Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws

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.

Automated summary

A framework for enforcing discrete maximum principles in discontinuous Galerkin discretizations is presented, applicable to scalar conservation laws and hyperbolic systems. Piecewise Bernstein polynomials are used as shape functions, a new invariant domain preserving DG scheme is designed and extended by subcell flux limiters for high-order bound preserving approximation. Numerical results for various benchmark problems are presented, considering linear and nonlinear scalar problems, Euler equations of gas dynamics, and the shallow water system.