Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws

TL;DR

This paper introduces a new framework for enforcing maximum principles in high-order discontinuous Galerkin methods for hyperbolic conservation laws, combining invariant domain preservation with advanced flux limiting techniques.

Contribution

It develops a novel, provably invariant DG scheme using Bernstein polynomials and subcell flux limiters, enabling high-order, bound-preserving solutions for complex conservation laws.

Findings

Successfully preserves bounds in scalar and system conservation laws.

Achieves high-order accuracy with Bernstein polynomial basis.

Demonstrates robustness on benchmark problems including Euler and shallow water equations.

Abstract

In this work we present a framework for enforcing discrete maximum principles in discontinuous Galerkin (DG) discretizations. The developed schemes are applicable to scalar conservation laws as well as hyperbolic systems. Our methodology for limiting volume terms is similar to recently proposed methods for continuous Galerkin approximations, while DG flux terms require novel stabilization techniques. Piecewise Bernstein polynomials are employed as shape functions for the DG spaces, thus facilitating the use of very high order spatial approximations. We discuss the design of a new, provably invariant domain preserving DG scheme that is then extended by state-of-the-art subcell flux limiters to obtain a high-order bound preserving approximation. The limiting procedures can be formulated in the semi-discrete setting. Thus convergence to steady state solutions is not inhibited by the algorithm. We present numerical results for a variety of benchmark problems. Conservation laws considered in this study are linear and nonlinear scalar problems, as well as the Euler equations of gas dynamics and the shallow water system.