Rigorous foundations of cell-vertex WENO stabilization with shock-capturing quadrature: entropy stability, invariant-domain preservation, and extension beyond structured meshes

The cell-vertex WENO (CV-WENO) stabilization and its shock-capturing quadrature variant (CV-WENO-SC), introduced by Vedral and Kuzmin in 2026, suppress spurious oscillations in high-order continuous and discontinuous Galerkin discretizations of hyperbolic conservation laws by injecting elementwise artificial viscosity whose magnitude is set by a WENO smoothness sensor, with candidate polynomials built at mesh vertices and dissipation redistributed across quadrature points inside each element. The method performs well numerically but lacks a rigorous theory: no discrete entropy inequality is proven, invariant-domain preservation for the continuous Galerkin formulation is not guaranteed, the free parameters (linear weights, cutoff thresholds, steepening exponent) are chosen heuristically, and all experiments are confined to uniform structured meshes in one and two dimensions. This problem asks for the missing analysis of three of these gaps, treating the heuristic determination of the free parameters as background context rather than a requested deliverable: a proof or refutation of (semi-)discrete entropy stability for the CV-WENO family, a provably invariant-domain-preserving continuous Galerkin variant incorporating CV-WENO, and the extension of the cell-vertex averaging and WENO quadrature to unstructured meshes and three dimensions.

Background

Hyperbolic conservation laws $\partial_t u+\nabla\cdot\mathbf{f}(u)=0$ govern wave propagation in fluids and other continua; their classical solutions develop discontinuities (shocks) in finite time even from smooth data, and the physically relevant weak solution is selected by an entropy condition. A numerical scheme is called entropy stable if it satisfies a discrete entropy inequality, i.e. the total entropy is nonincreasing at the discrete level up to boundary terms, which prevents convergence to nonphysical (entropy-violating) weak solutions; 'semi-discrete' refers to the spatial discretization alone, before time integration. Two high-order finite element frameworks are in play: continuous Galerkin (CG), where the approximation is a globally continuous piecewise polynomial, and discontinuous Galerkin (DG), where polynomials on separate elements are coupled through numerical fluxes such as the HLL (Harten-Lax-van Leer) Riemann-solver flux, a simple two-state approximate solver for the Riemann problem at element interfaces. CG discretizations of convection-dominated problems require stabilization to avoid oscillations and loss of optimal convergence order. WENO (weighted essentially non-oscillatory) methods suppress oscillations by taking a nonlinear, data-dependent weighted average of candidate polynomials, with weights that favor the smoothest candidates; Hermite WENO (HWENO) reconstructions additionally reuse derivative degrees of freedom as reconstruction data. The dissipation-based WENO stabilization framework of Kuzmin and Vedral (J. Comput. Phys. 487, 2023) adds to the CG or DG formulation an elementwise artificial-viscosity term whose strength is determined by a WENO smoothness sensor that measures the deviation of the finite element solution from an HWENO reconstruction, blending between a low-order dissipative operator near discontinuities and a consistent high-order operator in smooth regions. In the original version the candidate polynomials are built from von Neumann neighbors (elements sharing a face or edge), a 'cell-cell' procedure denoted CC-WENO. Vedral and Kuzmin (2026) introduced two upgrades: first, the cell-vertex procedure (CV-WENO), in which Hermite data of individual cells is sent to the cells' mesh vertices, averaged there, and sent back, so that vertex-neighbor information enters the reconstruction without enlarging stencils, mitigating mesh imprinting (the alignment of numerical artifacts with mesh directions); second, shock-capturing WENO quadrature (the CV-WENO-SC variant), in which the linear quadrature weights of the stabilization term are replaced by nonlinear WENO-type weights, concentrating dissipation at quadrature points near discontinuities while preserving each element's total dissipation rate $D_e^{\mathrm{orig}}$ exactly. Numerically the schemes achieve optimal $p+1$ convergence rates, behave entropy stably on the KPP (Kolmogorov-Petrovskii-Piskunov) stiff reaction-front propagation benchmark and the modified Sod shock-tube benchmark (a standard one-dimensional gas-dynamics Riemann problem), and resolve Euler flow problems robustly for polynomial degrees up to $p=32$ — but only on uniform structured one- and two-dimensional meshes. On the theoretical side, entropy stability is well understood for flux-based frameworks: entropy-conservative numerical fluxes combined with summation-by-parts (SBP) operators yield entropy-stable WENO finite-difference and spectral collocation schemes (Yamaleev and Carpenter, 2017), entropy-stable subcell shock capturing exists for split-form DG (Hennemann et al., 2021), and bound-preserving entropy-stable enriched Galerkin methods (Galerkin discretizations that enrich a continuous finite element space with discontinuous degrees of freedom) on (multi-)linear spaces with limiters were built by Kuzmin, Lee, and Yang (2025). None of these covers dissipation-based WENO stabilization, where the dissipation is a nonlinear, solution-dependent operator without the algebraic structure those proofs exploit; the source states that a (semi-)discrete entropy inequality 'cannot be formally proven' for its scheme, as is the case for most WENO schemes in the literature. Invariant-domain preservation (IDP) means that the discrete states remain in a prescribed convex admissible set, such as positive density and pressure for the Euler equations, or a discrete maximum principle for scalar laws. Tools for enforcing IDP include convex limiting (Guermond and Popov, 2017) and the positivity-preserving DG limiters of Zhang and Shu (2010); Kuzmin, Hajduk, and Vedral (J. Comput. Phys. 560, 2026) proved IDP for CG equipped with WENO-sensor-based nonlinear stabilization via element-based convex limiting and Bernstein basis functions under an explicit time-step restriction. The source's Remark 8 states that this machinery can render the CG formulation of CV-WENO provably IDP, but the specific combination is left undeveloped. Finally, the scheme's free parameters — the linear weights $\gamma_l^e$ of the WENO averaging, the cutoff thresholds $C_{D_e^{\mathrm{orig}}}$ and $C_{\gamma_e}$ that decide when dissipation redistribution is applied, and the steepening exponent $q$ — are prescribed heuristically (e.g. $q=1$, $C_{\gamma_e}=0.9$ in the source's experiments) with no selection theory.

