The adaptive flux reconstruction (AFR) scheme of Srinivasan and Nadarajah (arXiv:2511.01564; AIAA SciTech 2026) blends the discontinuous Galerkin (DG) and nonlinearly stable flux reconstruction (NSFR) methods for hyperbolic conservation laws by scaling the flux reconstruction filter parameter elementwise with a Persson–Peraire modal discontinuity sensor. The scheme retains provable discrete conservation and discrete entropy conservation under the Tadmor flux condition while damping shock oscillations and permitting larger CFL numbers than the DG variant, but its guarantees stop at stability: no convergence proof to the entropy solution exists, positivity of thermodynamic variables is supplied by an external Zhang–Shu-type limiter rather than being intrinsic to the adaptive lifting, and the sensor-to-parameter mapping (threshold s0s_0 and the piecewise sine-based transition of Eq. (11)) is heuristic. The open problem posed here is to establish, or refute, a rigorous reliability theory for AFR: convergence to the physically correct entropy solution under mesh refinement with explicit assumptions on the sensor mapping, limiter, and time discretization, together with a determination of whether admissibility is intrinsic or necessarily external. No later work addresses AFR; the concurrent 2026 convergence-theory results cover only the DG family (conditional Lax–Wendroff-type theorems, a priori error estimates) or smooth solutions of summation-by-parts discretizations, leaving the sensor-adapted FR lifting untouched. The problem is open.

Background

Hyperbolic conservation laws ∂tu+∇⋅f(u)=0\partial_t\mathbf{u}+\nabla\cdot\mathbf{f}(\mathbf{u})=\mathbf{0}, such as the compressible Euler equations, develop discontinuities (shocks) in finite time even from smooth data, and uniqueness is restored only within the class of entropy solutions — the weak solutions that satisfy an inequality ∂tη(u)+∇⋅q(u)≤0\partial_t\eta(\mathbf{u})+\nabla\cdot\mathbf{q}(\mathbf{u})\le 0 for every convex entropy pair (η,q)(\eta,q). A numerical scheme for such laws is physically trustworthy only if it converges to the entropy solution, not merely to some weak solution.

High-order discretizations approximate the solution on each mesh element by polynomials of degree pp. The discontinuous Galerkin (DG) method achieves p+1p+1 order accuracy on smooth problems but produces nonphysical high-frequency oscillations near shocks and suffers severely restricted stable time steps (small CFL numbers, where the CFL number bounds the ratio of time step to cell size relative to the fastest wave speed). Flux reconstruction (FR), introduced by Huynh, generalizes DG: the residual is corrected by correction functions, and a scalar parameter cc spans a continuous family of schemes — c=0c=0 recovers DG, while the largest stable value c+c_+ gives a more dissipative FR scheme that damps oscillations and allows larger CFL numbers at the price of a larger solution error. The FR parameter enters through a filter matrix KK added to the mass matrix MM, which modifies the lifting operator — the operator that lifts interface numerical-flux jumps into the element interior. Energy-stable FR (ESFR) and nonlinearly stable FR (NSFR) equip this framework with provable stability: using skew-symmetric split forms of the equations and two-point entropy-conservative numerical fluxes (fluxes built from pairs of states that satisfy a discrete chain-rule identity), the semidiscrete scheme conserves a discrete entropy whenever the two-point flux satisfies the Tadmor shuffle condition, a classical algebraic criterion for entropy conservation.

