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arxiv: 2607.06045 · v1 · pith:E6HM7JAA · submitted 2026-07-07 · math.NA · cs.NA

Invariant-domain-preserving limiting with Adaptive Mesh Refinement for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral Element Methods

Benjamin Bolm , Andr\'es M. Rueda-Ram\'irez , Dmitri Kuzmin , Gregor J. Gassner This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-08 18:07 UTCglm-5.2pith:E6HM7JAArecord.jsonopen to challenge →

classification math.NA cs.NA
keywords discontinuous Galerkin spectral element methodinvariant domain preservationadaptive mesh refinementmortar fluxconvex limitingcompressible Euler equationssubcell limiting
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0 comments X

The pith

Sparse mortar fluxes keep high-order simulations stable across hanging nodes

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a new type of interface flux for high-order discontinuous Galerkin simulations on adaptively refined meshes, where some grid cells are split into smaller ones and the cell boundaries no longer line up. At these nonconforming interfaces, standard high-order mortar fluxes can produce unphysical values such as negative density or pressure, which crashes simulations of compressible flows. The authors derive replacement fluxes that are provably conservative and preserve physical admissibility constraints (positivity, convex invariant sets) under an explicit time-step bound. The key mechanism is a sparsification strategy: instead of connecting every node on one side of the interface to every node on the other, they connect only nodes whose underlying subcells overlap, using characteristic functions of the LGL subcell grid as weights. This yields a compact stencil whose size does not grow with polynomial degree, avoids the excessive numerical diffusion of fully connected mortars, reduces exactly to the standard conforming-interface flux when no hanging nodes are present, and slots into existing graph-viscosity-based convex limiting frameworks via the discrete metric identities. The construction is verified through convergence tests and demonstrated on challenging compressible Euler problems including the Sedov blast, Double Mach reflection, and a Mach-2000 astrophysical jet.

Core claim

The central object is the sparse invariant-domain-preserving mortar flux defined by equations (31)-(32). Its local weights are computed as integrals of products of piecewise-constant characteristic functions of LGL subcells (equation 26), which are nonzero only when the subcells on opposite sides of a nonconforming interface physically overlap. This sparsification preserves conservation (equation 25), reduces to the standard conforming flux (equation 33), satisfies the discrete metric identities needed to fit into the graph-viscosity low-order update (equation 38), and yields a provably invariant-domain-preserving scheme under the time-step restriction (equation 44). The proof proceeds by re

What carries the argument

LGL subcell characteristic functions

If this is right

  • Adaptive mesh refinement can now be combined with provably positivity-preserving high-order DGSEM for compressible flow simulations, removing a major practical obstacle to using AMR in production shock-capturing codes.
  • The sparsification strategy via subcell characteristic functions could be extended to non-Cartesian meshes or to interfaces with refinement ratios other than 2-to-1, broadening applicability to general unstructured adaptive grids.
  • The IDP time-step restriction is more restrictive than the standard CFL condition, motivating the practical workaround of using low-order-solution-based bounds instead of bar-state bounds, which trades provable guarantees for computational efficiency.
  • The combined limiting strategy (equation 62), blending positivity limiting in smooth regions with local limiting at shocks via a smoothness indicator, offers a template for reducing numerical dissipation in high-order AMR simulations of complex flows.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The fact that the solution transfer operators during refinement and coarsening are not themselves IDP means the overall AMR scheme's guarantees are conditional on the Zhang-Shu post-processing limiter succeeding at every mesh adaptation event; a failure case for that limiter would break the chain of guarantees regardless of the mortar flux correctness.
  • The sparsification via subcell characteristic functions is conceptually independent of the specific choice of LGL nodes; an analogous construction might work for other nodal DG bases (e.g., Gauss or Gauss-Radau), though the diagonal mass matrix property and subcell structure that make LGL natural would need substitutes.
  • The sensitivity of results to minor setup differences (Remark 8) suggests that benchmark comparisons across codes using different AMR indicators or limiting strategies may produce visibly different solutions for problems like the astrophysical jet, complicating code verification.

