Rigorous fully discrete regularized-volume drift estimate

Establish a rigorous regularized-volume drift estimate for the fully discrete backward-Euler discontinuous Galerkin level-set transport scheme showing that the temporal discretization contribution is of order $O(\gamma+\tau_n^{2/3})$, while treating the volume change caused by subsequent level-set reinitialization as a separate numerical effect.

Background

The paper rigorously analyzes regularized-volume drift for continuous frozen-velocity transport and for the spatially semi-discrete discontinuous Galerkin scheme. It then considers the fully discrete update, in which backward-Euler time stepping is applied over an outer transport interval consisting of several inner time steps. The authors indicate that a first-order time-discretization argument combined with a narrow-band comparison should yield an additional contribution of order $O(\gamma+\tau_n^{2/3})$, but they do not prove this estimate. Reinitialization is performed after transport and changes the level-set profile away from the zero contour, so its induced volume jump is explicitly kept separate from the proposed fully discrete transport estimate.