Srinivasan and Nadarajah's AFR scheme adapts this family element by element: a modal discontinuity sensor of Persson and Peraire — which detects under-resolution by comparing the energy contained in the highest modal coefficients of the polynomial against the total — produces a scalar se=log⁡10Ses_e=\log_{10}S_e that is mapped by the piecewise transition function of Eq. (11) of the source to a value ε∈[0,1]\varepsilon\in[0,1] scaling the FR parameter as c=εc+c=\varepsilon c_+: ε=0\varepsilon=0 for se<s0−κs_e<s_0-\kappa, ε=12(1+sin⁡(π(se−s0)2))\varepsilon=\tfrac{1}{2}\left(1+\sin\left(\tfrac{\pi(s_e-s_0)}{2}\right)\right) for s0−κ<se≤s0+κs_0-\kappa<s_e\le s_0+\kappa, and ε=1\varepsilon=1 for se>s0+1s_e>s_0+1, with threshold s0=−4log⁡10ps_0=-4\log_{10}p. As written, the sine argument fixes a unit half-width, so the branches join consistently only for κ=1\kappa=1, and no numerical value of κ\kappa is given in the source. The scheme thus behaves like accurate DG in smooth regions and like robust FR near detected shocks. The authors prove that the adaptation of cc preserves local and global conservation and discrete entropy conservation in the (Mm+Km)(M_m+K_m)-norm under the Tadmor condition, and demonstrate p+1p+1 order on a smooth Gaussian-pulse test plus robustness on the Leblanc shock tube, shock diffraction, and double Mach reflection problems, all in one and two dimensions on quadrilateral elements with SSPRK3 (a third-order strong-stability-preserving Runge–Kutta) time stepping. Two features are explicitly acknowledged but unresolved: the implementation requires an external positivity-preserving limiter of Zhang and Shu type (a rescaling of the polynomial toward its cell mean so that density and pressure stay positive), and the authors state that stability does not imply convergence — no error or convergence analysis of AFR is given.

The theoretical backdrop makes the gap concrete. For linear problems the Lax equivalence theorem guarantees that a stable, consistent discretization converges; for nonlinear conservation laws no such theorem holds, and the classical Lax–Wendroff theorem gives only a conditional statement: if the discrete solutions converge (in a suitable sense) and the scheme is conservative and consistent, then the limit is a weak solution. Recent 2026 work has begun closing this gap for the DG family — a Lax–Wendroff-type theorem for fully discrete entropy-stable DG built from a Zhang–Shu-type scaling limiter (Liu, Guo, Jiang, and Sun), the first a priori error estimates for semidiscrete entropy-stable DG (Chen, Liu, and Shu), and convergence of entropy-stable summation-by-parts discretizations to smooth solutions (Worku, Del Rey Fernández, and Zingg; Ranocha) — but none of these covers FR schemes with a cc-parameter lifting, and none covers convergence through discontinuities for any high-order scheme with a solution-dependent (sensor-adapted) operator.

Problem Statement

Consider a hyperbolic system of conservation laws ∂tu+∇⋅f(u)=0\partial_t\mathbf{u}+\nabla\cdot\mathbf{f}(\mathbf{u})=\mathbf{0} admitting a convex entropy pair, discretized by the AFR scheme of Srinivasan and Nadarajah (arXiv:2511.01564): the NSFR semidiscretization (Eq. (7) of the source) in which the FR filter matrix KmK_m is built from the local parameter c=εc+c=\varepsilon c_+, where ε∈[0,1]\varepsilon\in[0,1] is the output of the Persson–Peraire modal sensor mapped through the piecewise sine-based transition of Eq. (11) (threshold s0=−4log⁡10ps_0=-4\log_{10}p; ε=0\varepsilon=0 below s0−κs_0-\kappa, the sine branch on s0−κ<se≤s0+κs_0-\kappa<s_e\le s_0+\kappa, and ε=1\varepsilon=1 above s0+1s_0+1; as written, consistent only for κ=1\kappa=1), advanced in time with SSPRK3 and coupled with a Zhang–Shu-type positivity-preserving limiter. The scheme is provably conservative and discretely entropy stable, but no convergence theory exists. The open problem, in two linked parts that together decide whether AFR reliably computes the physically correct solution:

(a) Convergence. Prove or disprove that the AFR solutions converge to the entropy solution of the underlying system as the mesh is refined (h→0h\to 0 at fixed polynomial degree pp); in particular, part (a) is resolved by a full convergence theorem or by a rigorous obstruction to convergence; the Lax–Wendroff-type theorem (any L1L^1-convergent subsequence has an entropy-solution limit) is conditional — it presupposes the convergence whose limit it identifies — and so counts only as a required intermediate step, not as completion of this part. The analysis must address the fully discrete scheme (SSPRK3 time stepping composed with the limiter and the sensor-adapted lifting), not only the semidiscrete form, and must state explicitly what assumptions on the sensor mapping are required.

(b) Intrinsic admissibility. For the compressible Euler equations — the system for which the source implements AFR together with the Zhang–Shu-type limiter — determine whether the adaptive lifting itself preserves the admissible set of states (positive density and positive internal energy), or prove that external positivity limiting is necessary.

