vlasov-Poisson方程及其连续变换的平方根表示。后者能保持粒子分布函数的正定性。我们通过解析方法推导了两种表述的守恒性质（包括粒子数、动量和能量守恒），并在以下基准问题上进行了数值验证：构造解、线性和非线性Landau阻尼、双流不稳定性、尾隆不稳定性以及离子声波。