Load-bearing premise

The solution transfer operators used when the mesh is refined or coarsened during AMR are not themselves invariant-domain-preserving; the authors rely on a Zhang-Shu positivity limiter as a post-processing fix, which is heuristic rather than guaranteed. If this limiter fails for some problem configuration, the overall scheme's physical-admissibility guarantee is broken at every refinement or coarsening event.

What would settle it

Construct a problem setup where the Zhang-Shu scaling limiter cannot restore admissible states after a mesh transfer (e.g., involving extreme density or pressure ratios at refinement boundaries), demonstrating that the overall AMR scheme produces non-physical values despite the mortar flux being correct.

Figures

Figures reproduced from arXiv: 2607.06045 by Andr\'es M. Rueda-Ram\'irez, Benjamin Bolm, Dmitri Kuzmin, Gregor J. Gassner.

Figure 1
Figure 1. Figure 1: Sketch of a conforming interface for a polynomial degree of 𝑁 = 3. Conforming meshes allow building a subcell LGL grid where every node has 2 𝑑 neighboring nodes based on the flux-differencing formulation. This holds for inner nodes as well as for nodes on interfaces. For that purpose, one high￾order and one low-order version of the flux are constructed and then blended together. This is explained in detai… view at source ↗
Figure 2
Figure 2. Figure 2: Sketch of a nonconforming interface for a polynomial degree of 𝑁 = 3. where {𝜑 − 𝑖 }𝑖∈− 𝑆 are the basis functions in the left element and {𝜑 + 𝑖 }𝑖∈+ 𝑆 in the right elements across 𝑆. For tensor￾product LGL elements, 𝜑𝑖 are 1D Lagrange basis functions. Note that the integrals are computed exactly using LGL quadrature with 𝑁 + 1 nodes. Since the basis functions on both sides of 𝑆 sum to unity, we have ∑ 𝑖… view at source ↗
Figure 3
Figure 3. Figure 3: Sketches of characteristic functions of LGL subcells for 𝑁 = 3, i.e., 𝑁 + 1 = 4 LGL nodes. Due to the nonnegativity of the characteristic functions, the local weights themselves are nonnegative as well. Moreover, similar to 𝜑 − and 𝜑 +, we have ∑ 𝑖∈− 𝑆 𝜓 − 𝑖 ≡ 1, ∑ 𝑗∈+ 𝑆 𝜓 + 𝑗 ≡ 1, (27) which yields ∑ 𝑖∈− 𝑆 ̃𝜔 𝑆 (𝑖,𝑗) = ∫𝑆 𝜓 + 𝑗 d𝑠 = 𝜔 + 𝑗,𝑆, ∀𝑗 ∈ + 𝑆 , ∑ 𝑗∈+ 𝑆 ̃𝜔 𝑆 (𝑖,𝑗) = ∫𝑆 𝜓 − 𝑖 d𝑠 = 𝜔 − 𝑖,𝑆, ∀𝑖 ∈… view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of node connections using the sparse approach for a polynomial degree of 𝑁 = 3. We refer to this as the IDP time step restriction. With that, the proof is complete. Remark 3. The proof uses a forward Euler step for the time integration. In order to achieve a higher order time integration, we use a strong-stability preserving (SSP) Runge–Kutta (RK) method which can be written as a convex combin… view at source ↗
Figure 5
Figure 5. Figure 5: Nonconforming mesh with base refinement level 4 used for the convergence tests. We quantify the 𝐿2 error of the solution as the mesh is further refined while the refined box is refined once more in every iteration. We perform convergence tests with polynomial degrees 𝑁 = 3 and 𝑁 = 4. We use CFL = 0.95 with the IDP time step restriction (44) in the following simulations. For the first simulation we enabled … view at source ↗
Figure 6
Figure 6. Figure 6: Density and pressure contours of the initial condition of the isentropic flow setup. A polynomial degree of 𝑁 = 3 and a mesh with base refinement level 5 with an additional refined box is used. Bolm, Rueda-Ramírez, Kuzmin, Gassner: Preprint submitted to Elsevier Page 16 of 30 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Density and pressure contours of the solution of the isentropic flow simulation at time 𝑡 = 0.1. A polynomial degree of 𝑁 = 3 and a mesh with base refinement level 5 with an additional refined box is used. (a) Density (b) Pressure [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Resulting density and pressure along the diagonal 𝑥 ′ slice for the simulation of the isentropic flow at time 𝑡 = 0.1. The simulation runs until 𝑡 = 0.1 and uses positivity limiting of density and pressure in the volume integral and the mortars. Due to the pure positivity limiting we use the low-order stability CFL condition (47) with CFL = 0.5 to reduce the number of time steps [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 9
