A high-order augmented basis positivity-preserving discontinuous Galerkin method for a Linear Hyperbolic Equation

TL;DR

The paper tackles positivity preservation in high-order discontinuous Galerkin schemes for linear hyperbolic equations by introducing an optimization-based augmentation of the finite element spaces. It enriches standard DG spaces with higher-order basis functions to guarantee a positive cell average of the unmodulated DG solution, enabling the simple Zhang-Shu limiter to maintain high-order accuracy without affecting cell averages. The method extends to polynomial degrees $k>1$ and to both (1+1) and (1+2) dimensions, including space-time DG formulations, with adaptive activation on troubled cells and the possibility of preprocessing augmented bases. Numerical experiments in 2D and 3D demonstrate high-order convergence, robustness to variable coefficients and wide CFL ranges, and effective positivity preservation when coupled with the limiter.

Abstract

This paper designs a high-order positivity-preserving discontinuous Galerkin (DG) scheme for a linear hyperbolic equation. The scheme relies on augmenting the standard polynomial DG spaces with additional basis functions. The purpose of these augmented basis functions is to ensure the preservation of a positive cell average for the unmodulated DG solution. As such, the simple Zhang and Shu limiter~\cite{zhang2010maximum} can be applied with no loss of accuracy for smooth solutions, and the cell average remains unaltered. A key feature of the proposed scheme is its implicit generation of suitable augmented basis functions. Nonlinear optimization facilitates the design of these augmented basis functions. Several benchmarks and computational studies demonstrate that the method works well in two and three dimensions. \keywords{discontinuous Galerkin \and High-order \and Positivity-preserving