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

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