A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method with the Discontinuous Galerkin Method for the Vlasov Dynamics
A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method with the Discontinuous Galerkin Method for the Vlasov Dynamics
复制标题
Vlasov动力学的局部宏观保守(LoMaC)低阶张量方法和间断伽辽金方法
DOI：
10.1007/s42967-023-00277-7
复制
发表时间：
2023
影响因子：
1.6
通讯作者：
Qiu, Jing-Mei
中科院分区：
文献类型：
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作者：
Guo, Wei;Ema, Jannatul Ferdous;Qiu, Jing-Mei
本文提出了一种新的局部宏观守恒（LoMaC）低秩张量方法，该方法在物理空间和相空间中采用间断Galerkin（DG）离散，并用于模拟Vlasov-Poisson（VP）系统. LoMaC性质是指在离散水平上宏观质量、动量和能量的精确局部守恒。最近开发的LoMaC低秩张量算法（arXiv：2207.00518）使用动力学通量矢量分裂同时演化质量，动量和能量的宏观守恒定律;然后通过将低秩动力学解投影到共享相同宏观可观测量的子空间上来实现LoMaC属性。本文是我们以前的工作的推广，但DG离散化，以利用其紧凑性和灵活性，在处理边界条件和其优越的上级精度在长期。该算法是开发在一个类似的方式为有限差分格式，通过观察DG方法可以被视为等同于在一个节点的方式。利用节点DG方法，假设张量化计算网格，将能够（i）基于运输项的DG迎风离散化导出不同节点的微分矩阵，以及（ii）基于节点DG网格点定义加权内积空间。通过解张量的分层Tucker（HT）分解和相应的保守投影算法，该算法可以推广到高维问题。在类似的精神中，该算法可以扩展到非结构化网格的节点上的DG方法，或者扩展到其他类型的离散化，例如，速度方向的谱方法。大量的数值结果进行展示的方法的有效性。
In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method with discontinuous Galerkin (DG) discretization for the physical and phase spaces for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum, and energy at the discrete level. The recently developed LoMaC low rank tensor algorithm (arXiv: 2207.00518) simultaneously evolves the macroscopic conservation laws of mass, momentum, and energy using the kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables. This paper is a generalization of our previous work, but with DG discretization to take advantage of its compactness and flexibility in handling boundary conditions and its superior accuracy in the long term. The algorithm is developed in a similar fashion as that for a finite difference scheme, by observing that the DG method can be viewed equivalently in a nodal fashion. With the nodal DG method, assuming a tensorized computational grid, one will be able to (i) derive differentiation matrices for different nodal points based on a DG upwind discretization of transport terms, and (ii) define a weighted inner product space based on the nodal DG grid points. The algorithm can be extended to the high dimensional problems by hierarchical Tucker (HT) decomposition of solution tensors and a corresponding conservative projection algorithm. In a similar spirit, the algorithm can be extended to DG methods on nodal points of an unstructured mesh, or to other types of discretization, e.g., the spectral method in velocity direction. Extensive numerical results are performed to showcase the efficacy of the method.
VLASOV-Poisson系统的高阶节能不连续Galerkin方法
DOI：
--
发表时间：
2012
期刊：
影响因子：
--
作者：
B. A. Dios;Soheil Hajian
通讯作者：
Soheil Hajian