Figure 9. Figure 9: Analysis of the mortar limiting in the simulation of the isentropic flow. break the simulation and therefore yields the full use of the stable low-order mortar flux at the affected mortars. The high-order flux in the volume integral is less affected by this. Another reason specifically in our implementation is the fact that we first apply the correction in the volume integral and only afterward at the mort… view at source ↗
Figure 10
Figure 10. Figure 10: Density contours and the used mesh for the simulation of the Kelvin-Helmholtz instability with polynomial degree of 7 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of the averaged limiting factors for the simulation of the Kelvin-Helmholtz instability. with 𝑟 = √ 𝑥 2 + 𝑦 2. We use the shock-capturing indicator developed by Hennemann et al. [25] applied as an AMR indicator for density and pressure to add refinement at shocks. The AMR setup employs between 2 4 = 16 and 2 7 = 128 elements per dimension. The polynomial degree is 𝑁 = 3 and the final time is 𝑡 =… view at source ↗
Figure 12
Figure 12. Figure 12: The second simulation uses local bounds based on the low-order solution, (a) Density contours and mesh at 𝑡 = 3. (b) Evolution of the limiting factors [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Results of the simulation of the Sedov blast wave using local bounds based on the low-order solution [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Resulting density and pressure along the diagonal 𝑥 = 𝑦. 3.5. Double Mach Reflection The next simulation uses the Double Mach Reflection setup based on [40]. It was also used in [41]. The initial setup contains a propagating shock of Mach 10 with an angle of 60°. It divides the spatial domain Ω = [0, 3.25] × [0, 1], which we have shortened slightly on the right side compared to the original setup, into a … view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of time step sizes of simulations with bar-state bounds and low-order bounds. This is the first setup with a non-periodic domain. In both 𝜉1 and positive 𝜉2 direction we consider a characteristic-based inflow/outflow boundary condition. For that, we use the initial condition propagated over time such that the post-shock values (left) are applied for 𝑥 < 1∕6 + (𝑦 + 20𝑡)∕√ 3 and the pre-shock val… view at source ↗
Figure 16
Figure 16. Figure 16: Density contours and resulting mesh of the simulation of the Double Mach reflection at 𝑡 = 0.2 with local limiting [PITH_FULL_IMAGE:figures/full_fig_p024_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Evolution of the limiting factors. 3.6. High-Mach Astrophysical Jet A common benchmark to test the robustness of the used scheme is the astrophysical jet with a Mach number of about 2000. The setup was originally proposed in [42]. The computational domain Ω = [−0.5, 0.5]2 contains a mono-atomic gas (𝛾 = 5∕3) and is at rest at the start of the simulation with 𝜌(𝑥, 𝑦) = 0.5, 𝑝(𝑥, 𝑦) = 0.4127, 𝑣1 (𝑥, 𝑦) = 0,… view at source ↗
Figure 18
Figure 18. Figure 18: Results of the simulation of the Double Mach reflection at 𝑡 = 0.2 using a combination of positivity limiting and local limiting. conditions at the bottom and top boundaries and characteristic-based inflow/outflow boundary condition on the left and right. We use the entropy-conserving and kinetic energy preserving flux of Chandrashekar [43] for the volume numerical fluxes. The simulation again uses the pr… view at source ↗
Figure 19
Figure 19. Figure 19: Results of the simulation of the astrophysical jet at 𝑡 = 0.0015. artifacts or carbuncles. The small-scale structures on both sides of the jet are symmetric, except for a few very minor irregularities. Remark 8. The results presented in this paper are, in some cases, highly sensitive to minor differences in the numerical setup. There are several reasons for this. Primarily, the chosen setups are inherentl… view at source ↗
read the original abstract

