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

A semi-Lagrangian discontinuous finite element scheme based on the characteristic Galerkin method (CSLDG) is investigated, which directly discretizes an integral invariant model derived from the coupling of the transport equation and its adjoint equation. First, the existence and stability of CSLDG are proven, along with the uniqueness of the numerical solution. Subsequently, in contrast to the commonly used interpolation-based dimensional splitting schemes (IBS) within the CSLDG framework, a separated-variable dimensional splitting approach based on the tensor product (SVS) is proposed and applied to the two-dimensional case. Numerical experiments show comparable accuracy between methods, but SVS demonstrates superior computational efficiency to IBS, especially on large-scale meshes.