论文标题
局部宏观保守（LOMAC）低级张量法，使用不连续的盖尔金方法用于Vlasov动力学
A Local Macroscopic Conservative (LoMaC) low rank tensor method with the discontinuous Galerkin method for the Vlasov dynamics
论文作者
论文摘要
在本文中，我们提出了一种新型的局部宏观保守（LOMAC）低级张量法，并针对模拟Vlasov-Poisson（VP）系统的物理空间和相位空间进行了不连续的Galerkin（DG）离散化。 LOMAC特性是指在离散水平上对宏观质量，动量和能量的确切局部保护。最近开发的LOMAC低级张量算法（ARXIV：2207.00518）同时使用动力学通量矢量拆分，同时进化了质量，动量和能量的宏观保护定律；然后，通过将低等级动力学解决方案投影到具有相同宏观可观察物的子空间上，可以实现LOMAC属性。 本文是我们以前的工作的概括，但是DG离散化以利用其在处理边界条件中的紧凑性和灵活性，并且从长远来看。该算法的开发方式与有限差异方案相似，通过观察DG方法可以以节点方式等效地观察。使用淋巴结DG方法，假设有张力计算网格，将能够（1）基于DG向上的运输项离散化来得出不同节点点的分化矩阵，并且（2）根据节点DG网格点定义加权内部产品空间。该算法可以通过溶液张量和相应的保守投影算法的分层塔克分解来扩展到高维问题。本着类似的精神，可以将算法扩展到非结构化网格的淋巴结或其他类型的离散化的DG方法，例如速度方向的光谱法。进行广泛的数值结果以展示该方法的功效。
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 (1) derive differentiation matrices for different nodal points based on a DG upwind discretization of transport terms, and (2) 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 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.
