A note on higher-order and nonlinear limiting approaches for continuously bounds-preserving discontinuous Galerkin methods

TL;DR

This paper extends a limiting approach for high-order discontinuous Galerkin methods to handle nonlinear constraints exactly, reducing unnecessary dissipation in simulations of compressible gas dynamics.

Contribution

It introduces a nonlinear limiting procedure that ensures exactness for nonlinear quasiconcave constraints, improving upon previous methods that were only exact for linear constraints.

Findings

Enhanced limiting approach for nonlinear constraints

Application to nonlinear pressure and entropy constraints

Reduction in numerical dissipation in simulations

Abstract

In (Dzanic, J. Comp. Phys., 508:113010, 2024), a limiting approach for high-order discontinuous Galerkin schemes was introduced which allowed for imposing constraints on the solution continuously (i.e., everywhere within the element). While exact for linear constraint functionals, this approach only imposed a sufficient (but not the minimum necessary) amount of limiting for nonlinear constraint functionals. This short note shows how this limiting approach can be extended to allow exactness for general nonlinear quasiconcave constraint functionals through a nonlinear limiting procedure, reducing unnecessary numerical dissipation. Some examples are shown for nonlinear pressure and entropy constraints in the compressible gas dynamics equations, where both analytic and iterative approaches are used.