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

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.

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Cited by in corpus (6)

• Algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws
• Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting
• A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations
• A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws
• Spectral analysis of high order continuous FEM for hyperbolic PDEs on triangular meshes: influence of approximation, stabilization, and time-stepping
• Local subcell monolithic DG/FV convex property preserving scheme on unstructured grids and entropy consideration