Continuously bounds-preserving discontinuous Galerkin methods for hyperbolic conservation laws

• Published in: Journal of computational physics Vol. 508; no. C; p. 113010
• Main Author: Dzanic, T.
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.07.2024; Elsevier
• Subjects: Bounds-preserving; Discontinuous Galerkin; High-order; Hyperbolic conservation laws; Limiting; Positivity-preserving; Hyperbolic conservation laws; Discontinuous Galerkin; Bounds-preserving; Limiting; Positivity-preserving; High-order
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2024.113010

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. •Novel bounds-preserving stabilization method for discontinuous Galerkin schemes.•Ensures constraints are satisfied across entire solution polynomial.•Applicable to any basis, mesh, and approximation order.