Locally bound-preserving enriched Galerkin methods for the linear advection equation

Abstract

In this work, we introduce algebraic flux correction schemes for enriched (P 1 ⊕ P 0 and Q 1 ⊕ P 0 ) Galerkin discretizations of the linear advection equation. The piecewise-constant component stabilizes the continuous Galerkin approximation without introducing free parameters. However, violations of discrete maximum principles are possible in the neighborhood of discontinuities and steep fronts. To keep the cell averages and the degrees of freedom of the continuous P 1 /Q 1 component in the admissible range, we limit the fluxes and element contributions, the complete removal of which would correspond to first-order upwinding. The first limiting procedure that we consider in this paper is based on the flux-corrected transport (FCT) paradigm. It belongs to the family of predictor-corrector algorithms and requires the use of small time steps. The second limiting strategy is monolithic and produces nonlinear problems with well-defined residuals. This kind of limiting is well suited for stationary and time-dependent problems alike. The need for inverting consistent mass matrices in explicit strong stability preserving Runge-Kutta time integrators is avoided by reconstructing nodal time derivatives from cell averages. Numerical studies are performed for standard 2D test problems.

Bound-preserving flux limiting for high-order explicit Runge-Kutta time discretizations of hyperbolic conservation laws

“…Using the GMC criterion (17) to define the bounds Q ±,GMC i for algorithm ( 21)-( 23), we propose an alternative definition of α ij . The GMC formula provides the BP property (31) under a time step restriction which depends on the scaling factor γ ≥ 0 and reduces to (26) for γ = 0.…”

Enriched Galerkin method for the shallow-water equations

“…Since no limiter was used, all approximations exhibit spurious oscillations close to the discontinuities. We expect the results to improve and the numerical solutions to become bound-preserving if a limiter, such as the one developed in Kuzmin et al (2020), is utilized (at least) for the free surface elevation.…”

“…Popular methods designed particularly for DG discretizations include the edge-based Barth-Jesperson limiter (Barth and Jespersen 1989), and its vertex-based counterparts (Kuzmin 2010;Aizinger 2011). Limiters for an EG discretization of the linear advection equation have recently been proposed in Kuzmin et al (2020). The approach therein deviates from classical DG slope limiters but rather fits in the framework of algebraic flux correction (Kuzmin 2012), which only recently has been extended to the DG setting (Anderson et al 2017;.…”

“…The approach therein deviates from classical DG slope limiters but rather fits in the framework of algebraic flux correction (Kuzmin 2012), which only recently has been extended to the DG setting (Anderson et al 2017;. Numerical solutions based on the methods in Kuzmin et al (2020) can be proven to satisfy discrete maximum principles under CFL-like time step restrictions, which makes the approach superior to geometrical slope limiting.…”

A subcell-enriched Galerkin method for advection problems

“…Similarly to CG approximations, EG methods for hyperbolic equations may develop spurious oscillations. Kuzmin et al [21] proposed several algebraic flux correction schemes to ensure the validity of local maximum principles. Limiting techniques of this kind have also been successfully applied to CG [18,27] and DG [10] discretizations.…”