A High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations

Abstract

Abstract. In this paper, we develop a class of high order conservative semi-Lagrangian (SL) discontinuous Galerkin (DG) methods for solving multi-dimensional linear transport equations. The methods rely on a characteristic Galerkin weak formulation, leading to L 2 stable discretizations for linear problems. Unlike many existing SL methods, the high order accuracy and mass conservation of the proposed methods are realized in a non-splitting manner. Thus, the detrimental splitting error, which is known to significantly contaminate long term transport simulations, will be not incurred. One key ingredient in the scheme formulation, borrowed from CSLAM [Lauritzen, Nair & Ullrich, 2010], is the use of Green's theorem which allows us to convert volume integrals into a set of line integrals. The resulting line integrals are much easier to approximate with high order accuracy, hence facilitating the implementation. Another novel ingredient is the construction of quadratic curves in approximating sides of upstream cell, leading to quadratic-curved quadrilateral upstream cells. Formal third order accuracy is obtained by such a construction. The desired positivity-preserving property is further attained by incorporating a high order bound-preserving filter. To assess the performance of the proposed methods, we test and compare the numerical schemes with a variety of configurations for solving several benchmark transport problems with both smooth and nonsmooth solutions. The efficiency and efficacy are numerically verified.

A High Order Semi-Lagrangian Discontinuous Galerkin Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model Without Operator Splitting

“…In this paper, we propose a class of high order semi-Lagrangian discontinuous Galerkin (SLDG) methods for the two-dimensional (2D) time dependent incompressible Euler equation in the vorticity stream-function formulation and the guiding center Vlasov model. This is a continuation of our previous research effort on the development of high order non-splitting SLDG methods for 2D linear transport equations [4] and the Vlasov-Poisson (VP) system [5].…”

“…In [4,5], we formulated a class of high order conservative SLDG schemes for solving 2D transport problems with application to the VP system. To the authors' best knowledge, such a method is the first SLDG scheme in the literature that is high order accurate (up to third order accurate), unconditionally stable, mass conservative and free of splitting error for 2D transport simulations.…”

“…Note that a similar algorithm can be formulated for the 2D incompressible Euler model in vorticity stream-function formulation as well. We start by reviewing the high order truly multi-dimensional SLDG framework originally proposed in [4] in a linear setting. Then we describe how to incorporate the high order characteristics tracing scheme proposed in [28] in the same SLDG framework for the nonlinear model problem.…”

A semi-Lagrangian discontinuous Galerkin (DG) – local DG method for solving convection-diffusion equations

“…Similar to the 1D case and the strategy in [3], we consider the adjoint problem for the test function ψ(x, y, t) satisfying…”

“…In this paper, we propose to evolve the convection term by the SLDG method recently proposed in [3], and treat the diffusion term by the local DG (LDG) method coupled with a diagonally implicit RK (DIRK) method along dynamic characteristics elements. The proposed method is termed as the SLDG-LDG method.…”