{"id":"c035eb52-efa5-4b66-bc37-e7d9b4a3c9cc","arxiv_id":"2507.09131","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A modified Zhang-Shu limiter that checks solution nodes as well as quadrature nodes enables NSFR to capture strong shocks robustly, with larger flux reconstruction parameters giving essentially oscillation-free solutions.","lead":"This paper modifies a positivity-preserving limiter for the nonlinearly stable flux reconstruction (NSFR) method so it can handle shock waves without extra limiting. Tests on 1D and 2D compressible flow problems show the method stays stable, keeps density and pressure positive, and allows larger time steps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Section 3.1 assertion that enlarging the checked node set to include xi_r,3 preserves the Wang-Shu limiter's guarantees is unproven, and for GL flux nodes the pure GL x GL set is not checked at all, so the positivity guarantee is not established for uncollocated runs.","rationale":"The reader identified the lack of proof for the limiter modification as the weakest assumption, and I agree. My stress-test adds a more specific point: for GL flux nodes the pure GL x GL quadrature set is not among the sets checked by Algorithm 1, so the positivity guarantee is even weaker for the uncollocated runs than for the default GLL runs. However, the central claim for the headline cases (shock diffraction, double Mach reflection, astrophysical jet) uses GLL flux nodes, where xi_r,3 coincides with the volume quadrature nodes, and the extensive numerical evidence supports the empirical robustness claim. The concern is therefore addressable by an analytical proof plus a diagnostic run, which matches a conditional recommendation rather than a rejection.","tokens_in":30500,"tokens_out":15071,"duration_ms":189144,"concrete_test":"Independently re-derive the Wang-Shu proof for the enlarged checked set: verify that choosing theta_2 as the minimum over {xi_r,1, xi_r,2, xi_r,3} maps every checked nodal state into G by convexity of G, and show that the added solution-node minima are O(h^{p+1}) close to the cell average in smooth regions so the limiter is inactive asymptotically. Separately, instrument the GL-flux runs (Shu-Osher, SVSW) to record min rho and P on the pure GL x GL node set after every RK stage; if any negative value appears before failure, the limiter guarantee does not cover uncollocated flux nodes and the claims in Sections 4.3.3 and 4.5.4 need qualification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Algorithm 1 changes the minima entering theta_1 and theta_2 from the two mixed quadrature sets to {xi_r,1, xi_r,2, xi_r,3}. The text states that 'this modification preserves the properties of the limiter' but gives no proof. Three properties are needed: conservativity (true by affine scaling about the cell average), positivity at the checked nodes (plausible from convexity of the admissible set G), and high-order accuracy (requires the minimum over the enlarged set to be O(h^{p+1}) close to the cell average in smooth regions). None of these is derived. More concretely, the checked sets never include the pure GL x GL nodes, which are exactly the volume flux nodes when GL flux quadrature is used; the GL runs in Sections 4.3.3 and 4.5.4 therefore rely on the small CFL (0.05, 0.07) rather than on a limiter guarantee. If any of the missing properties fails, the paper's central claim that the modified limiter 'guarantees positivity' while preserving high-order accuracy for the reported tests is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the Zhang-Shu/Wang-Shu positivity-preserving limiter to the nonlinearly stable flux reconstruction (NSFR) scheme for the compressible Euler equations. The main modification is to include the solution-node set (xi_{r,3}, the GLL tensor product) in the computation of the minimum density and pressure that enter the two scaling parameters theta_1 and theta_2, in addition to the two mixed quadrature sets xi_{r,1} and xi_{r,2}. The authors assert that this preserves the limiter's properties of positivity, high-order accuracy, and conservativity, and they verify the method on a wide range of one- and two-dimensional problems: a low-density accuracy test, Sod and Shu-Osher shock tubes, the Leblanc shock tube, strong vortex-shock interaction, shock diffraction, double Mach reflection, and Mach 80 and Mach 2000 astrophysical jets. The paper also studies the influence of the two-point flux, the choice of GLL versus GL flux quadrature nodes, the flux reconstruction correction parameter, and numerically investigates a CFL condition for positivity of the CHRA two-point flux.","tokens_in":30802,"tokens_out":8109,"duration_ms":99872,"significance":"If the central claim holds, the paper makes a useful practical contribution: it shows that NSFR, equipped with a modified bound-preserving limiter, can run strongly shocked compressible flows without TVD or subcell limiting, while maintaining high-order accuracy in smooth regions. The numerical evidence is substantial and well organized: convergence tables (Tables 2 and 3) demonstrate the expected order on fine grids, the CFL study (Tables 4, 6, and 7) provides useful quantitative guidance for the CHRA flux, and the parameter study of the FR correction parameter gives a clear picture of the trade-off between oscillation control and accuracy. The comparisons against standard DG are also informative, particularly the entropy behavior in the strong vortex-shock