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.

💡 Research Summary

The paper presents a rigorous mathematical analysis and an efficient implementation strategy for a semi‑Lagrangian discontinuous Galerkin (SLDG) scheme derived from the characteristic Galerkin method, referred to as CSLDG. The authors start by coupling the linear scalar transport equation (U_t + \nabla!\cdot(AU)=0) with its adjoint (\psi_t + A!\cdot!\nabla\psi =0) to obtain the integral invariant (\frac{d}{dt}\int_{\Omega(t)}U\psi,dx = 0). This invariant is discretized directly, avoiding the usual reconstruction steps of conventional semi‑Lagrangian methods.

A set of assumptions on the velocity field (A(x,t)) is introduced: continuity, continuously differentiable components, and a uniformly bounded gradient (|\nabla_x A|_{op}\le L_A). Under these conditions the flow map (D_A^{T\to t}) and the associated transport operator (\Phi_A^{T\to t}) are globally well‑posed and invertible. The Jacobian determinant of the flow map is shown to be (\exp!\bigl(-\int_T^t \nabla!\cdot A,d\tau\bigr)), which guarantees that the change‑of‑variables in the integral invariant is mathematically sound.

The spatial domain is partitioned into a possibly non‑uniform mesh (\mathcal{T}_h) and the time interval (