Continuously bounds-preserving discontinuous Galerkin methods for hyperbolic conservation laws

TL;DR

This work tackles the challenge of enforcing physical bounds for high-order DG discretizations of hyperbolic conservation laws in a way that remains valid for arbitrary evaluation points. It introduces a novel continuous limiting framework built on modified constraint functionals and an element-local optimization that computes a limiting factor, ensuring the entire solution polynomial satisfies convex invariants. The method preserves high-order accuracy across scalar transport to compressible Euler flows, and numerical experiments demonstrate continuous bounds satisfaction, robustness near shocks, and only modest overhead compared with discrete limiting. The approach enables robust remapping, adaptive mesh refinement, and overset/moving-mesh contexts where evaluation points do not align with limiting nodes, with potential extensions to local bounds and more complex invariant sets.

Abstract

For finite element approximations of transport phenomena, it is often necessary to apply a form of limiting to ensure that the discrete solution remains well-behaved and satisfies physical constraints. However, these limiting procedures are typically performed at discrete nodal locations, which is not sufficient to ensure the robustness of the scheme when the solution must be evaluated at arbitrary locations (e.g., for adaptive mesh refinement, remapping in arbitrary Lagrangian--Eulerian solvers, overset meshes, etc.). In this work, a novel limiting approach for discontinuous Galerkin methods is presented which ensures that the solution is continuously bounds-preserving (i.e., across the entire solution polynomial) for any arbitrary choice of basis, approximation order, and mesh element type. Through a modified formulation for the constraint functionals, the proposed approach requires only the solution of a single spatial scalar minimization problem per element for which a highly efficient numerical optimization procedure is presented. The efficacy of this approach is shown in numerical experiments by enforcing continuous constraints in high-order unstructured discontinuous Galerkin discretizations of hyperbolic conservation laws, ranging from scalar transport with maximum principle preserving constraints to compressible gas dynamics with positivity-preserving constraints.

Figures (21)

• Figure 1: Schematic of the limiting procedure for a vector-valued solution $\mathbf{u}(x) = [u_1(x), u_2(x)]^T$ showcasing the minimal limiting necessary for the limited solution $\widehat{\mathbf{u}}(x)$ to be continuously bounds-preserving.
• Figure 1: Convergence in the $L^1$ norm for the advecting waveforms problem at varying approximation order with continuously bounds-preserving limiting and global max principle bounds ($u \in [0,1]$).
• Figure 2: Schematic for an example system and constraints showing that limiting based on the extremum of the constraint functional ($g^*$) does not guarantee that the limited solution is continuously bounds-preserving.
• Figure 2: Convergence in the $L^1$ norm for the advecting waveforms problem at varying approximation order with discretely bounds-preserving limiting and global max principle bounds ($u \in [0,1]$).
• Figure 3: Schematic of the tolerance correction $\Delta h$ computed at the end of the optimization process.

Theorems & Definitions (4)

• Theorem 3.1: Bounds preservation
• Corollary 3.1.1
• Theorem 3.2: Continuity
• Lemma 3.3