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Optimization-based Cell Average Limiters as Metric Projections: Constrained Evolution and Convergence Analysis for Convection-diffusion Equations
Seungmin Lee, Sanghyun Lee, Chen Liu
TL;DR
High-order DG methods can fail to preserve admissibility, and repeated application of the successful cell average limiter lacked convergence analysis. The paper models conservation and admissibility through a differential inclusion and a proximal limiter step. For heat and convection-diffusion equations, it proves stability properties and conditional optimal-order and unconditional suboptimal-order error estimates.
Problem
High-order DG methods do not generally preserve admissibility, and convergence under repeated application of the cell average limiter had not been analyzed.
Source: §1 · pp. 1–2Method
The paper encodes conservation and admissibility in a differential inclusion whose Lie splitting pairs an implicit Euler DG PDE step with a backward Euler proximal limiter step.
Source: §1 · p. 3Results
The limiter is nonexpansive in L2 and does not increase cell-average total variation; for heat and convection-diffusion equations, estimates are O(τ+h^(k+1)) under τ≥c h^(2(k+1)/3) and unconditionally O(τ+h^k).
Source: §1 · p. 3Takeaways & Limitations
The analysis explains the limiter's accuracy in repeated use and shows the reverse CFL becomes more permissive for higher polynomial degrees k≥3.
Source: §1 · p. 4, §6 · p. 24Rigorous error analysis is restricted to the heat and convection-diffusion equations, and does not establish an error estimate for applying the additional Zhang–Shu scaling at every time step.
Source: §1 · p. 3, §5 · p. 19
Abstract
from arXiv · showhide
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 $O(\tau+h^{k+1})$ estimate for implicit Euler time stepping with DG discretization of polynomial degree $k$ under a reverse CFL condition, together with an unconditional $O(\tau+h^{k})$ estimate. Numerical experiments confirm the predicted convergence rates and the enforcement of the bounds.
1 Introduction
High-order DG methods provide accuracy and flexibility but may violate admissibility, while repeated cell-average limiting had lacked a convergence analysis. This paper frames the limiter as a projection in a constrained evolution and analyzes its stability and convergence for heat and convection-diffusion equations.
1 Introduction
- Prior bound-preserving approaches include Zhang–Shu scaling, flux limiting, and conservative redistribution, each with distinct constraints or construction requirements.The Zhang–Shu approach uses a more restrictive bound-preserving CFL; flux limiting needs a suitable low-order scheme, while redistribution balances mass locally.Source: §1 · p. 2
- Repeated limiting had no known convergence analysis, despite successful applications of the optimization-based cell average limiter across several PDE systems.The limiter minimizes distance to high-order DG cell averages subject to conservation and invariant-domain constraints.Source: §1 · p. 2
- The limiter is independent of the PDE and time discretization, and its elementwise optimization has O(N_el) complexity and negligible cost relative to PDE advancement.The elementwise constraints decouple, leaving one unknown per element and a single linear conservation constraint; direct bounds at all degrees of freedom would be more costly.Source: §1 · pp. 2–3
- The framework encodes conservation and admissibility as a differential inclusion, splitting evolution into a DG PDE step and a backward Euler proximal limiter step.For parabolic equations, the sequential stages form a Lie splitting of the constrained evolution.Source: §1 · p. 3
- The limiter is identified as the metric projection onto the discrete admissible set and is proved nonexpansive in L2.The paper also reports L2 stability and no increase in total variation of cell averages.Source: §1 · p. 3
- For heat and convection-diffusion equations, implicit Euler DG with degree k and limiting has an O(τ+h^(k+1)) estimate under a reverse CFL and an unconditional O(τ+h^k) estimate.The reverse CFL is τ≥c h^(2(k+1)/3); the condition becomes more permissive for k≥3 than τ≥c h².Source: §1 · pp. 3–4
2 Constrained evolution and discretization
The paper encodes conservation and admissibility as a convex constraint in a differential inclusion, then discretizes the constrained evolution by splitting PDE advancement from projection onto the admissible set. The resulting limiter preserves global mass and admissibility while the high-order DG solution is reconstructed.
