Optimal L^2 Error Estimates for CSLDG Ideal Numerical Solutions in Time-Dependent Compressible Periodic Flows

Abstract

We study the characteristic Galerkin-type semi-Lagrangian discontinuous finite element (CSLDG) method for conservative linear transport equations. When the characteristic maps and upstream regions are exact, each update of the ideal CSLDG scheme is equivalent to exact conservative transport of the current numerical solution followed by L^2 projection onto the background DG space. The conventional estimate obtained by accumulating projection errors over successive time steps loses one order when Δt\asymp h, but this loss does not represent the actual accuracy of the scheme. For smooth velocity fields \A(\x,t) on a common periodic lattice, we construct local correctors used only in the analysis on uniform Cartesian meshes, assuming that Δt/h is fixed and that every component of the local Courant vector remains uniformly separated from the integers throughout the space-time domain. We prove strict contraction of the reference-cell transport-projection operator on the zero-mean polynomial space, establish uniform solvability of the corrector equation and Lipschitz estimates with respect to its parameter, and control the difference between the exact flow map and a local frozen translation through weak integral moments. This yields a one-step residual of order O(h^{K+2}), and, while retaining the step-by-step update and the standard L^2-projected initial data, gives, for the P^K approximation, an optimal O(h^{K+1}) error bound. The velocity may depend on both space and time, and incompressibility is not required. An appendix extends the analysis to a compact-support setting, in which the velocity field and the solution need not share a common spatial period.