Adaptive hierarchic transformations for dynamically $p$-enriched slope-limiting over discontinuous Galerkin systems of generalized equations

We study a family of generalized slope limiters in two dimensions for Runge-Kutta discontinuous Galerkin (RKDG) solutions of advection--diffusion systems. We analyze the numerical behavior of these limiters applied to a pair of model problems, comparing the error of the approximate solutions, and discuss each limiters advantages and disadvantages. We then introduce a series of coupled $p$-enrichment schemes that may be used as standalone dynamic $p$-enrichment strategies, or may be augmented via any in the family of variable-in-$p$ slope limiters presented.