Problem Statement

The open problem is to complete the mathematical foundation of the cell-vertex WENO (CV-WENO) and shock-capturing quadrature (CV-WENO-SC) stabilizations introduced by Vedral and Kuzmin (2026). The common scientific objective uniting the requested results is to upgrade this shock-capturing methodology for high-order continuous and discontinuous Galerkin discretizations of hyperbolic conservation laws from empirically validated to provably reliable, and to establish its validity beyond the restricted setting of the source. Three linked questions, each explicitly acknowledged as unresolved or deferred by the source, define the completion boundary. (1) Entropy stability: prove or disprove that the semi-discrete CV-WENO and CV-WENO-SC schemes satisfy a discrete entropy inequality — the source's acknowledgment that such an inequality 'cannot be formally proven' concerns the schemes in general, and the purely numerical evidence to be replaced by a rigorous guarantee spans a two-dimensional scalar benchmark (KPP) and a one-dimensional Euler system (modified Sod) — with any refutation exhibiting a concrete entropy-violating configuration of the scheme. (2) Invariant-domain preservation for CG: construct a continuous Galerkin formulation incorporating CV-WENO stabilization for which preservation of a convex invariant domain (positive density and pressure for the Euler equations, or a discrete maximum principle for scalar laws) is proven under an explicit time-step restriction, and analyze the accuracy and time-step cost of the combined scheme. (3) Geometric generality: extend the cell-vertex averaging procedure and the shock-capturing WENO quadrature to unstructured meshes and to three space dimensions, maintaining the accuracy and shock-capturing behavior demonstrated on structured meshes and, where established in (1) and (2), carrying the entropy and invariant-domain guarantees over to the extended setting. Components (1) and (2) supply the rigorous guarantees for the schemes exactly as defined in the source; component (3) removes the structured-mesh restriction under which all current evidence and analysis is confined.

