arXiv:2610.33449v1 Announce Type: new Abstract: The optimization-based cell average limiter enforces global conservation and bounds, and has been successfully applied to discontinuous Galerkin (DG) discretizations of a wide range of nonlinear systems, from phase-field models to the compressible Navier-Stokes equations. In this paper, taking the heat and convection-diffusion equations as prototypes, we analyze the convergence of the scheme resulting from cell average limiting. Encoding conservation and admissibility in a differential inclusion identifies the limiter as a backward Euler proximal step in a Lie splitting of the constrained evolution. This proximal step is the metric projection onto the convex admissible set, and we prove that it is nonexpansive and stable in the $L^2$ norm, and that it does not increase the total variation of the cell averages. Combined with the standard error analysis, nonexpansiveness yields an $O(\tau+h^{k+1})$ estimate for implicit Euler time stepping with DG discretization of polynomial degree $k$ under a reverse CFL condition, together with an unconditional $O(\tau+h^{k})$ estimate. Numerical experiments confirm the predicted convergence rates and the enforcement of the bounds.