Error Analysis of Time-Dependent Test Functions in the Semi-Lagrangian Discontinuous Finite Element Scheme Based on the Characteristic Galerkin Method

This paper studies the time-dependent test-function error in the characteristic Galerkin-type semi-Lagrangian discontinuous finite element (CSLDG) method caused by numerical integration errors of the characteristic ODE solver, and its effect on convergence. Unlike classical finite element methods and standard DG methods, the test functions in CSLDG are constructed by characteristic backtracking. As a result, ODE errors affect not only the upstream integration region but also enter directly into the discrete weak formulation through perturbations of the test function support. Neglecting mesh geometric errors, we introduce an ideal auxiliary solution and an auxiliary solution retaining the test-function error, and derive the corresponding error equation and recursive relations. First, based on a global time-dependent test-function error analysis, we show that the optimal $L^2$ convergence order of the $P^K$-CSLDG method is preserved provided that the ODE solver order satisfies $D\ge 2K+3+d$. We then propose a new analysis based on the propagation of local modal coefficient errors, which controls the error cell by cell through the actual modal update formulas and recovers the global $L^2$ estimate. This leads to an improved sufficient condition $D\ge K+1+\frac{d}{2}$. The results show that the required ODE accuracy depends strongly on the analytical route, and that the modal-coefficient-based analysis is both closer to the actual implementation and more consistent with numerical observations.