interaction case. The main weakness is that the key property-preservation claim for the modified limiter is asserted rather than proved, and the statement that positivity is 'guaranteed' is stronger than what is actually established for uncollocated GL flux-node runs.","major_comments":[{"comment":"The sentence 'this modification preserves the properties of the limiter' is an assertion, not a proof. The three properties claimed for the enlarged node set {xi_{r,1}, xi_{r,2}, xi_{r,3}}—conservativity, positivity at the checked nodes, and high-order accuracy—are not derived. Conservativity follows immediately from the affine scaling about the cell average, and positivity at the checked nodes is plausible from convexity of the admissible set, but neither is shown. Accuracy requires an argument that the minimum over the enlarged set is not far enough from the cell average in smooth regions to degrade the convergence order. Because the abstract and conclusion state that the modified limiter 'guarantees positivity' and preserves the properties of the limiter, the manuscript should either provide this proof or explicitly downgrade the claim to a numerically verified modification.","section":"§3.1, Algorithm 1"},{"comment":"The positivity guarantee is only asserted for the checked node sets. When GL flux quadrature is used, the pure GL x GL nodes are not included in any of the sets xi_{r,1}, xi_{r,2}, or xi_{r,3}; these are precisely the volume flux nodes where the solution is evaluated for the two-point flux. The GL runs in Sections 4.3.3 and 4.5.4 therefore rely on the reduced CFL values (0.05 and 0.07, respectively) rather than on a limiter guarantee. The conclusion's statement that the modified limiter 'guarantees positivity for the entire duration of the test' is too strong for these configurations. Please state the guarantee precisely—positivity at the solution nodes and the mixed quadrature sets—and note that the GL flux-node runs are supported by the reduced CFL, not by the limiter alone.","section":"§4.3.3, §4.5.4"}],"minor_comments":[{"comment":"Table 2 uses initial data '1 + 0.999sin(x+y)' in its caption while the text of Section 4.1 specifies '1 + 0.995sin(x+y)' for both the initial condition and exact solution; please reconcile the two values.","section":"§4.1, Tables 2 and 3"},{"comment":"Line 3 of Algorithm 1 reads 'rho(xi_{r,})' where it should read 'rho(xi_{r,1})', and line 6 uses 'theta' where the text uses 'theta_1'; please correct these typos so the algorithm can be followed unambiguously.","section":"Algorithm 1"},{"comment":"The caption for Figure 11 states 'at t=0.2s', but the Leblanc shock tube is run to a final time t=1e-4 in Section 4.4; this is either a typo or an unexplained inconsistency.","section":"Figure 11 caption"},{"comment":"The caption for Figure 17 states 'at t=1.8s', but the strong vortex-shock interaction case is run to t=0.7s in Section 4.5; please correct the time label.","section":"Figure 17 caption"},{"comment":"References [1] and [36] appear to be the same paper ('On maximum-principle-satisfying high order schemes for scalar conservation laws'); citing the same work twice with different numbers should be fixed.","section":"References"},{"comment":"The CFL verification tables list time steps and computed CFL values but do not indicate which rows failed; please mark the 'last successful' and 'first failing' rows explicitly so the reader can verify the claimed CFL bounds of 1 (Eq. 29) and 0.5 (Eq. 30).","section":"Tables 4, 6, and 7"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid numerical study with a potentially useful practical modification, but the two major comments concern the central claim of guaranteed positivity with the modified limiter. The first is a missing proof for the property-preservation claim; the second is an overstatement of the guarantee for uncollocated GL runs. Both are addressable in revision. The manuscript would also benefit from a clearer statement of what is proved and what is numerically verified, especially because the title and abstract promise an investigation rather than a new theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a useful numerical study, honestly presented, and the main caveat is exactly where the stress test points: the modified limiter's properties are asserted, not proved, and the pure GL x GL node set is never checked. I would send it to review, but with requests to tighten the limiter claim and publish code/data.\n\nWhat is new: the extension of the Wang-Shu limiter to include the solution-node set xi_r,3 in the minimum density/pressure evaluation, plus a broad parameter sweep over FR correction parameter c, two-point fluxes, quadrature nodes, and CFL thresholds. The numerical work is extensive and mostly well done: convergence tables show the expected high orders; the modified limiter runs shock diffraction, double Mach reflection, and the Mach 2000 jet; comparisons with Strong DG show NSFR running without a TVD limiter; and c>0 reduces oscillations while permitting larger time steps. The CFL study is careful, with two definitions of dx and clear failure thresholds, and the empirical confirmation of Eqs. (29)-(30) is a useful data point.