We present an invariant-domain-preserving (IDP) treatment of nonconforming interfaces for Legendre--Gauss--Lobatto Discontinuous Galerkin Spectral Element Methods (LGL-DGSEM) with adaptive mesh refinement (AMR) on Cartesian meshes. The proposed methodology extends recently developed convex limiting and graph-viscosity frameworks for DGSEM to meshes containing hanging nodes. Starting from a conservative mortar formulation, we derive low-order interface fluxes that satisfy the requirements of invariant-domain-preserving discretizations. To avoid the excessive diffusion associated with fully connected mortar couplings, a sparsification strategy based on LGL subcell characteristic functions is introduced, yielding compact interface stencils. The resulting mortar fluxes remain conservative, reduce to the standard conforming formulation on matching interfaces, and naturally fit into graph-viscosity-based low-order schemes used for convex limiting. The proposed construction provides the missing ingredient required to combine high-order DGSEM discretizations, invariant-domain-preserving limiting, and adaptive mesh refinement within a unified framework for nonlinear hyperbolic conservation laws. We provide numerical verifications of the properties of the proposed scheme and run challenging simulations that require positivity limiting and shock-capturing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 9 minor

Summary. The paper constructs an invariant-domain-preserving (IDP) low-order mortar flux for Legendre–Gauss–Lobatto Discontinuous Galerkin Spectral Element Methods (LGL-DGSEM) on nonconforming Cartesian interfaces arising from adaptive mesh refinement (AMR). The key technical contribution is a sparsification strategy (Eqs. 26–28) that replaces the dense all-to-all mortar coupling with compact, subcell-based weights derived from characteristic functions of LGL subcells. The resulting fluxes are shown to be conservative (Eq. 25), to reduce to the standard conforming formulation on matching interfaces (Eq. 33), to satisfy discrete metric identities (Eqs. 38–39), and to yield an IDP scheme under a derived time-step restriction (Eq. 44) via a bar-state convex-combination argument (Eqs. 40–43). The method is integrated into an existing convex-limiting/FCT framework and tested on advection, isentropic flow, Kelvin–Helmholtz, Sedov blast, Double Mach reflection, and a high-Mach astrophysical jet.

Significance. The paper addresses a genuine gap: existing IDP/convex-limiting DGSEM frameworks require sparse low-order operators, and the extension to nonconforming (hanging-node) interfaces has not been available. The sparsification via LGL subcell characteristic functions is a natural and parameter-free construction, and the IDP proof via bar-states is clean and follows established convex-limiting theory. The convergence tests (Tables 1, 3, 4) show expected orders in smooth regimes, and the challenging Euler simulations (Sedov blast, Double Mach, Mach-2000 jet) demonstrate practical robustness. The explicit weight tables (Tables 5, 6) and the illustrative computation in Appendix A aid reproducibility. The main limitation, acknowledged by the authors, is that the solution transfer during AMR refinement/coarsening (Section 2.4) is not IDP and relies on a Zhang–Shu scaling limiter as a heuristic post-processing fix.

