论文标题
基于物理学的适应性基于vlasov-Poisson方程的光谱方法，基于速度空间中不对称加权的爱好扩展
Physics-based adaptivity of a spectral method for the Vlasov-Poisson equations based on the asymmetrically-weighted Hermite expansion in velocity space
论文作者
论文摘要
我们为1D-1V Vlasov-Poisson系统提出了一种频谱方法，其中速度空间中的离散化基于非对称加权的HERMITE功能，通过缩放$α$动态适应了速度变量的$ u $。具体而言，每次适应性标准在每次即时选择$α$和$ u $的新值都基于当时获得的离散vlasov-Poisson系统的数值解决方案。一旦固定了HERMITE参数的新值$α$和$ U $，则更新了HERMITE的扩展，并在下一个时间步骤进一步发展了离散系统。该过程在所需的时间间隔上迭代应用。自适应算法的关键方面是：与保留总质量，动量和能量的不同值相关的近似空间之间的映射；以及根据物理学的注意事项更新$α$和$ U $的适应性标准，将HERMITE参数与每个等离子体物种的平均速度和温度有关。为了离散空间坐标，我们依靠傅立叶函数，并使用隐式中点规则进行时间步进。所得的数值方法本质上具有流体运动耦合的特性，其中膨胀的低阶项类似于血浆的宏观描述的流体矩，而动力学物理学则通过添加更多的光谱术语来保留。此外，该方案的特征是对周期性边界条件中离散中相关的总质量，动量和能量的保护。一组数值实验证实，自适应方法在数值解决方案的准确性和稳定性方面优于非自适应方法。
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 $α$ and shifting $u$ of the velocity variable. Specifically, at each time instant an adaptivity criterion selects new values of $α$ 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 $α$ 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 $α$ 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.
