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

TL;DR

This paper introduces adaptive hierarchic transformations for dynamic p-enrichment in slope-limiting techniques applied to Runge-Kutta discontinuous Galerkin methods, enhancing solution accuracy for advection-diffusion systems.

Contribution

It develops a novel family of generalized slope limiters and coupled p-enrichment schemes for improved DG solutions of complex PDEs.

Findings

Analyzed the error behavior of new limiters on model problems.

Compared advantages and disadvantages of various limiters.

Proposed flexible p-enrichment strategies for DG methods.

Abstract

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 limiter's 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.