major comments (3)
  1. Section 2.4: The solution transfer operators used during AMR refinement and coarsening are not invariant-domain-preserving, and the authors acknowledge this explicitly. The Zhang–Shu scaling limiter is a heuristic post-processing step, not a provable IDP mechanism. This means the overall scheme's IDP guarantee is broken at every refinement/coarsening event, regardless of the mortar flux correctness. Since AMR is a central feature of the paper's contribution, this gap is load-bearing for the claim of a unified IDP-AMR framework. The authors should either (a) clearly scope the IDP claim to the time-stepping update only, with the transfer step explicitly excluded from the guarantee, or (b) provide a more rigorous argument for why the Zhang–Shu limiter suffices (e.g., conditions under which it is guaranteed to restore admissibility). As written, the abstract and conclusion claim an IDP-AMR '
  2. Remark 6 and Section 3.4: The IDP time-step restriction (44) is acknowledged to be more restrictive than the standard low-order CFL condition (47), and the authors describe a practical workaround that bypasses the bar-state bounds entirely, using low-order-solution-based bounds instead. This workaround is used in several numerical experiments (isentropic flow: CFL=0.5 with Eq. 47; KHI: CFL=0.2 with Eq. 47; Sedov with low-order bounds: CFL=0.4 with Eq. 47). When this workaround is used, the scheme is no longer provably IDP. The paper should clearly state in each experiment's description which CFL condition and which bound type are used, and should avoid presenting results obtained under the non-IDP workaround as validation of the IDP scheme. A table summarizing which experiments use the provably IDP configuration would help.
  3. Section 2.3, Remark 7: The mortar limiting uses one limiting factor per mortar, and the admissible correction budget at a node is divided among all incident mortar contributions by scaling each anti-diffusive flux by n_i (the number of active mortar contributions). This is a conservative heuristic, but it is potentially suboptimal and may introduce more dissipation than necessary. The authors should discuss whether this division strategy preserves the formal accuracy of the high-order scheme in smooth regions (i.e., whether alpha_S -> 0 at the designed rate as the mesh is refined). The convergence results in Table 2 (local limiting, EOC ~ 1) suggest significant dissipation, but it is unclear how much of this is due to the mortar limiting strategy versus the local bounds themselves.
minor comments (9)
  1. The notation switches between local 2D indices (ij) and global indices (i) throughout Section 2. While the authors note this, a brief reminder at key transitions (e.g., before Eq. 11) would improve readability.
  2. Equation (45): The notation (Nj)^+ is introduced without explicit definition. It would help to state that this refers to the matching node on the neighbor element across the interface.
  3. Table 2: The EOC values for local limiting with N=3 converge to approximately 0.97, which is close to first order. The authors state this was 'expected,' but a brief explanation of why first order (rather than a higher sub-optimal order) is the expected rate for local limiting would be informative.
  4. Section 3.2: The isentropic flow test uses gamma=3, while the rest of the paper uses gamma=1.4. This should be stated more prominently, as it affects the near-vacuum conditions.
  5. Figure 9a: The y-axis label is unclear. Specifying that it shows the weighted average of limiting factors would help.
  6. Section 3.5: The domain is shortened to [0, 3.25] compared to the original [0, 4]. This should be noted as a deviation from the standard benchmark.
  7. Remark 8 discusses sensitivity of results to minor setup differences. While this is an honest disclosure, it would benefit from a more specific statement about which results are affected and to what degree.
  8. References: The self-citations [17, 18] are appropriate but frequent. The authors should ensure that the present contribution is clearly distinguished from [18], which also deals with monolithic convex limiting for LGL-DGSEM.
  9. No reproducible code or data is shipped with the paper ('Access to data and software is granted upon request'). For a methodological paper with extensive numerical experiments, providing a reproducibility artifact would strengthen the contribution.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The three major comments are well-taken, and we agree that the manuscript needs revisions to clarify the scope of the IDP guarantee and to label experiments more transparently. We address each comment below.

read point-by-point responses

  1. Referee: Section 2.4: The solution transfer operators used during AMR refinement and coarsening are not invariant-domain-preserving, and the authors acknowledge this explicitly. The Zhang–Shu scaling limiter is a heuristic post-processing step, not a provable IDP mechanism. This means the overall scheme's IDP guarantee is broken at every refinement/coarsening event, regardless of the mortar flux correctness. Since AMR is a central feature of the paper's contribution, this gap is load-bearing for the claim of a unified IDP-AMR framework. The authors should either (a) clearly scope the IDP claim to the time-stepping update only, with the transfer step explicitly excluded from the guarantee, or (b) provide a more rigorous argument for why the Zhang–Shu limiter suffices.

    Authors: The referee is correct on the substance. The solution transfer operators (interpolation for refinement, L2 projection for coarsening) are not IDP, and the Zhang–Shu scaling limiter we apply is a heuristic positivity-stabilizing post-processing step, not a provable mechanism for restoring admissibility in all cases. We do not have a rigorous argument that would elevate option (b) to a theorem, so we will adopt option (a). Specifically, we will revise the abstract, the introduction, and the conclusion to scope the IDP guarantee explicitly to the time-stepping update (i.e., the semi-discrete low-order scheme and the convex-limiting correction stage), and to state clearly that the solution transfer step is excluded from the IDP guarantee. Section 2.4 already contains the disclaimer ('We do not claim that this transfer operator is invariant-domain preserving in the strict sense'), but we will strengthen this language and add a forward reference from the abstract and conclusion so that the scope is unambiguous. We will also add a brief remark noting that constructing a provably IDP transfer operator for high-order DGSEM is an open problem and a target for future work. revision: yes

