Hamiltonian Spectral Methods for the Vlasov-Maxwell Equations

这个项目探讨了保持汉密尔顿结构的 Vlasov-Maxwell方程的数值方法。 根本 将谱型方法应用于无穷维问题的若干方面 哈密顿系统，使所得的半离散（连续的， 时间）系统精确地保持系统哈密顿量和一些其他的 正在研究守恒定律。 函数逼近 定义了一个算子来映射真无限维相空间 有限维数值近似，以及嵌入 运算符被定义为将该数值相空间映射回 精确相空间 这些运算符一起允许泊松括号 在无限维相空间上， 数值相空间;然后可以在 数值相空间使用精确的数值近似 汉米尔顿。 由此产生的有限维哈密顿系统 表示一种半离散数值方法， 系统能量。 的数值守恒的必要条件 卡西米尔和动量也被推导出来。 这些技术 也被推广到非孤立的哈密顿系统， 允许一般边界条件。 应用该等 弗拉索夫-麦克斯韦方程组的技术将允许数值等离子体 动力学理论模型，精确地保存能量，粒子和 动量，即使在空间、速度和时间的有限膨胀下。

This project explores the preservation of the Hamiltonian structure of Vlasov-Maxwell equations in a numerical method. The fundamental aspects of applying spectral type methods to infinite dimensional Hamiltonian systems so that the resulting semi-discrete (continuous in time) system exactly preserves the system Hamiltonian and some other conservation laws are being investigated. A functional approximation operator is defined to map the true infinite dimensional phase space to a finite dimensional numerical approximation, and an imbedding operator is defined to map this numerical phase space back to the exact phase space. Together these operators allow the Poisson bracket on the infinite dimensional phase space to be transferred to the numerical phase space; Hamiltonian dynamics can then be defined on the numerical phase space using a numerical approximation to the exact Hamiltonian. The resulting finite dimensional Hamiltonian system represents a semi-discrete numerical method that exactly conserves the system energy. Conditions necessary for the numerical conservation of Casimirs and momenta are also being derived. These techniques are also being extended to Hamiltonian systems which are not isolated and allow general boundary conditions. The application of these techniques to the Vlasov-Maxwell equations will allow numerical plasma kinetic theory models which exactly conserve energy, particles and momentum, even at a finite expansion in space, velocity and time.