A Local Macroscopic Conservative(LoMaC)Low Rank Tensor Method with the Discontinuous Galerkin Method for the Vlasov Dynamics  

作　　者：Wei Guo Jannatul Ferdous Ema Jing-Mei Qiu 

机构地区：[1]Department of Mathematics and Statistics,Texas Tech University,Lubbock,TX,70409,USA [2]Department of Mathematical Sciences,University of Delaware,Newark,DE,19716,USA

出　　处：《Communications on Applied Mathematics and Computation》2024年第1期550-575,共26页应用数学与计算数学学报（英文）

基　　金：supported by the NSF(Grant Nos.the NSF-DMS-1818924 and 2111253);the Air Force Office of Scientific Research FA9550-22-1-0390 and Department of Energy DE-SC0023164;supported by the NSF(Grant Nos.NSF-DMS-1830838 and NSF-DMS-2111383);the Air Force Office of Scientific Research FA9550-22-1-0390.

摘　　要：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.

关 键 词：Hierarchical Tucker(HT)decomposition Conservative SVD Energy conservation Discontinuous Galerkin(DG)method 

分 类 号：O17[理学—数学]