  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.
