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

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.

Figures (3)

• Figure 1: Comparison of the linear and nonlinear limiting approaches for pressure (left) and entropy (right) constraints for a static discontinuity problem computed with a $\mathbb P_9$ approximation. Pressure limiting computed using analytic formulation, entropy limiting computed using iterative formulation. Dotted line represents zero contour for the constraint functional.
• Figure 1: Comparison of the $L^{\infty}$ norm of the pressure error using linear and nonlinear limiting for the near-vacuum isentropic Euler vortex after one flow-through of the domain ($t = 20$) computed with a $\mathbb P_4$ approximation.
• Figure 2: Comparison of the $L^{\infty}$ norm of the pressure error using linear and nonlinear limiting for the near-vacuum isentropic Euler vortex after one flow-through of the domain ($t = 20$) computed with a $\mathbb P_5$ approximation.