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Scalable direct vlasov solver with discontinuous Galerkin method on unstructured mesh

机译:非结构网格上具有不连续Galerkin方法的可扩展直接vlasov求解器

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This paper presents the development of parallel direct Vlasov solvers with discontinuous Galerkin (DG) method for beam and plasma simulations in four dimensions. Both physical and velocity spaces are in two dimesions (2P2V) with unstructured mesh. Contrary to the standard particle-in-cell (PIC) approach for kinetic space plasma simulations, i.e., solving Vlasov-Maxwell equations, direct method has been used in this paper. There are several benefits to solving a Vlasov equation directly, such as avoiding noise associated with a finite number of particles and the capability to capture fine structure in the plasma. The most challanging part of a direct Vlasov solver comes from higher dimensions, as the computational cost increases as N2d, where d is the dimension of the physical space. Recently, due to the fast development of supercomputers, the possibility has become more realistic. Many efforts have been made to solve Vlasov equations in low dimensions before; now more interest has focused on higher dimensions. Different numerical methods have been tried so far, such as the finite difference method, Fourier Spectral method, finite volume method, and spectral element method. This paper is based on our previous efforts to use the DG method. The DG method has been proven to be very successful in solving Maxwell equations, and this paper is our first effort in applying the DG method to Vlasov equations. DG has shown several advantages, such as local mass matrix, strong stability, and easy parallelization. These are particularly suitable for Vlasov equations. Domain decomposition in high dimensions has been used for parallelization; these include a highly scalable parallel two-dimensional Poisson solver. Benchmark results have been shown and simulation results will be reported.
机译:本文介绍了采用不连续Galerkin(DG)方法的并行直接Vlasov解算器的开发,该方法可在四个维度上对射束和等离子体进行仿真。物理空间和速度空间都位于具有非结构化网格的两个维度(2P2V)中。与用于动力学空间等离子体模拟的标准单元中粒子(PIC)方法相反,即求解Vlasov-Maxwell方程,本文采用了直接方法。直接求解Vlasov方程有几个好处,例如避免与有限数量的粒子相关的噪声以及捕获等离子体中精细结构的能力。直接Vlasov求解器最具挑战性的部分来自更高的维度,因为计算成本随着N2d的增加而增加,其中d是物理空间的维度。近来,由于超级计算机的快速发展,这种可能性变得更加现实。以前,人们已经为解决低维Vlasov方程做出了许多努力。现在,更多的兴趣集中在更高的维度上。迄今为止,已经尝试了不同的数值方法,例如有限差分法,傅立叶光谱法,有限体积法和光谱元素法。本文基于我们先前使用DG方法的努力。事实证明,DG方法在求解麦克斯韦方程组方面非常成功,这是我们将DG方法应用于Vlasov方程的首次尝试。 DG具有许多优点,例如局部质量矩阵,稳定性强和易于并行化。这些特别适合于Vlasov方程。高维域分解已用于并行化;其中包括高度可扩展的并行二维Poisson求解器。已显示基准测试结果,并将报告模拟结果。

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