Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws

TL;DR

This paper introduces a novel flux limiting technique for high-order Bernstein finite element methods, ensuring stability and maximum principles in scalar hyperbolic conservation laws while maintaining computational efficiency.

Contribution

It develops a monolithic flux limiting strategy that preserves invariant domains and local maximum principles in high-order Bernstein finite element discretizations.

Findings

Ensures local maximum principles under CFL conditions.

Maintains compact stencil property preventing non-nearest neighbor mass exchange.

Demonstrates effectiveness through numerical tests on 2D problems.

Abstract

This work extends the concepts of algebraic flux correction and convex limiting to continuous high-order Bernstein finite element discretizations of scalar hyperbolic problems. Using an array of adjustable diffusive fluxes, the standard Galerkin approximation is transformed into a nonlinear high-resolution scheme which has the compact sparsity pattern of the piecewise-linear or multilinear subcell discretization. The representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under CFL-like conditions. In contrast to predictor-corrector approaches based on the flux-corrected transport methodology, the proposed flux limiting strategy is monolithic; i.e., limited antidiffusive terms are incorporated into the well-defined residual of a nonlinear (semi-)discrete problem. A stabilized high-order Galerkin discretization is recovered if no limiting is performed. In the limited version, the compact stencil property prevents direct mass exchange between nodes that are not nearest neighbors. A formal proof of sparsity is provided for simplicial and box elements. The involved element contributions can be calculated efficiently making use of matrix-free algorithms and precomputed element matrices of the reference element. Numerical studies for $\mathbb{Q}_2$ discretizations of linear and nonlinear two-dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions.