Analysis of algebraic flux correction for semi-discrete advection problems

TL;DR

This paper analyzes the stability and error properties of algebraic flux correction schemes with convex limiting for semi-discrete advection problems, providing theoretical guarantees and numerical validation.

Contribution

It offers a rigorous stability and error analysis for flux correction schemes, including a new limiter enforcing coercivity, and validates findings through numerical experiments.

Findings

Proves global-in-time existence and 1/2 convergence rate for semi-discrete advection.

Shows stabilized fluxes improve accuracy and coercivity enforcement can be often omitted.

Numerical experiments confirm theoretical predictions and method effectiveness.

Abstract

We present stability and error analysis for algebraic flux correction schemes based on monolithic convex limiting. For a continuous finite element discretization of the time-dependent advection equation, we prove global-in-time existence and the worst-case convergence rate of 1/2 w.r.t. the L2 error of the spatial semi-discretization. Moreover, we address the important issue of stabilization for raw antidiffusive fluxes. Our a priori error analysis reveals that their limited counterparts should satisfy a generalized coercivity condition. We introduce a limiter for enforcing this condition in the process of flux correction. To verify the results of our theoretical studies, we perform numerical experiments for simple one-dimensional test problems. The methods under investigation exhibit the expected behavior in all numerical examples. In particular, the use of stabilized fluxes improves the accuracy of numerical solutions and coercivity enforcement often becomes redundant.