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Sparse low-order mortar fluxes let high-order DG methods keep invariant domains on hanging-node adaptive meshes.

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2026-07-14 16:07 UTC pith:E6HM7JAA

load-bearing objection Clean, correctly scoped fix that finally lets IDP/convex-limiting LGL-DGSEM run on Cartesian AMR with hanging nodes.

arxiv 2607.06045 v2 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
classification math.NA cs.NA MSC 65M6065M7076M1035L65
keywords DGSEMinvariant domain preservationconvex limitingmortar methodsadaptive mesh refinementgraph viscosityEuler equationspositivity preservation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

High-order discontinuous Galerkin spectral-element schemes are accurate and efficient, yet they can produce non-physical states near shocks or rarefactions. Invariant-domain-preserving limiters fix that on conforming meshes by blending a carefully designed low-order graph-viscosity scheme with the high-order residual. Adaptive mesh refinement, however, introduces hanging nodes whose standard mortar couplings destroy the low-order structure and the proof of domain preservation. This paper constructs a new set of interface fluxes for exactly those nonconforming faces: they remain conservative, collapse to ordinary conforming fluxes when nodes match, and use LGL-subcell characteristic functions so that each node couples only to a few nearby nodes on the opposite side. The resulting sparse low-order operator fits the existing convex-limiting framework, so positivity of density and pressure (and optional local bounds) can be enforced under a CFL restriction even while the mesh is refined and coarsened. Numerical tests confirm high-order accuracy in smooth regions and robust shock-capturing on classic Euler problems that demand both adaptivity and limiting.

Core claim

On Cartesian meshes with at most one-level hanging-node interfaces, the sparsified mortar fluxes built from LGL-subcell characteristic-function weights are conservative, reduce to the standard local-Lax–Friedrichs flux on conforming faces, fit the graph-viscosity low-order form, and therefore yield an invariant-domain-preserving semi-discrete scheme under an explicit CFL restriction.

What carries the argument

The sparsified low-order mortar fluxes (31)–(32) whose interface weights are integrals of LGL-subcell characteristic functions; these weights produce a compact stencil that still satisfies the discrete metric identities and the bar-state convexity argument.

Load-bearing premise

The construction and the IDP proof hold only for Cartesian meshes, equal polynomial degree on both sides of a mortar, and at most a single level of refinement difference; the operators that transfer the solution during mesh refinement or coarsening are only positivity-stabilized, not proved fully invariant-domain-preserving.

What would settle it

On a 2-to-1 Cartesian mortar with equal polynomial degree, replace the characteristic-function weights by the dense L2-projection mortar weights and check whether the resulting low-order scheme still keeps density and pressure non-negative for every CFL-stable time step on a near-vacuum isentropic vortex or Sedov blast; if it does not, the sparsification claim is essential.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper constructs sparse, conservative, invariant-domain-preserving (IDP) mortar fluxes for nonconforming interfaces in LGL-DGSEM on Cartesian meshes with hanging nodes (equal polynomial degree, one-level 2-to-1 refinement). Starting from an all-to-all central mortar integral, the authors replace the dense L2-projection weights by non-negative overlap integrals of LGL-subcell characteristic functions, add a local Lax–Friedrichs graph-viscosity term, and obtain interface fluxes (31)–(32) that reduce exactly to the standard conforming LLF flux on matching faces, satisfy discrete metric identities, fit the graph-viscosity low-order form (11), and therefore admit the standard bar-state convex-combination argument for IDP under the CFL restriction (44). The fluxes are then blended with high-order L2-mortar fluxes via a posteriori FCT-type limiting; a Zhang–Shu scaling limiter is applied after AMR transfers. Numerical tests (density-wave and isentropic-flow convergence, Kelvin–Helmholtz, Sedov, Double Mach, high-Mach jet) confirm conservation, expected orders under pure positivity limiting, and robustness on adaptively refined meshes.

