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arxiv: 2604.11959 · v1 · submitted 2026-04-13 · 💻 cs.CE

An Embedded Boundary Scheme for Three-Dimensional Flow Over Terrain on a Staggered Mesh

Soonpil Kang , Ann S. Almgren , Mahesh Natarajan , Aaron M. Lattanzi , Jeffrey D. Mirocha , Katie Lundquist , Jordan Musser , Weiqun Zhang This is my paper

Pith reviewed 2026-05-10 16:12 UTC · model grok-4.3

classification 💻 cs.CE
keywords embedded boundarystaggered meshterrain flowsmall cell instabilityweighted state redistributionadaptive mesh refinementfluid dynamicsERF model
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The pith

An embedded boundary scheme enables three-dimensional flow over terrain on staggered meshes by storing separate geometric data for cell centers and face velocities.

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

The paper establishes that an embedded boundary method can be extended to staggered meshes for simulating fluid flow over complex terrain, where velocities live on cell faces and thermodynamic variables on centers. It does this by building and storing multiple copies of the geometry information—one set for centers and one for each velocity component—while extending the weighted state redistribution scheme to suppress small-cell instabilities near the boundary. The approach is coded into the ERF model, which already supports performance portability and adaptive mesh refinement, and is checked by running the same test problems with both the new EB method and conventional terrain-following coordinates. A reader would care because the method removes the need for grids that conform to the terrain, thereby avoiding the coordinate distortions that arise in steep regions while retaining the staggered-mesh discretization that many atmospheric models already use.

Core claim

The central claim is that an embedded boundary approach can be adapted for staggered meshes in three-dimensional flow over terrain by constructing separate instances of the geometric information for cell-centered quantities and for each face-centered velocity component, together with an extension of the weighted state redistribution scheme that stabilizes the small cells created by the boundary cut, and that this combination produces results that match those obtained with terrain-following coordinates when both are run inside the ERF model.

What carries the argument

Multiple stored instances of embedded-boundary geometric data—one for cell centers and one for each velocity component on faces—plus the weighted state redistribution scheme extended to staggered meshes to remove small-cell instability.

If this is right

  • Flow over arbitrary terrain can be simulated on Cartesian staggered meshes without the grid distortions introduced by terrain-following coordinates.
  • The ERF model gains an embedded-boundary option that preserves its existing performance portability and adaptive mesh refinement capabilities.
  • Small cells cut by the embedded boundary no longer trigger instability once the weighted state redistribution scheme is applied on the staggered layout.
  • Direct numerical comparisons become possible between embedded-boundary and terrain-following solutions on identical problems, providing a practical validation route.

Where Pith is reading between the lines

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

  • The same geometric bookkeeping could be reused for other staggered discretizations in computational fluid dynamics beyond atmospheric modeling.
  • Because the method keeps the underlying Cartesian mesh, it may simplify coupling to additional physics modules that already assume regular grids.
  • Extending the validation to cases with steeper slopes or moving boundaries would test whether the small-cell treatment remains robust when the cut fraction changes rapidly.

Load-bearing premise

That agreement between embedded-boundary and terrain-following simulations on the same test cases is enough to confirm the accuracy and stability of the new staggered-mesh scheme.

What would settle it

A clear mismatch in velocity or pressure fields, or the appearance of instability, in the embedded-boundary runs but not in the terrain-following runs for any of the standard validation cases would show the scheme does not yet deliver equivalent results.

Figures

Figures reproduced from arXiv: 2604.11959 by Aaron M. Lattanzi, Ann S. Almgren, Jeffrey D. Mirocha, Jordan Musser, Katie Lundquist, Mahesh Natarajan, Soonpil Kang, Weiqun Zhang.

Figure 1
Figure 1. Figure 1: Layout of the staggered grid with cell-centered and face-centered variables and their control volumes. The [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Construction of cut-cell geometric data on a staggered grid. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bilinear interpolation of the advective flux on the cut face. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Neighborhood stencils for calculating the gradient of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Fitted, embedded boundary (EB), and EB with AMR grids for Witch of Agnesi test. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Horizontal velocity in time at three locations ( [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Instantaneous horizontal and vertical velocity fields in the Witch of Agnesi test, computed using the [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Horizontal velocity profiles along vertical lines at [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Vertical velocity profiles along vertical lines at [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of x-momentum with and without applying the weighted state redistribution scheme. The terrain profile is indicated by the white line. 6.2. Flow over a hemisphere This test case represents three-dimensional potential flow over a hemisphere for which an exact solution exists. The flow is assumed to be inviscid and irrotational. We model this problem using 15 [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 11
Figure 11. Figure 11: Computational grid around an embedded hemisphere. [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Velocity field around an embedded hemisphere. [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Exact and numerical solutions for x-velocity along the z-axis at x = 5 and x = 6. (a) x-velocity (b) y-velocity [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Exact and numerical solutions for the streamwise [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: To assess the EB scheme, the surface of the cylinder is misaligned with the underlying [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Vortical structures in the wake of square cylinders with different aspect ratios. Vortices are identified [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Instantaneous out-of-plane vorticity: z-vorticity at the mid-height (z = 0.5h) and y-vorticity mid-plane (y = 0) for flow past square cylinders of varying aspect ratios. The vortex shedding behavior is characterized by the Strouhal number. The Strouhal number, St, is defined as f d/u∞, where f is the frequency of vortex shedding. The frequency is determined as the dominant peak in the power spectrum of th… view at source ↗
Figure 18
Figure 18. Figure 18: Time-averaged streamwise velocity along the centerline at the mid-height of the cylinder with different [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Time-averaged streamwise and transverse velocity profiles in the transverse direction at [PITH_FULL_IMAGE:figures/full_fig_p020_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Time-averaged streamwise and vertical velocity profiles in the vertical direction at [PITH_FULL_IMAGE:figures/full_fig_p021_20.png] view at source ↗
read the original abstract