\n\nWhere it is soft: Section 3.1 says \"this modification preserves the properties of the limiter\" without proof. Conservativity is immediate, and positivity at the checked nodes is plausible, but high-order accuracy (the minimum over the enlarged set being O(h^{p+1}) close to the cell average in smooth regions) and the stronger guarantee are not derived. Algorithm 1's Ensure only covers xi_r,3, and the pure GL x GL volume flux nodes are never included in the checked sets, so the GL runs at CFL 0.05 and 0.07 rely on the small CFL rather than on a limiter guarantee. The conclusion's phrase \"guarantees positivity for the entire duration\" overstates what is shown. There is also no code or data, which makes reproduction harder than it should be for a numerical study. One minor textual inconsistency: the Mach 2000 section says \"ten times c+\" but gives c=3.76e-2, which is a factor of about 1000 off from c+=2.87e-5.\n\nOverall, the central practical claim, that this modification makes NSFR usable for strong shock problems, is supported by the benchmark suite. The unproven guarantee is a real gap, but it is one the authors can plausibly close with a proof or by explicitly reframing the claim as empirical. The paper is worth serious referee time.","headline":"Useful, honestly presented numerical study of NSFR shock-capturing, but the modified limiter's guarantees are asserted rather than proved and the GL flux-node case is not covered by the checked node sets.","tokens_in":31262,"tokens_out":3365,"would_cite":true,"duration_ms":40422,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M60","65M70","76N15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that adding solution-node checks to the positivity-preserving limiter lets the nonlinearly stable flux reconstruction method handle shock-dominated compressible flows without TVD limiting, while keeping high-order…","keywords":["nonlinearly stable flux reconstruction","positivity-preserving limiter","bound-preserving limiter","compressible Euler equations","shock capturing","flux reconstruction parameter","two-point flux","entropy stability"],"falsifier":"Construct or find a cell state where density and pressure are positive on both tensored quadrature node sets and on the solution-node set at the start of a stage, but where after one SSPRK3 stage with the CHRA flux some solution node has negative density or pressure; such a case would refute the claimed guarantee. A direct way to look is to run the shock-diffraction or Mach 2000 jet setup with the modified limiter and monitor the solution-node minima at every Runge-Kutta stage for any negative value.","tokens_in":30300,"feed_emoji":"💥","tokens_out":6324,"duration_ms":74112,"temperature":0.7,"pith_summary":"The paper argues that a small change to a standard positivity-preserving limiter makes the nonlinearly stable flux reconstruction (NSFR) method reliable for shock-dominated compressible flows. The change is to evaluate minimum density and pressure not only on the two tensor-product quadrature node sets used by the original limiter, but also on the solution nodes themselves. With that addition, the paper claims, the limiter still preserves high-order accuracy, conservation, and positivity, while preventing the negative density or pressure that previously appeared at solution nodes. If true, NSFR can run demanding tests — shock diffraction, double Mach reflection, and a Mach 2000 astrophysical jet — without any TVD-type limiter, at larger time steps than standard DG methods. The paper supports the claim with a suite of one- and two-dimensional Euler tests rather than a proof.","feed_headline":"Extra node check keeps shock flows positive without TVD","feed_subtitle":"Checking density and pressure at solution nodes lets a stable high-order scheme run Mach 2000 jets and shock diffraction.","key_machinery":"The load-bearing object is the modified positivity-preserving limiter, given as Algorithm 1. It computes cell-averaged density and pressure from two tensor-product quadrature rules, then forms two scaling factors: $\\theta_1$ scales the density toward the cell average so that the minimum density over all three node sets is at least $\\varepsilon = 10^{-13}$, and $\\theta_2$ scales the whole state vector toward the cell average so that the minimum pressure over the same three sets is positive. The cell average is untouched by both scalings, which is what conserves mass and maintains accuracy; the novel step is including the solution-node set $\\xi^{r,3} = \\xi^{\\alpha} \\otimes \\eta^{\\alpha}$ in the min computations.","core_discovery":"The central discovery is that robustness of the positivity-preserving limiter is governed by where the minima are sampled. The paper modifies the two-stage scaling procedure so that the minimum density $\\rho_{\\min}$ and minimum pressure $P_{\\min}$ are computed over three node sets: the two tensored quadrature sets used to compute cell averages, plus the Gauss-Lobatto solution node set. Because the scaling factors $\\theta_1$ and $\\theta_2$ are applied directly at the solution nodes, checking those nodes ensures that the limited polynomial is admissible exactly where the scheme evaluates it. The paper demonstrates, case by case, that this modification lets the NSFR scheme complete shock-dominated simulations that fail with the original limiter, at grid resolutions and CFL numbers where the unmodified approach produces nonphysical values.","pith_inferences":["The same node-set enlargement could be applied to maximum-principle limiters for scalar conservation laws, where checking solution nodes instead of only quadrature nodes may similarly prevent bound violations at the interpolation points.","Because the modification is validated numerically rather than proved, a natural