Significance. The construction supplies the missing low-order interface operator that allows existing convex-limiting / graph-viscosity DGSEM frameworks to be used with hanging-node AMR. The algebraic derivation is self-contained once the standard LGL-DGSEM and graph-viscosity machinery are granted; conservation, reduction to the conforming case, and the IDP property under the stated CFL follow by direct verification without free parameters. Implementations are provided in the open-source Trixi.jl framework, and the numerical suite includes both order verification and genuinely challenging Euler problems that require positivity and shock capturing. Within the declared Cartesian, equal-degree, one-level setting the result is a clean and useful building block for high-order adaptive simulations of nonlinear hyperbolic systems.

minor comments (6)
  1. Section 2.2, after (33): a short explicit statement that the sparsified weights remain non-negative and form a partition of unity on both sides of a 2-to-1 interface would make the subsequent bar-state argument completely self-contained without reference to the appendix.
  2. Remark 6 and Section 3.4: the practical use of the less restrictive low-order CFL together with low-order-solution bounds is well motivated, but a one-sentence clarification that the resulting scheme is no longer provably IDP (only positivity-preserving in practice) would avoid any ambiguity for readers who skip the remark.
  3. Section 2.4: the Zhang–Shu transfer limiter is correctly described as positivity-stabilizing rather than fully IDP; a brief remark that a fully IDP transfer operator remains open would be helpful.
  4. Figures 9b, 12a, 13a, 16, 18a: the colour scales for limiting factors and indicator variables are not always labelled; adding a colour bar or a short legend would improve readability.
  5. A few typographical slips remain (e.g., “adaptivemeshrefinement”, missing spaces after commas in the abstract and introduction, “theso-called”). A careful proof-reading pass would remove them.
  6. Table 5 and the accompanying text in Appendix A give a useful concrete example of the sparse weights; a one-line reference to this table already in Section 2.2 would help readers who want an immediate illustration of the stencil.

Circularity Check

0 steps flagged

No significant circularity: the sparsified IDP mortar fluxes are derived algebraically from first principles and reduce to known conforming operators by direct identity, without fitted parameters or load-bearing self-citation chains.

full rationale

The central construction (Section 2.2) begins from a conservative all-to-all mortar surface integral, replaces the dense L2 weights by non-negative LGL-subcell characteristic-function weights that obey the partition-of-unity identities, adds a standard local-Lax–Friedrichs graph-viscosity term, and rewrites the result in the already-established graph-viscosity residual form. Conservation follows at once from antisymmetry of the normals and the weight identities; reduction to the conforming LLF flux is the elementary Kronecker-delta collapse of the characteristic functions on matching interfaces; the bar-state convex-combination argument and the associated CFL restriction are the standard ones already proved for conforming meshes. All steps are purely algebraic identities that hold under the paper’s explicitly declared hypotheses (Cartesian meshes, equal polynomial degree, 2-to-1 hanging-node interfaces). Prior self-citations supply only the volume-limiting machinery and the general IDP framework; they are not used to force the mortar fluxes themselves. No parameters are fitted to data, no uniqueness theorem is imported to exclude alternatives, and no known empirical pattern is merely renamed. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The paper rests on standard SBP/DGSEM theory, the Guermond–Popov bar-state IDP framework, and the authors’ prior subcell limiting operators. The only genuinely new mathematical objects are the sparse mortar weights and the mortar-level FCT blending; no free parameters are fitted to data.

axioms (3)
  • domain assumption The low-order graph-viscosity operator with local Lax–Friedrichs viscosity yields bar-states that remain inside any convex invariant set of the hyperbolic system.
    Invoked throughout Section 2.2 to transfer the IDP property from bar-states to the fully discrete scheme; taken from Guermond–Popov theory.
  • standard math LGL quadrature and the associated diagonal-norm SBP operators satisfy the discrete metric identities on Cartesian elements.
    Used to cancel the self-flux terms and obtain the pure bar-state form (42).
  • ad hoc to paper Characteristic functions of the LGL subcells form a partition of unity and produce non-negative overlap weights.
    Definition (26) and the subsequent sparsity argument; the non-negativity is essential for the graph-viscosity coefficients to remain non-negative.
invented entities (1)
  • Sparse mortar weights defined by integrals of LGL-subcell characteristic functions no independent evidence
    purpose: Replace dense all-to-all mortar couplings by compact, degree-independent stencils while preserving conservation and the IDP algebraic structure.
    Introduced in Section 2.2; no independent experimental evidence outside the numerical tests of this paper.

pith-pipeline@v1.1.0-grok45 · 33428 in / 2280 out tokens · 24179 ms · 2026-07-14T16:07:10.864516+00:00 · methodology

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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.

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 ↗

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