We analyze the anti-symmetric properties of 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, and bump-on-tail instability.

翻译：我们分析了单质点Vlasov-Poisson方程谱离散的反对称性质。该离散方法采用对称加权Hermite基函数进行速度方向的谱展开、空间方向中心有限差分以及时间方向隐式龙格-库塔积分器。所提出的离散方案保持了Vlasov方程中平流算子的反对称结构，从而得到稳定的数值方法。我们将该离散方法应用于两种表述形式：标准Vlasov-Poisson方程及其连续变换后的平方根表示，后者能保持粒子分布函数的正定性。我们解析推导了两种表述的守恒性质，包括粒子数、动量和能量守恒，并通过以下基准问题进行了数值验证：人为构造解、线性与非线性朗道阻尼、双流不稳定性以及束尾不稳定性。