  2. Referee: Remark 6 and Section 3.4: The IDP time-step restriction (44) is acknowledged to be more restrictive than the standard low-order CFL condition (47), and the authors describe a practical workaround that bypasses the bar-state bounds entirely, using low-order-solution-based bounds instead. This workaround is used in several numerical experiments (isentropic flow: CFL=0.5 with Eq. 47; KHI: CFL=0.2 with Eq. 47; Sedov with low-order bounds: CFL=0.4 with Eq. 47). When this workaround is used, the scheme is no longer provably IDP. The paper should clearly state in each experiment's description which CFL condition and which bound type are used, and should avoid presenting results obtained under the non-IDP workaround as validation of the IDP scheme. A table summarizing which experiments use the provably IDP configuration would help.

    Authors: We agree. The distinction between the provably IDP configuration (bar-state bounds + CFL condition (44)) and the practical workaround (low-order-solution-based bounds + CFL condition (47)) is important and should be stated explicitly for each experiment. We will add a summary table (new Table) listing, for each numerical experiment: the bound type (bar-state local bounds, low-order-solution-based bounds, or positivity-only), the CFL condition used ((44) or (47)), the CFL number, and whether the configuration is provably IDP. We will also add a sentence at the beginning of each experiment's description cross-referencing this table. To be precise about the current state of the manuscript: the advection convergence test (Section 3.1) uses the IDP CFL condition (44) with CFL=0.95; the isentropic flow test (Section 3.2) uses the low-order CFL (47) with CFL=0.5 and positivity limiting (non-IDP workaround); the KHI (Section 3.3) uses CFL (47) with CFL=0.2 and positivity limiting (non-IDP workaround); the Sedov blast (Section 3.4) presents both configurations — bar-state bounds with CFL (44) at CFL=0.95 (provably IDP) and low-order-solution-based bounds with CFL (47) at CFL=0.4 (non-IDP workaround); the Double Mach reflection (Section 3.5) uses bar-state bounds with CFL (44) at CFL=0.9 (provably IDP); and the astrophysical jet (Section 3.6) uses bar-state bounds with CFL (44) at CFL=0.9 (provably IDP). We will make all of this explicit in the revised text and ensure that results obtained under the non-IDP workaround are not presented as validation of the IDP property, but rather as demonstrations of practical robustness. revision: yes

  3. Referee: Section 2.3, Remark 7: The mortar limiting uses one limiting factor per mortar, and the admissible correction budget at a node is divided among all incident mortar contributions by scaling each anti-diffusive flux by n_i (the number of active mortar contributions). This is a conservative heuristic, but it is potentially suboptimal and may introduce more dissipation than necessary. The authors should discuss whether this division strategy preserves the formal accuracy of the high-order scheme in smooth regions (i.e., whether alpha_S -> 0 at the designed rate as the mesh is refined). The convergence results in Table 2 (local limiting, EOC ~ 1) suggest significant dissipation, but it is unclear how much of this is due to the mortar limiting strategy versus the local bounds themselves.

    Authors: The referee raises a valid question about whether the n_i-scaling strategy preserves formal high-order accuracy in smooth regions. We do not have a formal proof that alpha_S -> 0 at the designed rate under mesh refinement when the n_i-scaling is used. However, we can offer the following observations. First, the EOC~1 result in Table 2 (local limiting with N=3) is consistent with what is observed in the conforming case with local limiting — see, e.g., [17, Table 4] where the conforming subcell limiting with local bounds also yields EOC~1. Since the conforming case has no mortar limiting at all, the reduced order of convergence is primarily attributable to the local bounds themselves (which enforce discrete maximum principles and thus introduce O(h) dissipation), not to the mortar limiting strategy per se. Second, the advection convergence test with positivity limiting only (Table 1) achieves full high-order convergence (EOC~4 for N=3), which shows that the mortar flux construction and the mortar limiting framework do not degrade accuracy when the local bounds are inactive. Third, the n_i-scaling is a conservative heuristic in the sense that it guarantees the sum of all mortar corrections stays within the admissible interval, but it is indeed potentially suboptimal. A tighter analysis could use a Zalesak-type synchronized limiting across all mortars incident on a node, but this would require a coupled solve and is significantly more complex to implement. We will add a remark to Section 2.3 discussing these points: (i) the EOC~1 is consistent with the conforming local-limiting case and is dominated by the local bounds, not the mortar strategy; (ii) the n_i-scaling is conservative but potentially suboptimal; and (iii) a synchronized Zalesak-type mortar limiter is a natural, revision: no