This paper describes an embedded boundary (EB) approach for simulating three-dimensional fluid flow on a staggered mesh where the velocity components are defined on cell faces and the thermodynamic state is defined on cell centers. Most EB approaches assume that all components of the solution, including the velocity, are co-located. To compute solution quantities on faces as well as cell centers, we construct and store multiple instances of the geometric information, one for the quantities stored at cell centers and one for each velocity component. In addition, we extend the weighted state redistribution (WSRD) scheme to staggered meshes to address the small-cell instability issue. This new approach is implemented in the Energy Research and Forecasting (ERF) model that provides performance portability and adaptive mesh refinement. We validate the new EB method by comparing EB simulations to those computed using terrain-following coordinates.

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

2 major / 0 minor

Summary. The paper describes an embedded boundary (EB) approach for three-dimensional fluid flow over terrain on a staggered mesh, with velocity components on cell faces and thermodynamic state on cell centers. Multiple geometric instances are stored for the different variable locations, and the weighted state redistribution (WSRD) scheme is extended to staggered meshes to address small-cell instabilities. The method is implemented in the Energy Research and Forecasting (ERF) model supporting performance portability and adaptive mesh refinement, and is validated by comparing EB results to terrain-following coordinate simulations.

Significance. If the central claims hold, the work would be a useful contribution to computational fluid dynamics for atmospheric and environmental modeling, as it enables EB treatment on staggered grids without the grid distortions and potential instabilities of terrain-following coordinates on steep slopes. The implementation within the ERF model, which provides performance portability and adaptive mesh refinement, strengthens its applicability to scalable, real-world simulations.

major comments (2)
  1. Abstract: The validation claim ('We validate the new EB method by comparing EB simulations to those computed using terrain-following coordinates') supplies no quantitative error metrics, convergence rates, grid resolutions, or description of test cases (e.g., slope angles or flow regimes), so the accuracy and stability of the staggered-mesh EB construction and extended WSRD cannot be assessed from the provided information.
  2. Validation description: Direct agreement between EB and terrain-following results on identical cases does not isolate the truncation error or stability properties of the new staggered EB discretization (multiple geometric instances plus extended WSRD), because terrain-following coordinates themselves introduce errors and possible instabilities precisely on steep slopes where EB is intended to be advantageous; independent tests such as manufactured solutions or grid-convergence studies against an analytic reference are needed to confirm the method.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: Abstract: The validation claim ('We validate the new EB method by comparing EB simulations to those computed using terrain-following coordinates') supplies no quantitative error metrics, convergence rates, grid resolutions, or description of test cases (e.g., slope angles or flow regimes), so the accuracy and stability of the staggered-mesh EB construction and extended WSRD cannot be assessed from the provided information.

    Authors: We agree that the abstract would benefit from greater specificity. In the revised manuscript we will expand the abstract to include the grid resolutions employed, the slope angles and flow regimes of the test cases, and quantitative measures of agreement (e.g., L2 error norms) between the embedded-boundary and terrain-following results. revision: yes

  2. Referee: Validation description: Direct agreement between EB and terrain-following results on identical cases does not isolate the truncation error or stability properties of the new staggered EB discretization (multiple geometric instances plus extended WSRD), because terrain-following coordinates themselves introduce errors and possible instabilities precisely on steep slopes where EB is intended to be advantageous; independent tests such as manufactured solutions or grid-convergence studies against an analytic reference are needed to confirm the method.

    Authors: We acknowledge that comparisons with terrain-following coordinates alone do not fully isolate the truncation error of the new staggered EB discretization and WSRD extension, since terrain-following grids themselves incur geometric errors on steep slopes. Our validation was intended to demonstrate practical consistency with an established method on representative terrain cases. To address the referee’s concern, the revised manuscript will add a grid-convergence study on successively refined meshes, using a high-resolution reference solution to quantify convergence rates and stability for the staggered EB scheme. While analytic manufactured solutions are difficult to construct for arbitrary terrain, the added convergence analysis will provide independent evidence of the method’s accuracy and small-cell stability. revision: yes

Circularity Check

0 steps flagged

No circularity: independent algorithmic construction with external validation

full rationale

The paper constructs a staggered-mesh EB scheme by defining multiple geometric instances per velocity component and extending WSRD for small-cell handling; these steps are presented as direct algorithmic extensions without reducing to self-definition or fitted inputs. Validation consists of direct numerical comparison against terrain-following coordinate results on the same test cases, which is an independent external reference method rather than a self-referential loop. No load-bearing self-citations, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation are invoked in the provided derivation chain. The central claims remain self-contained algorithmic descriptions with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new physical entities or fitted constants; it rests on standard finite-volume discretization assumptions for staggered grids and the geometric cut-cell constructions common to embedded-boundary methods.

axioms (1)
  • standard math Standard finite-volume conservation and interpolation properties hold when geometric information is stored separately for cell centers and each velocity face component.
    Implicit foundation of any staggered-grid embedded-boundary discretization.

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Reference graph

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