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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.

Transport problems arise across diverse fields of science and engineering. Semi-Lagrangian (SL) discontinuous Galerkin (DG) methods are a class of high-order deterministic transport solvers that enjoy advantages of both the SL approach and the DG spatial discretization. In this paper, we review existing SLDG methods to date and compare numerically their performance. In particular, we make a comparison between the splitting and non-splitting SLDG methods for multi-dimensional transport simulations. Through extensive numerical results, we offer a practical guide for choosing optimal SLDG solvers for linear and nonlinear transport simulations.

In this paper, we present error estimates of the integral deferred correction method constructed with stiffly accurate implicit Runge–Kutta methods with a nonsingular matrix Ain its Butcher table representation, when applied to stiff problems characterized by a small positive parameter ε. In our error estimates, we expand the global error in powers of εand show that the coefficients are global errors of the integral deferred correction method applied to a sequence of differential algebraic systems. A study of these errors and of the remainder of the expansion yields sharp error bounds for the stiff problem. Numerical results for the van der Pol equation are presented to illustrate our theoretical findings. Finally, we study the linear stability properties of these methods.