Energy conserving discontinuous Galerkin spectral element method for the Vlasov-Poisson system
Energy conserving discontinuous Galerkin spectral element method for the Vlasov-Poisson system
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DOI：
10.1016/j.jcp.2014.09.010
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发表时间：
2014-12
期刊：
影响因子：
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通讯作者：
Éric Madaule;M. Restelli;E. Sonnendrücker
中科院分区：
文献类型：
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作者：
Éric Madaule;M. Restelli;E. Sonnendrücker
我们提出了一种新的、节能的、谱元的、不连续的Galerkin方法，用于在任意维度上用笛卡尔网格逼近Vlasov-Poisson系统。该方法是在[4]中提出的方法的基础上衍生出来的，并做了两个修改：通过一个合适的投影算子作用于泊松问题的解来获得能量守恒，而不是通过求解多个泊松问题，并且在有限元公式中出现的所有积分都用高斯-洛巴托正交逼近，从而得到谱元公式。所得到的方法具有以下特性：精确的能量守恒（不包括时间离散带来的误差）、稳定性（由于使用了逆风数值通量）、高阶精度和高局域性。对于时间离散化，我们同时考虑了龙格-库塔方法和指数积分器，并给出了一维和二维情况下的结果（分别为相空间中的二维和四维）。
We propose a new, energy conserving, spectral element, discontinuous Galerkin method for the approximation of the Vlasov–Poisson system in arbitrary dimension, using Cartesian grids. The method is derived from the one proposed in [4], with two modifications: energy conservation is obtained by a suitable projection operator acting on the solution of the Poisson problem, rather than by solving multiple Poisson problems, and all the integrals appearing in the finite element formulation are approximated with Gauss–Lobatto quadrature, thereby yielding a spectral element formulation. The resulting method has the following properties: exact energy conservation (up to errors introduced by the time discretization), stability (thanks to the use of upwind numerical fluxes), high order accuracy and high locality. For the time discretization, we consider both Runge–Kutta methods and exponential integrators, and show results for 1D and 2D cases (2D and 4D in phase space, respectively).