Numerical Analysis and Dimension Splitting for A Semi-Lagrangian Discontinuous Finite Element Scheme Based on the Characteristic Galerkin Method

A characteristic Galerkin-type semi-Lagrangian discontinuous Galerkin methods (CSLDG) is investigated, which directly discretizes an integral invariant model derived from the coupling of a transport equation and its dual equation. First, the existence and uniqueness of the CSLDG numerical solutions are proven, along with the stability of the numerical scheme. Subsequently, in contrast to the commonly used interpolation-based dimensional splitting schemes within the CSLDG framework, a separated-variable dimensional splitting approach based on the tensor product is proposed and applied to the two-dimensional case.