next step is a proof that positivity at the three node sets implies positivity of the scaled polynomial at every solution node under the given CFL condition; that would turn the robustness claim into a theorem.","The paper's evidence that larger FR parameters suppress oscillations suggests an adaptive strategy: pick $c$ locally from a shock sensor, using larger values near discontinuities and $c_{DG}$ in smooth flow, which the authors note but do not implement.","The numerically established CFL bound for the CHRA flux could be tested for the other two-point fluxes considered in the paper to see whether the same $\\Delta t/\\Delta x$ condition is flux-independent."],"forward_implications":["NSFR with the modified limiter can run the Sod shock tube, Shu-Osher, and strong vortex-shock wave interaction at CFL numbers where standard DG fails, and in some cases without the limiter at all.","The limiter preserves the expected order of accuracy: the 2D low-density convergence test reaches the designed rates at $p=2$ and $p=3$ as the grid is refined.","Increasing the flux-reconstruction parameter from $c_{DG}$ toward $c_{+}$ damps oscillations and overshoots and raises the maximum stable CFL, at the cost of extra dissipation in smooth regions.","The Chandrashekar-Ranocha two-point flux numerically satisfies the CFL condition $\\frac{\\Delta t}{\\Delta x}\\max(|u|+c) \\le 1$ for positivity, independent of polynomial degree, in both one and two dimensions.","With a sufficiently strong FR parameter, the Mach 80 and Mach 2000 astrophysical jet cases run to completion with positivity preserved and no TVD limiting."],"supporting_citations":[{"why":"Supplies the original positivity-preserving limiter for the 2D Euler equations that the paper modifies.","marker":"[43]"},{"why":"Provides the improved two-stage scaling form whose solution-node limitation this paper addresses.","marker":"[44]"},{"why":"Establishes the maximum-principle-satisfying high-order scheme foundation behind bound-preserving limiters.","marker":"[1]"},{"why":"Proves nonlinear stability of the NSFR split form that the scheme is built on.","marker":"[22]"},{"why":"Provides the provably stable flux reconstruction formulation on curvilinear elements used by the method.","marker":"[23]"},{"why":"Supplies the two-point flux differencing and entropy stable formulation for uncollocated nodes used in NSFR.","marker":"[8]"},{"why":"Defines the CHRA two-point flux used in all test cases.","marker":"[45]"},{"why":"Gives the detailed derivation and proofs of the NSFR discretization that the limiter is coupled to.","marker":"[42]"}],"fun_headline_variants":["Node-based limiter check makes shock flows positive","Sampling at solution nodes preserves positivity in shocks","Limiter fix extends stable flux reconstruction to shock flows","Shock-capturing improved with node-aware positivity limiter","Node checks strengthen bound-preserving limiter for shocks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central premise is that checking the minimum density and pressure on the solution nodes in addition to the quadrature nodes preserves the limiter's guarantees of positivity, accuracy, and conservation; the paper asserts this and verifies it numerically, but does not prove it.","fun_headline_variants_meta":{"raw":{"variants":["Node-based limiter check makes shock flows positive","Sampling at solution nodes preserves positivity in shocks","Limiter fix extends stable flux reconstruction to shock flows","Shock-capturing improved with node-aware positivity limiter","Node checks strengthen bound-preserving limiter for shocks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000351,"raw_usage":{"total_tokens":1957,"prompt_tokens":1028,"completion_tokens":929,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":644,"completion_tokens_details":{"reasoning_tokens":853}},"tokens_in":644,"tokens_out":929,"duration_ms":8157,"temperature":1.0,"reasoning_tokens":853,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:03:33.956001+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct or find a cell state where density and pressure are positive on both tensored quadrature node sets and on the solution-node set at the start of a stage, but where after one SSPRK3 stage with the CHRA flux some solution node has negative density or pressure; such a case would refute the claimed guarantee. A direct way to look is to run the shock-diffraction or Mach 2000 jet setup with the modified limiter and monitor the solution-node minima at every Runge-Kutta stage for any negative value.","supporting_citations":[{"cited_title":"Zhang, C.-W","cited_arxiv_id":null,"evidence_quote":"Supplies the original positivity-preserving limiter for the 2D Euler equations that the paper modifies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the improved two-stage scaling form whose solution-node limitation this paper addresses."},{"cited_title":"Cicchino, S","cited_arxiv_id":null,"evidence_quote":"Proves nonlinear stability of the NSFR split form that the scheme is built on."},{"cited_title":"Ranocha, G","cited_arxiv_id":null,"evidence_quote":"Defines the CHRA two-point flux used in all test cases."},{"cited_title":"Cicchino, Weight-adjusted nonlinearly stable flux reconstruction high-order methods for compressible flows in curvi- linear coordinates (2024)","cited_arxiv_id":null,"evidence_quote":"Gives the detailed derivation and proofs of the NSFR discretization that the limiter is coupled to."}],"review_version":1}