Optimization-based Cell Average Limiters as Metric Projections: Constrained Evolution and Convergence Analysis for Convection-diffusion Equations
Abstract
The optimization-based cell average limiter enforces global conservation and bounds, and has been successfully applied to discontinuous Galerkin (DG) discretizations of a wide range of nonlinear systems, from phase-field models to the compressible Navier-Stokes equations. In this paper, taking the heat and convection-diffusion equations as prototypes, we analyze the convergence of the scheme resulting from cell average limiting. Encoding conservation and admissibility in a differential inclusion identifies the limiter as a backward Euler proximal step in a Lie splitting of the constrained evolution. This proximal step is the metric projection onto the convex admissible set, and we prove that it is nonexpansive and stable in the L2 norm, and that it does not increase the total variation of the cell averages. Combined with the standard error analysis, nonexpansiveness yields an O(τ+hk+1) estimate for implicit Euler time stepping with DG discretization of polynomial degree k under a reverse CFL condition, together with an unconditional O(τ+hk) estimate. Numerical experiments confirm the predicted convergence rates and the enforcement of the bounds.
Related
- Stochastic Augmented Lagrangian Framework with Second-Order Convergence Guarantees for Nonconvex Expectation-Constrained Optimization
- Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions
- Efficient ADMM for Soft-Constrained Quadratic Programs via Generalized Slack Projections
- A new convergence proof for an iterative scheme for elliptic Cauchy problems based on alternating orthogonal projections
- Convergence of Random Products of Projections Under Infinite-Periodic Selections
BibTeX
@misc{lee2026optimizationbased,
title = {{Optimization-based Cell Average Limiters as Metric Projections: Constrained Evolution and Convergence Analysis for Convection-diffusion Equations}},
author = {Seungmin Lee and Sanghyun Lee and Chen Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33449},
eprint = {2610.33449},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33449},
}
RIS
TY - UNPB
TI - Optimization-based Cell Average Limiters as Metric Projections: Constrained Evolution and Convergence Analysis for Convection-diffusion Equations
AU - Seungmin Lee
AU - Sanghyun Lee
AU - Chen Liu
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33449
UR - https://arxiv.org/abs/2610.33449
AB - The optimization-based cell average limiter enforces global conservation and bounds, and has been successfully applied to discontinuous Galerkin (DG) discretizations of a wide range of nonlinear systems, from phase-field models to the compressible Navier-Stokes equations. In this paper, taking the heat and convection-diffusion equations as prototypes, we analyze the convergence of the scheme resulting from cell average limiting. Encoding conservation and admissibility in a differential inclusion identifies the limiter as a backward Euler proximal step in a Lie splitting of the constrained evolution. This proximal step is the metric projection onto the convex admissible set, and we prove that it is nonexpansive and stable in the L^2 norm, and that it does not increase the total variation of the cell averages. Combined with the standard error analysis, nonexpansiveness yields an O(τ+h^k+1) estimate for implicit Euler time stepping with DG discretization of polynomial degree k under a reverse CFL condition, together with an unconditional O(τ+h^k) estimate. Numerical experiments confirm the predicted convergence rates and the enforcement of the bounds.
ER -