2.1 Encoding admissibility in a differential inclusion
- The admissible set A combines conservation of the initial mass with pointwise membership in a convex set G, and the PDE is written as a differential inclusion using its indicator subgradient.For the studied diffusion and convection-diffusion equations, the maximum principle ensures the exact solution remains in A, making the formulations equivalent when the inclusion has a unique solution.Source: §2.1 · p. 4
2.2 Operator splitting and numerical scheme
- The Lie splitting first advances the unconstrained PDE and then applies a subgradient-flow step that projects the intermediate state onto the admissible set.The projection step is independent of the time-step size.Source: §2.2 · pp. 6, 8
- The discrete projection minimizes distance to the DG solution’s cell averages over constraints enforcing conservation and admissible bounds.The strongly convex optimization problem has a unique minimizer.Source: §2.2 · pp. 7–8
- The scheme is globally conservative, and the exact solution’s cell averages remain in the discrete admissible set.After limiting cell averages, high-order moments are adjusted using the Zhang–Shu scaling limiter; the paper’s analysis focuses on the cell-average limiter alone.Source: §2.2 · p. 8
3 Properties of the cell average limiter
The cell-average limiter is a projection onto a closed convex admissible set, which yields accuracy preservation, L2 stability, and nonexpansiveness. It also preserves the ordering of cell averages and does not increase their total variation.
Accuracy preservation and stability
- The limiter’s projection does not worsen the L2 approximation error to an admissible exact solution, preserving high-order spatial accuracy.For a fixed time step, an O(h^(k+1)) DG error remains O(h^(k+1)) after limiting.Source: §3 · p. 10
- Under an admissible exact solution and a conservative algorithm, the limiter does not increase the L2 norm of the limited solution or its cell averages.The stability result relies on both stated assumptions.Source: §3 · p. 11
Total variation
- The limiter preserves the order of cell averages and does not increase their total variation.This total-variation result concerns the postprocessing step alone, not the full scheme, since the DG step need not be total-variation diminishing.Source: §3 · p. 12
Nonexpansiveness
- The cell-average limiter is nonexpansive in the L2 norm, so differences between inputs cannot grow under postprocessing.The result applies at a single time step and is independent of the time-stepping scheme.Source: §3 · pp. 12–13
4 Error analysis
The error analysis studies the cell-average limiter alone, without Zhang–Shu scaling postprocessing. For the heat and convection-diffusion equations, it establishes optimal-order estimates under a reverse CFL condition and an unconditional suboptimal estimate for convection-diffusion.
4 Error analysis
- The analysis isolates the repeated-limiting cost in the cell average of the elliptic projection error, charged once per time step; controlling its accumulation yields the reverse CFL condition.The reverse CFL condition is not needed for convergence, but permits the additional spatial-order factor in the conditional estimate.Source: §4 · pp. 13–14
4.1 Example 1. Heat equation
- For k ≥ 1, assuming u ∈ L∞(0,T;H^{k+1}), ∂t u ∈ L²(0,T;H^{k+1}), and ∂tt u ∈ L²(0,T;L²), if τ ≥ c h^{2(k+1)/3}, then max_{0≤n≤N_T} ||U_h^n−u^n||_{L²} ≤ C(τ+h^{k+1}), with C independent of h and τ.Source: §4.1 · p. 15
4.2 Example 2. Convection-diffusion equation
- Under τ ≥ c h^{2(k+1)/3}, the convection-diffusion error is bounded by C(τ + h^{k+1}) for k ≥ 1 under the stated regularity and velocity-field assumptions.The velocity field is divergence-free and has vanishing normal component on the boundary.Source: §4.2 · p. 17
- Without a time-step restriction, the convection-diffusion error is bounded by C(τ + h^k) under the stated regularity and velocity-field assumptions.The proof replaces the elliptic projection used for the conditional estimate with the L2 projection.Source: §4.2 · p. 19
5 Numerical experiments
The experiments assess an efficient limiter solver, convergence for the heat and convection-diffusion equations, and bound preservation in solid-body rotation. They report optimal convergence rates in the heat and rotating Gaussian tests and conservation and bound preservation in solid-body rotation.