Completion of the problem is a set of rigorous theorems, or definitive counterexamples or obstructions, covering the interplay of the sensor-adapted lifting, the entropy structure, and the limiter. The companion acknowledged directions from the source — an empirical sensor-optimality and robustness study across broader problem classes, and the extension and validation to three-dimensional, curvilinear, and mixed-element meshes — are parallel methodological directions and are not part of this problem's completion boundary.

References

  1. Sai Shruthi Srinivasan and Siva Nadarajah, 'An Adaptive Flux Reconstruction Scheme for Robust Shock Capturing', arXiv:2511.01564 (2025), https://arxiv.org/abs/2511.01564
  2. Sai Shruthi Srinivasan and Sivakumaran Nadarajah, 'An Adaptive Flux Reconstruction Scheme for Robust Shock Capturing', AIAA SCITECH 2026 Forum (2026), doi:10.2514/6.2026-2679, https://doi.org/10.2514/6.2026-2679
  3. Sai Shruthi Srinivasan and Siva Nadarajah, 'Investigation of shock-capturing with bound-preserving limiters for the nonlinearly stable flux reconstruction method', arXiv:2507.09131 (2025); journal version with Julien Brillon in Computers & Fluids 317 (2026) 107192, doi:10.1016/j.compfluid.2026.107192, https://arxiv.org/abs/2507.09131
  4. Carolyn M.V. Pethrick and Siva Nadarajah, 'Fully-discrete nonlinearly-stable flux reconstruction methods for compressible flows', Journal of Computational Physics 533 (2025) 113984, doi:10.1016/j.jcp.2025.113984; arXiv:2409.00208, https://arxiv.org/abs/2409.00208
  5. Carolyn M. V. Pethrick and Siva Nadarajah, 'A Space-time Approach to Entropy-Stable Discontinuous Galerkin and Flux Reconstruction', arXiv:2604.19900 (2026), https://arxiv.org/abs/2604.19900
  6. Yuchang Liu, Wei Guo, Yan Jiang, Zheng Sun, 'A limiter-based approach to construct high-order fully-discrete entropy stable explicit DG schemes for hyperbolic conservation laws', arXiv:2603.27478 (2026), https://arxiv.org/abs/2603.27478
  7. Xu-Kun Chen, Yong Liu, Chi-Wang Shu, 'The error estimate of entropy-stable discontinuous Galerkin methods for hyperbolic conservation laws', arXiv:2608.06642 (2026), https://arxiv.org/abs/2608.06642
  8. Hendrik Ranocha, 'Convergence of entropy-conservative summation-by-parts discretizations to smooth solutions of hyperbolic conservation laws', arXiv:2607.27049 (2026), https://arxiv.org/abs/2607.27049
  9. Zelalem Arega Worku, David C. Del Rey Fernández, David W. Zingg, 'Convergence of entropy-stable continuous summation-by-parts discretizations of symmetric hyperbolic conservation laws', arXiv:2603.18369 (2026), https://arxiv.org/abs/2603.18369
  10. Jielin Yang and Guosheng Fu, 'An entropy-stable oscillation-eliminating DGSEM for the Euler equations on curvilinear meshes', arXiv:2602.16732 (2026), https://arxiv.org/abs/2602.16732
  11. Xiangxiong Zhang and Chi-Wang Shu, 'On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes', Journal of Computational Physics 229 (2010) 8918-8934, doi:10.1016/j.jcp.2010.08.016, https://doi.org/10.1016/j.jcp.2010.08.016
  12. Per-Olof Persson and Jaime Peraire, 'Sub-Cell Shock Capturing for Discontinuous Galerkin Methods', 44th AIAA Aerospace Sciences Meeting and Exhibit (2006), doi:10.2514/6.2006-112, https://doi.org/10.2514/6.2006-112
  13. Alexander Cicchino and Siva Nadarajah, 'Discretely nonlinearly stable weight-adjusted flux reconstruction high-order method for compressible flows on curvilinear grids', Journal of Computational Physics 521 (2025) 113532, doi:10.1016/j.jcp.2024.113532, https://doi.org/10.1016/j.jcp.2024.113532
  14. H.T. Huynh, 'Shock capturing via limiting for high-order methods including discontinuous Galerkin and flux reconstruction', Computers & Fluids 300 (2025) 106743, doi:10.1016/j.compfluid.2025.106743, https://doi.org/10.1016/j.compfluid.2025.106743