We propose a spectral method for the 1D-1V Vlasov-Poisson system where the discretization in velocity space is based on asymmetrically-weighted Hermite functions, dynamically adapted via a scaling $\alpha$ and shifting $u$ of the velocity variable. Specifically, at each time instant an adaptivity criterion selects new values of $\alpha$ and $u$ based on the numerical solution of the discrete Vlasov-Poisson system obtained at that time step. Once the new values of the Hermite parameters $\alpha$ and $u$ are fixed, the Hermite expansion is updated and the discrete system is further evolved for the next time step. The procedure is applied iteratively over the desired temporal interval. The key aspects of the adaptive algorithm are: the map between approximation spaces associated with different values of the Hermite parameters that preserves total mass, momentum and energy; and the adaptivity criterion to update $\alpha$ and $u$ based on physics considerations relating the Hermite parameters to the average velocity and temperature of each plasma species. For the discretization of the spatial coordinate, we rely on Fourier functions and use the implicit midpoint rule for time stepping. The resulting numerical method possesses intrinsically the property of fluid-kinetic coupling, where the low-order terms of the expansion are akin to the fluid moments of a macroscopic description of the plasma, while kinetic physics is retained by adding more spectral terms. Moreover, the scheme features conservation of total mass, momentum and energy associated in the discrete, for periodic boundary conditions. A set of numerical experiments confirms that the adaptive method outperforms the non-adaptive one in terms of accuracy and stability of the numerical solution.

翻译：针对一维一速Vlasov-Poisson系统，我们提出一种谱方法，其速度空间离散基于非对称加权埃尔米特函数，并通过速度变量的缩放参数α和平移参数u实现动态自适应。具体而言，在每个时间步，自适应准则根据该时间步离散Vlasov-Poisson系统的数值解，选取α和u的新值。待埃尔米特参数α和u的新值确定后，更新埃尔米特展开，并进一步推进离散系统进入下一时间步。该过程在所需时间区间内迭代执行。自适应算法核心要点包括：一方面，在保持总质量、动量与能量守恒的前提下，建立不同埃尔米特参数值对应逼近空间之间的映射关系；另一方面，基于物理考量（将埃尔米特参数与各等离子体组分的平均速度和温度相关联）制定α和u的更新准则。空间坐标离散采用傅里叶函数，时间推进使用隐式中点法则。最终数值方法内禀流-动耦合特性，其中展开的低阶项类似等离子体宏观描述的流体矩，而高阶谱项则保留动力学物理。此外，该格式在周期边界条件下离散层面保持总质量、动量与能量守恒。系列数值实验证实，自适应方法在数值解的精度与稳定性方面均优于非自适应方法。