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

Abstract

In this work, we modify a continuous Galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures. Discrete entropy conditions are used to determine the minimal amount of entropy stabilization and constrain antidiffusive corrections of a property-preserving low-order scheme. The addition of a second-order entropy dissipative component to the antidiffusive part of a nearly entropy conservative numerical flux is generally insufficient to prevent violations of local bounds in shock regions. Our monolithic convex limiting technique adjusts a given target flux in a manner which guarantees preservation of invariant domains, validity of local maximum principles, and entropy stability. The new methodology combines the advantages of modern entropy stable / entropy conservative schemes and their local extremum diminishing counterparts. The process of algebraic flux correction is based on inequality constraints which provably provide the desired properties. No free parameters are involved. The proposed algebraic fixes are readily applicable to unstructured meshes, finite element methods, general time discretizations, and steady-state residuals. Numerical studies of explicit entropy-constrained schemes are performed for linear and nonlinear test problems.

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

“…Let u(x, t) be a scalar quantity of interest depending on the space location x ∈ R d , d ∈ {1, 2, 3} and time instant t ≥ 0. Consider an initial-boundary value problem of the form [17,33] ∂u ∂t…”

“…Restricting the monolithic convex limiting strategy proposed in [33] to f e ij , we will correct the bar statesū e ij of the low-order IDP scheme (17)…”

“…A locally bound-preserving IDP approximation to a given target flux f e ij is given by [33] f e, * ij =…”

“…To facilitate a direct comparison with the P 1 /Q 1 version of algebraic flux correction schemes and variational approaches to shock capturing, let us now consider the solid body rotation benchmark [27,33,34,38]. In this 2D experiment, we solve the unsteady linear advection equation ∂u ∂t + ∇ · (vu) = 0 in Ω = (0, 1) 2 using the divergence-free velocity field v(x, y) = 2π(0.5 − y, x − 0.5) to rotate a slotted cylinder, a sharp cone, and a smooth hump around the center (0.5, 0.5) of the domain Ω. Homogeneous boundary conditions are prescribed on Γ − .…”

“…This approach, which is described in §5, involves the solution of small sparse linear systems on each macroelement. The IDP property of the corresponding discrete problem is shown using the proof techniques developed in [17,33]. In §6 and §7, we discuss the optional stabilization techniques for the high-order target flux and Hessian-based smoothness indicators that preserve the high-order accuracy near smooth local extrema.…”