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Convex limiting for finite elements and its relationship to residual distribution

Dmitri Kuzmin

TL;DR

The paper surveys element-based convex limiting techniques for continuous finite element discretizations of nonlinear hyperbolic problems and reveals their deep connection to residual distribution (RD) methods. It articulates two main convex-limiting paradigms—a flux-corrected transport (FCT)-style two-stage approach and a monolithic convex limiting (MCL) method—that enforce inequality constraints to keep intermediate states within a convex invariant set $\mathcal{G}$; the framework covers both scalar and system limits, using mechanisms such as scaling factors $\alpha^e$ and clip-and-scale operations. An Euler equations example (double Mach reflection) demonstrates that MCL combined with clip-and-scale can yield nonoscillatory, positivity-preserving solutions with higher accuracy than the base low-order Rusanov scheme. The residual-distribution viewpoint clarifies the relationship between low-order diffusion and antidiffusive corrections and points to pathways for extending these ideas to finite-volume formulations and more general meshes, fostering a unified development across discretization paradigms.

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.

Convex limiting for finite elements and its relationship to residual distribution

TL;DR

The paper surveys element-based convex limiting techniques for continuous finite element discretizations of nonlinear hyperbolic problems and reveals their deep connection to residual distribution (RD) methods. It articulates two main convex-limiting paradigms—a flux-corrected transport (FCT)-style two-stage approach and a monolithic convex limiting (MCL) method—that enforce inequality constraints to keep intermediate states within a convex invariant set ; the framework covers both scalar and system limits, using mechanisms such as scaling factors and clip-and-scale operations. An Euler equations example (double Mach reflection) demonstrates that MCL combined with clip-and-scale can yield nonoscillatory, positivity-preserving solutions with higher accuracy than the base low-order Rusanov scheme. The residual-distribution viewpoint clarifies the relationship between low-order diffusion and antidiffusive corrections and points to pathways for extending these ideas to finite-volume formulations and more general meshes, fostering a unified development across discretization paradigms.

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.
Paper Structure (8 sections, 20 equations, 1 figure)

This paper contains 8 sections, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Element-based Rusanov method
  3. Residual distribution
  4. Convex limiting
  5. Limiting for scalars
  6. Limiting for systems
  7. Example: Euler equations
  8. Concluding remarks

Figures (1)

  • Figure 1: Double Mach reflection: density $\rho_h$ calculated using $\mathbb P_1$ elements and $h=1/128$.