Convex limiting for finite elements and its relationship to residual distribution

TL;DR

This paper reviews recent convex limiting techniques for finite element discretizations of nonlinear hyperbolic problems, highlighting their interpretation as residual distribution methods and their role in enforcing maximum principles.

Contribution

It introduces and compares two convex limiting approaches—flux-corrected transport and monolithic convex limiting—and relates them to residual distribution schemes.

Findings

Convex limiting techniques can enforce discrete maximum principles.

Residual distribution interpretation provides new insights into stabilization methods.

Two main convex limiting strategies are analyzed and compared.

Abstract

We review some recent advances in the field of element-based algebraic stabilization for continuous finite element discretizations of nonlinear hyperbolic problems. The main focus is on multidimensional convex limiting techniques designed to constrain antidiffusive element contributions rather than fluxes. We show that the resulting schemes can be interpreted as residual distribution methods. Two kinds of convex limiting can be used to enforce the validity of generalized discrete maximum principles in this context. The first approach has the structure of a localized flux-corrected transport (FCT) algorithm, in which the computation of a low-order predictor is followed by an antidiffusive correction stage. The second option is the use of a monolithic convex limiting (MCL) procedure at the level of spatial semi-discretization. In both cases, inequality constraints are imposed on scalar functions of intermediate states that are required to stay in convex invariant sets.