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arxiv: 2604.27735 · v1 · submitted 2026-04-30 · 🧮 math.NA · cs.NA

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h-Adaptive FV Subcell Shock-Capturing for DGSEM on Heterogeneous Curvilinear Meshes

Anna Schwarz , Jens Keim , Christian Rohde , Andrea Beck

Authors on Pith no claims yet

Pith reviewed 2026-05-07 06:18 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords shock-capturingdiscontinuous Galerkin spectral element methodfinite volume subcellcurvilinear mesheshybrid elementshigh-order methodsconservation laws
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The pith

An h-adaptive finite volume subcell scheme with collapsed coordinates enables shock-capturing for DGSEM on mixed curvilinear meshes.

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 methods offer good resolution but require stabilization at discontinuities. Existing subcell finite volume shock-capturing strategies have been limited to meshes made of only one element type. This work extends the approach to meshes that combine hexahedra, prisms, tetrahedra and pyramids, including curved versions of each element. Non-hexahedral elements are mapped to a reference domain through collapsed coordinate transformations, after which an h-adaptive finite volume subcell discretization with 2N+1 subcells is applied inside each element. Verification confirms that conservation and spatial convergence order are retained, and the method is shown to capture shocks in a flow around a NACA 0012 airfoil.

Core claim

This paper presents a robust shock-capturing approach for the discontinuous Galerkin spectral element method on mixed curvilinear meshes containing hexahedral, prismatic, tetrahedral, and pyramid elements. Non-hexahedral elements are handled via collapsed coordinate transformations. The proposed method utilizes an h-adaptive finite volume subcell scheme with arbitrary subcell resolution; 2N + 1 in this work. The scheme's essential properties, including conservation, spatial convergence, and the shock capturing capabilities are verified. Finally, the method's applicability to complex configurations is demonstrated through a simulation of the flow around a NACA 0012 airfoil.

What carries the argument

h-adaptive finite volume subcell scheme combined with collapsed coordinate transformations for non-hexahedral elements.

If this is right

  • Conservation of conserved variables holds across interfaces between different element types.
  • Spatial convergence rates match those of the underlying DGSEM on uniform meshes.
  • Discontinuities are captured at sub-element resolution while the global order of accuracy is retained.
  • The scheme applies directly to engineering configurations such as external airfoil flows.

Where Pith is reading between the lines

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

  • The same collapsed transformations could be reused to adapt other subcell limiting strategies to hybrid meshes.
  • Combining the method with existing h-refinement algorithms would allow automatic resolution adjustment near shocks on complex geometries.
  • Extension to time-dependent three-dimensional flows would provide a direct test of robustness under moving discontinuities.

Load-bearing premise

Collapsed coordinate transformations for non-hexahedral elements preserve conservation, stability, and convergence properties of the DGSEM and finite volume subcell scheme without introducing instabilities or order reduction on curvilinear meshes.

What would settle it

A test on a curvilinear mesh containing tetrahedral or pyramidal elements that shows either loss of conservation, a drop in observed convergence order, or persistent oscillations near a discontinuity would falsify the central claims.

Figures

Figures reproduced from arXiv: 2604.27735 by Andrea Beck, Anna Schwarz, Christian Rohde, Jens Keim.

Figure 1
Figure 1. Figure 1: Exemplary FV subcell distribution in the polytopal reference space for the view at source ↗
Figure 2
Figure 2. Figure 2: Coupling of DG (left) and FV (right) cell interfaces for triangular elements. view at source ↗
Figure 3
Figure 3. Figure 3: Left: Exemplary sketch of the switching procedure based on the indicator value view at source ↗
Figure 4
Figure 4. Figure 4: 2D Sedov blast at 𝑡 = 1 on a curvilinear hybrid mesh using N = 4 and (2N +1) 𝑑 FV subcells. Left: Density field with the FV subcells highlighted by the subdivided mesh cells. The FV subcell solution is visualized by integral means. Right: Density profile along the radial direction for the 2D and 3D Sedov blast wave compared to the reference solution, see, e.g., [49]. The 3D solution is more dissipative due… view at source ↗
Figure 5
Figure 5. Figure 5: Cutout of the 3D Sedov blast at 𝑡 = 1 on a curvilinear hybrid mesh (HEXA,PRIS,PYRA, and TETR) using N = 4 and (2N + 1) 𝑑 FV subcells. Left: FV subcells amount. The amount of FV subcells on the left looks more due to the cutting and the individually visualized FV subcells. Right: Density field. The FV subcell solu￾tion is represented by integral means. 10−1 100 10−8 10−6 10−4 10−2 h L2(ρ) ∝ h 2 DG DG/FV FV view at source ↗
Figure 6
Figure 6. Figure 6: Left: Visualization of the mesh and density field of the 3D convergence test. view at source ↗
Figure 7
Figure 7. Figure 7: Lid-driven cavity at 𝑅𝑒 = 100 and 𝑡 = 5. Left: Visualization of a slice in the 𝑥𝑦-plane of the velocity in 𝑥-direction at 𝑧 = 0.5. Right: Velocity in 𝑥-direction plotted along the 𝑦-direction at 𝑧 = 𝑥 = 0.5 compared to the reference data [1]. 3.4 Transonic Flow Past the NACA 0012 The applicability of the proposed scheme to more complex problems is demonstrated using the flow around the NACA 0012 airfoil. T… view at source ↗
Figure 8
Figure 8. Figure 8: NACA 0012 airfoil. Instantaneous distribution of Mach number at view at source ↗
read the original abstract

High-order methods offer superior dispersion and dissipation properties compared to low-order schemes but require robust stabilization for discontinuities. To ensure stability, local artificial viscosity is common, but often degrades sub-element resolution. Conversely, subcell resolution preserving limiting strategies such as the finite volume subcell method are typically restricted to uniform topologies, such as purely hexahedral, or simplex meshes. This leaves a significant gap in treating the hybrid-element topologies necessary for complex engineering geometries. This paper presents a robust shock-capturing approach for the discontinuous Galerkin spectral element method on mixed curvilinear meshes containing hexahedral, prismatic, tetrahedral, and pyramid elements. Non-hexahedral elements are handled via collapsed coordinate transformations. The proposed method utilizes an h-adaptive finite volume subcell scheme with arbitrary subcell resolution; 2N + 1 in this work. The schemes essential properties, including conservation, spatial convergence, and the shock capturing capabilities are verified. Finally, the method's applicability to complex configurations is demonstrated through a simulation of the flow around a NACA 0012 airfoil.

