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 Q 2 discretizations of linear and nonlinear two-dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions.

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

“…The highly dissipative LLF flux (22) satisfies not only the entropy condition (12) but also the assumptions of Theorem 2. The corresponding IDP bar states are given bȳ…”

“…The AFC methodology modifies a standard Galerkin discretization by adding artificial diffusion operators and limited antidiffusive fluxes. The convex limiting techniques proposed in [14,19,22] are applicable to nonlinear hyperbolic problems and lead to high-order IDP approximations. However, additional inequality constraints must be taken into account to ensure entropy stability.…”

“…As shown by Guermond et al [15], entropy stability is an essential requirement for convergence of AFC schemes to correct weak solutions of nonlinear hyperbolic problems. Residual-based entropy viscosity [14,15] was found to be a good way to stabilize flux-corrected continuous Galerkin (CG) approximations [14,22]. However, it involves a free parameter and does not guarantee entropy stability.…”

A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations

“…Another limiting strategy introduced by Kuzmin and colleagues is the algebraic flux-correction strategy [18]. This was generalized by Guermond, Popov, and colleagues using a low-order discretization based on a artificial "graph viscosity" term.…”

Bounds-constrained polynomial approximation using the Bernstein basis

“…One particular application of bounds-constrained approximation comes in the numerical solution of partial differential equations, especially hyperbolic equations [30]. Some approaches to limiting in discontinuous Galerkin (DG) methods for hyperbolic PDE explicitly utilize the geometric properties of Bernstein polynomials to enforce maximum principles and other invariant properties [18,23]. These methods are monolithic, with a built-in limiting process.…”