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In this paper, we develop a class of high order conservative semi-Lagrangian(SL) discontinuous Galerkin (DG) methods for solving multi-dimensional lineartransport equations. The methods rely on a characteristic Galerkin weakformulation, leading to $L^2$ stable discretizations for linear problems.Unlike many existing SL methods, the high order accuracy and mass conservationof the proposed methods are realized in a non-splitting manner. Thus, thedetrimental splitting error, which is known to significantly contaminate longterm transport simulations, will be not incurred. One key ingredient in thescheme formulation, borrowed from CSLAM [Lauritzen, Nair, and Ullrich, 2010],is the use of Green's theorem which allows us to convert volume integrals intoa set of line integrals. The resulting line integrals are much easier toapproximate with high order accuracy, hence facilitating the implementation.Another novel ingredient is the construction of quadratic curves inapproximating sides of upstream cell, leading to quadratic-curved quadrilateralupstream cells. Formal third order accuracy is obtained by such a construction.The desired positivity-preserving property is further attained by incorporatinga high order bound-preserving filter. To assess the performance of the proposedmethods, we test and compare the numerical schemes with a variety ofconfigurations for solving several benchmark transport problems with bothsmooth and nonsmooth solutions. The efficiency and efficacy are numericallyverified.

A high order conservative semi-Lagrangian discontinuous Galerkin method for two-dimensional transport simulations