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Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws
Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws-2020 Using the theoretical framework of algebraic flux correction and invariant domain preserving schemes, we introduce
a monolithic approach to convex limiting in continuous finite element schemes for linear advection equations, nonlinear
scalar conservation laws, and hyperbolic systems. In contrast to flux-corrected transport (FCT) algorithms that apply limited
antidiffusive corrections to bound-preserving low-order solutions, our new limiting strategy exploits the fact that these solutions
can be expressed as convex combinations of bar states belonging to a convex invariant set of physically admissible solutions.
Each antidiffusive flux is limited in a way which guarantees that the associated bar state remains in the invariant set and
preserves appropriate local bounds. There is no free parameter and no need for limit fluxes associated with the consistent mass
matrix of time derivative term separately. Moreover, the steady-state limit of the nonlinear discrete problem is well defined and
independent of the pseudo-time step. In the case study for the Euler equations, the components of the bar states are constrained
sequentially to satisfy local maximum principles for the density, velocity, and specific total energy in addition to positivity
preservation for the density and pressure. The results of numerical experiments for standard test problems illustrate the ability
of built-in convex limiters to resolve steep fronts in a sharp and nonoscillatory manner. Keywords: Hyperbolic conservation laws | Positivity preservation | Invariant domains | Finite elements | Algebraic flux correction | Convex limiting |
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