References

1. Joshua Vedral, Dmitri Kuzmin (2026). Cell-vertex WENO schemes with shock-capturing quadrature for high-order finite element discretizations of hyperbolic problems. Computers & Fluids 317, 107201. DOI: 10.1016/j.compfluid.2026.107201; https://doi.org/10.1016/j.compfluid.2026.107201 (arXiv:2601.16911; https://arxiv.org/abs/2601.16911)
2. Dmitri Kuzmin, Joshua Vedral (2023). Dissipation-based WENO stabilization of high-order finite element methods for scalar conservation laws. Journal of Computational Physics 487, 112153. DOI: 10.1016/j.jcp.2023.112153; https://doi.org/10.1016/j.jcp.2023.112153
3. Joshua Vedral (2023). Dissipative WENO stabilization of high-order discontinuous Galerkin methods for hyperbolic problems. arXiv:2309.12019; https://arxiv.org/abs/2309.12019
4. Joshua Vedral, Andreas Rupp, Dmitri Kuzmin (2025). Strongly consistent low-dissipation WENO schemes for finite elements. Applied Numerical Mathematics 210, pp. 64-81. DOI: 10.1016/j.apnum.2024.12.008; https://doi.org/10.1016/j.apnum.2024.12.008
5. Dmitri Kuzmin, Hennes Hajduk, Joshua Vedral (2026). An element-based convex limiting framework for continuous Galerkin methods with nonlinear stabilization. Journal of Computational Physics 560, 114952. DOI: 10.1016/j.jcp.2026.114952; https://doi.org/10.1016/j.jcp.2026.114952 (arXiv:2509.04673; https://arxiv.org/abs/2509.04673)
6. Dmitri Kuzmin, Manuel Quezada de Luna (2020). Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws. Computers & Fluids 213, 104742. DOI: 10.1016/j.compfluid.2020.104742; https://doi.org/10.1016/j.compfluid.2020.104742
7. Sebastian Hennemann, Andrés M. Rueda-Ramírez, Florian J. Hindenlang, Gregor J. Gassner (2021). A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations. Journal of Computational Physics 426, 109935. DOI: 10.1016/j.jcp.2020.109935; https://doi.org/10.1016/j.jcp.2020.109935
8. Xiangxiong Zhang, Chi-Wang Shu (2010). On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. Journal of Computational Physics 229, pp. 8918-8934. DOI: 10.1016/j.jcp.2010.08.016; https://doi.org/10.1016/j.jcp.2010.08.016
9. Jean-Luc Guermond, Bojan Popov (2017). Invariant Domains and Second-Order Continuous Finite Element Approximation for Scalar Conservation Equations. SIAM Journal on Numerical Analysis 55, pp. 3120-3146. DOI: 10.1137/16M1106560; https://doi.org/10.1137/16M1106560
10. Dmitri Kuzmin, Sanghyun Lee, Yi-Yung Yang (2025). Bound-preserving and entropy stable enriched Galerkin methods for nonlinear hyperbolic equations. Journal of Computational Physics 541, 114323. DOI: 10.1016/j.jcp.2025.114323; https://doi.org/10.1016/j.jcp.2025.114323 (arXiv:2411.19160; https://arxiv.org/abs/2411.19160)
11. Bartolomeo Fanizza, Florent Renac (2026). Invariant domain preserving limiting of time explicit and time implicit discretizations for systems of conservation laws. arXiv:2605.08038; https://arxiv.org/abs/2605.08038
12. Murtazo Nazarov (2026). High-Order Invariant-Domain Preserving Continuous Finite Elements via Graph-Poisson Convex Limiting. arXiv:2606.23397; https://arxiv.org/abs/2606.23397
13. Nail K. Yamaleev, Mark H. Carpenter (2017). A family of fourth-order entropy stable nonoscillatory spectral collocation schemes for the 1-D Navier-Stokes equations. Journal of Computational Physics 331, pp. 90-107. DOI: 10.1016/j.jcp.2016.11.039; https://doi.org/10.1016/j.jcp.2016.11.039