Analysis of algebraic flux correction schemes for semi-discrete advection problems

Abstract

AbstractBased on recent developments regarding the analysis of algebraic flux correction schemes, we consider a locally bound-preserving discretization of the time-dependent advection equation. Specifically, we analyze a monolithic convex limiting scheme based on piecewise (multi-)linear continuous finite elements in the semi-discrete formulation. To stabilize the discretization, we use low order time derivatives in the definition of raw antidiffusive fluxes. Our analytical investigation reveals that their limited counterparts should satisfy a certain compatibility condition. The conducted numerical experiments suggest that this prerequisite is satisfied unless the size of mesh elements is vastly different. We prove global-in-time existence of semi-discrete approximations and derive an a priori error estimate for finite time intervals with a worst-case convergence rate of $$\frac{1}{2}$$ 1 2 w. r. t. the $${\textrm{L}}^2$$ L 2  error. This rate is optimal in the setting under consideration because we allow all correction factors of the flux-corrected scheme to become zero. In this case, the algorithm reduces to the bound-preserving discrete upwinding method but the limited counterpart of this scheme converges much faster, in practice. Additional numerical experiments are performed to verify the provable convergence rate for a few variants of the scheme.