Circularity Check

0 steps flagged

No significant circularity found; the derivation is parameter-free and self-contained.

full rationale

The paper's central derivation chain is not circular. The sparse mortar fluxes (Eqs. 31-32) are constructed from local weights defined by LGL subcell characteristic functions (Eq. 26), which are geometric quantities determined by the subcell layout — not fitted to target data. Conservation (Eq. 25) follows from the symmetry of the weights and n_S^+ = -n_S^- (Eq. 24 vs. 21). The discrete metric identity (Eq. 38) extends to the nonconforming case because the partition of unity (Eq. 28) holds for characteristic functions by construction, yielding Eq. (39) = (1/2) ω_{i,S}^- n_S^- — the same value the volume SBP terms cancel. The bar-state form (Eq. 42) is a standard algebraic rearrangement of the low-order update (Eq. 11) using the metric identity, and the IDP property follows from the convexity of bar-states (Eq. 40) under the CFL restriction (Eq. 44), which is derived, not fitted. Self-citations [17, 18] provide the conforming-case foundation, but the nonconforming extension is independently derived: the sparsification (Eq. 26), the weight properties (Eq. 28), the metric identity extension (Eq. 39), and the IDP proof are all carried out in this paper. The conforming reduction (Eq. 33) is a consistency check, not a circular definition. The solution transfer gap (Section 2.4) is an acknowledged practical limitation, not a circularity in the mortar flux derivation. No step reduces to its inputs by construction or by unverified self-citation chain. Score 1 reflects minor self-citation to [17, 18] for the conforming-case framework, which is not load-bearing for the nonconforming extension's independent derivation.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

No new physical entities, particles, forces, or dimensions are introduced. The method is purely algorithmic. The free parameters (beta, CFL, threshold) are standard user choices in the convex limiting literature, not fitted constants. The axioms are either standard mathematical results or domain restrictions clearly stated in the paper.

free parameters (3)
  • beta (positivity lower bound scaling) = 0.1
    Used in Eq. (50) to set density/pressure lower bounds as a fraction of the low-order solution. Not fitted to data but chosen by hand; standard in the convex limiting literature.
  • CFL number = varies (0.2-0.95)
    Chosen per simulation; not a derived constant. The IDP restriction (44) is derived, but the CFL multiplier applied in practice is user-chosen.
  • global positivity threshold = 1e-10 or 1e-6
    Used in the Zhang-Shu limiter for solution transfer (Section 2.4). Chosen as a small constant; problem-dependent.
axioms (4)
  • standard math The invariant domain G is convex and the bar-states lie in G if the nodal states do (convexity of LLF/HLL intermediate states).
    Invoked in the IDP proof (Section 2.2, Eq. 40). This is a standard result from Guermond et al. [11] and Harten-Lax-van Leer [28].
  • standard math SSP-RK methods can be written as convex combinations of forward Euler steps.
    Invoked in Remark 3 to extend the forward Euler IDP proof to higher-order time integration. Standard SSP theory.
  • domain assumption Nonconforming interfaces are 2-to-1 connections with same polynomial degree on both sides.
    Stated in Section 2.2. Restricts generality; 3D and multi-level nonconformity are not covered.
  • ad hoc to paper The Zhang-Shu scaling limiter sufficiently stabilizes solution transfer during AMR.
    Section 2.4: the transfer operator is not IDP, and the limiter is a heuristic fix. The paper acknowledges this is not a strict guarantee.

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