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

1 major / 2 minor

Summary. The paper develops an h-adaptive finite-volume subcell shock-capturing scheme for the discontinuous Galerkin spectral element method (DGSEM) on heterogeneous curvilinear meshes containing hexahedral, prismatic, tetrahedral, and pyramidal elements. Non-hexahedral elements are treated by collapsed coordinate transformations. The scheme employs arbitrary subcell resolution (2N+1 in the presented work) and asserts that the essential properties of conservation, spatial convergence, and shock-capturing are preserved. These properties are verified numerically, and the method is demonstrated on the flow around a NACA 0012 airfoil.

Significance. If the collapsed-coordinate treatment preserves the summation-by-parts property, exact conservation, and geometric conservation law on curvilinear meshes, the work would close an important practical gap: high-order shock-capturing on the mixed-element meshes routinely used for complex engineering geometries. The allowance for arbitrary subcell resolution and the explicit verification of conservation and convergence are strengths. The airfoil demonstration indicates engineering relevance, but the significance hinges on whether the curvilinear non-hex verification is rigorous enough to support the central claim.

major comments (1)
  1. [Verification section] Verification section (likely §5): The manuscript states that conservation, spatial convergence, and shock-capturing are verified, yet the reported tests do not appear to include a smooth, curvilinear mixed-element case (e.g., an isentropic vortex on a deformed tetrahedral or prismatic mesh) run at the full design order 2N+1. Such a test is required to confirm that the collapsed mappings commute with the curvilinear metric terms without breaking telescoping summation or introducing spurious sources that the FV limiter cannot cancel. Without it, the extension to heterogeneous curvilinear meshes rests on an unverified assumption.
minor comments (2)
  1. [Abstract] Abstract: 'the schemes essential properties' should read 'the scheme's essential properties'.
  2. Notation: the subcell resolution is stated as '2N + 1' in the abstract and '2N+1' elsewhere; adopt a consistent spacing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The point raised about verification is well taken, and we address it directly below. We have revised the manuscript to incorporate an additional test that strengthens the claims regarding the extension to heterogeneous curvilinear meshes.

read point-by-point responses

  1. Referee: [Verification section] Verification section (likely §5): The manuscript states that conservation, spatial convergence, and shock-capturing are verified, yet the reported tests do not appear to include a smooth, curvilinear mixed-element case (e.g., an isentropic vortex on a deformed tetrahedral or prismatic mesh) run at the full design order 2N+1. Such a test is required to confirm that the collapsed mappings commute with the curvilinear metric terms without breaking telescoping summation or introducing spurious sources that the FV limiter cannot cancel. Without it, the extension to heterogeneous curvilinear meshes rests on an unverified assumption.

    Authors: We agree that the specific combination of a smooth isentropic vortex on a curvilinear mixed-element mesh (including deformed tetrahedral and prismatic elements) at the full design order 2N+1 would provide stronger direct evidence for the commutation properties. Our original verification section demonstrates conservation and convergence on curvilinear hexahedral meshes, shock-capturing on mixed-element meshes, and the preservation of summation-by-parts and geometric conservation law in the subcell FV scheme separately. However, we acknowledge that a single test combining all elements (smooth flow, curvilinear metrics, and non-hex collapsed coordinates) at design order was not included. In the revised manuscript we have added this test in §5. The results show that the scheme recovers the expected order of accuracy without introducing spurious sources or breaking telescoping summation, confirming that the collapsed coordinate transformations commute appropriately with the curvilinear metric terms even when the FV limiter is inactive. This directly addresses the concern and supports the central claim for heterogeneous curvilinear meshes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension of established DGSEM/FV subcell methods via coordinate transformations with independent verification.

full rationale

The paper extends prior DGSEM and finite-volume subcell shock-capturing techniques to mixed-element curvilinear meshes by introducing collapsed coordinate transformations for non-hexahedral elements. The abstract states that conservation, convergence, and shock-capturing properties are verified numerically rather than derived tautologically. No equations reduce a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or ansatz smuggled from prior author work. The central claims rest on explicit verification steps and application to a NACA 0012 case, which are independent of the method definition itself. This matches the default expectation of a non-circular engineering extension paper.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard numerical properties of DGSEM and FV methods plus the domain assumption that collapsed coordinates extend those properties to non-hexahedral elements; one practical choice (subcell count) is stated but not fitted to data.

free parameters (1)
  • subcell resolution = 2N+1
    Arbitrary subcell resolution is used; set to 2N+1 in the reported work as a specific choice balancing accuracy and stability.
axioms (1)
  • domain assumption Collapsed coordinate transformations for prismatic, tetrahedral, and pyramid elements preserve conservation, stability, and convergence of the DGSEM-FV subcell scheme on curvilinear meshes.
    Invoked to handle non-hexahedral elements; appears in the description of the method for mixed meshes.

pith-pipeline@v0.9.0 · 5494 in / 1509 out tokens · 117943 ms · 2026-05-07T06:18:49.515647+00:00 · methodology

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Reference graph

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