5 Numerical experiments
- The section combines convergence studies using manufactured solutions that trigger the limiter with a solid-body rotation test of conservative bound preservation; experiments also apply Zhang–Shu scaling after cell average limiting.The additional scaling modifies only high-order moments, preserving cell averages and global conservation, but its repeated application is not covered by the paper's error estimate.Source: §5 · p. 19
5.1 Efficient splitting solver
- The preferred splitting solver is parallel-friendly, using vectorizable cutoff, scaling, and addition operations plus one vector inner product; an exact breakpoint search is harder to implement on distributed-memory systems.On a uniform mesh, the iteration recovers the formula from prior work [29].Source: §5.1 · p. 20
5.2 Convergence study for heat equation
- Without limiting, the heat-equation solution exceeded the bounds by up to 2.678 × 10−3; with limiting, it stayed in [0, 1] at every resolution while retaining optimal convergence rates.The tables compare simulations with and without the cell average limiter in Q2 and Q3 spaces.Source: §5.2 · p. 21, Table 1 · p. 21, Table 2 · p. 21
5.3 Rotating Gaussian hill
- Third-order Q2 and fourth-order Q3 convergence rates were obtained for the rotating Gaussian hill while the limiter with [m, M] = [0, +∞) was active.The L2 errors and rates are reported for successively refined meshes.Source: §5.3 · p. 21, Table 3 · p. 22
5.4 Solid body rotation
- In solid-body rotation, the solution remained in [0, 1] and conserved mass throughout; Q2 at h = 1/200 and Q3 at h = 1/160 were visually indistinguishable at roughly comparable degrees of freedom.The coarser Q2 result showed noticeable numerical diffusion, and reducing the time step produced only modest improvement.Source: §5.4 · p. 23, Figure 2 · p. 23
6 Conclusions
The paper frames conservation and admissibility as a constrained evolution and uses a proximal limiter step to enforce them. The limiter is nonexpansive, supporting error estimates and the observed accuracy and bound preservation in the tested equations.
6 Conclusions
- The limiter is a metric projection onto the convex admissible set, making it nonexpansive and stable in L2, and it does not increase total variation of cell averages.The Lie-split scheme applies an implicit Euler DG step followed by the limiter as a backward Euler proximal step.Source: §6 · p. 24
- For heat and convection-diffusion equations, the scheme has an O(τ + h^(k+1)) error estimate under the reverse CFL condition τ ≥ c h^(2(k+1)/3), and an unconditional O(τ + h^k) estimate.Source: §6 · p. 24
- Numerical tests confirm the predicted convergence rates and prescribed bounds for the heat equation and rotating Gaussian hill, while solid-body rotation validates conservation and bound preservation.Source: §6 · p. 24
A Proof of Theorem 10
The proof appendix establishes a bound for the SIPG bilinear form and uses projection and discrete-error estimates to complete the theorem's proof.
A Proof of Theorem 10
- The SIPG bilinear form satisfies |a_D(v, v_h)| ≤ C∥v∥_*∥v_h∥_DG for v ∈ H2(T_h) and v_h ∈ V_h, with C independent of h.Source: A Proof of Theorem 10 · p. 24
- The L2 projection preserves cell averages, placing the projected solution in Λ and leaving it unchanged under both Q and P_h.Source: A Proof of Theorem 10 · p. 25
- The proof concludes by summing the discrete estimates over time steps and taking the maximum over n.Source: A Proof of Theorem 10 · p. 26