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arxiv: 2604.28152 · v1 · submitted 2026-04-30 · 🧮 math.NA · cs.NA· physics.flu-dyn

Recognition: unknown

Beyond first-order accuracy in continuous-forcing immersed boundary methods, and their well-conditioned projection-based solution

Diederik Beckers , H. Jane Bae , Andres Goza
Authors on Pith no claims yet

Pith reviewed 2026-05-07 07:21 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.flu-dyn
keywords immersed boundary methodscontinuous forcinghigher-order accuracyprojection methodsNavier-StokesPoisson equationsmoothed indicatornumerical methods
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The pith

Continuous-forcing immersed boundary methods can reach second-order accuracy by restoring neglected terms through composite interior and exterior fields.

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

Standard continuous-forcing immersed boundary methods are typically limited to first-order accuracy. This paper reframes the problem by using a smoothed indicator function to express the solution as a combination of distinct interior and exterior fields. Through this view, it becomes clear that prior methods left out certain terms in the governing equations and constraints. Restoring those terms produces a method that achieves second-order accuracy in Poisson problems and nearly second-order in Navier-Stokes flows. The approach also fits into a projection method that improves conditioning and lessens sensitivity to geometry discretization.

Core claim

When cast through the composite-solution lens enabled by a smoothed indicator function, prior continuous-forcing IB methods neglect terms in the governing and constraint equations that restrict the solution to first-order accuracy. Incorporating these terms systematically improves accuracy without heuristic corrections. The resulting formulation is embedded in a projection-based solution process that mitigates spurious surface stresses and reduces sensitivity to geometric resolution.

What carries the argument

The smoothed-indicator composite decomposition of the solution into interior and exterior fields, which identifies the previously neglected terms in the equations for restoration.

If this is right

  • Canonical Poisson problems exhibit second-order convergence.
  • Incompressible Navier-Stokes simulations achieve slightly sub-second-order performance.
  • Spurious surface stresses from ill-conditioned systems are reduced.
  • Geometric resolution sensitivity decreases.
  • Further extensions may enable second-order or higher accuracy.

Where Pith is reading between the lines

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

  • The method could extend to other problems involving smoothed interfaces, such as multiphase flows.
  • Verification in fully three-dimensional turbulent flows would provide additional evidence of practical accuracy gains.
  • Combining the formulation with higher-order discretizations might achieve even better convergence rates.
  • This perspective implies that accuracy in IB methods is more a matter of complete equation formulation than inherent limitations of smoothing.

Load-bearing premise

The smoothed indicator function allows the solution to be cleanly decomposed into interior and exterior fields so that neglected terms can be restored without creating new errors that cap the order of accuracy.

What would settle it

A mesh refinement study on a Poisson problem where the observed convergence rate remains first-order rather than improving to second-order would falsify the accuracy improvement claim.

Figures

Figures reproduced from arXiv: 2604.28152 by Andres Goza, Diederik Beckers, H. Jane Bae.

Figure 1
Figure 1. Figure 1: Two-dimensional example of a surface Γ immersed in a rectangular domain Ω. Γ is discretized with a set of IB points (red squares) and Ω is discretized with a Cartesian grid (gray dots). The position vector of the grid point indicated by the blue star is x(i, j) . to Γ, enabling the use of solvers optimized for Cartesian grids on simple geometries and avoiding expensive mesh (re-)generation, which is partic… view at source ↗
Figure 2
Figure 2. Figure 2: Three-dimensional grid cell. The labeled locations in the right panel each share the same index tuple ( view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of three different IB solutions on 16 grid cells for the 1D Dirichlet Poisson example. (Left) Solution of the prototypical IB equation (25) with the analytical forcing strength (green, stars) and the forcing strength obtained using Eq. (38) (red, diamonds). (Right) Solution (blue, triangles) of the proposed IB equation (24) with the forcing strength obtained using Eq. (31). The exact solution is… view at source ↗
Figure 4
Figure 4. Figure 4: Error of the numerical solution computed over the entire domain (solid lines) and over the domain excluding cells within the support of the view at source ↗
Figure 5
Figure 5. Figure 5: (Left) Analytical solution to the 2D Poisson equation on an unbounded domain subject to a sinusoidal Dirichlet interface condition on a view at source ↗
Figure 6
Figure 6. Figure 6: Infinity norm (left) and 2-norm (center) of the numerical solution error computed over all cell centers (solid lines) and over all cell centers view at source ↗
Figure 7
Figure 7. Figure 7: (Left) Condition number of the Schur complement versus the surface-to-grid spacing ratio for the proposed method ( view at source ↗
Figure 8
Figure 8. Figure 8: The inner cylinder Γ1 rotates with a constant angular velocity ω, while the outer cylinder Γ2 remains stationary. In this study, we set the outer radius to be twice the inner radius, corresponding to a radius ratio κ B R1/R2 = 0.5. The analytical solution to this problem, non-dimensionalized using the length scale R1, velocity scale ωR1, and radius ratio κ is vθ(r) =    r, r ≤ 1, κ 2 1 − κ 2 view at source ↗
Figure 8
Figure 8. Figure 8: (Left) Diagram of the circular Couette flow problem. (Center and right) Comparison of the vertical velocity solution on the line along view at source ↗
Figure 9
Figure 9. Figure 9: (Infinity norm (left) and 2-norm (right) of the velocity solution error with respect to their analytical values for the circular Couette view at source ↗
Figure 10
Figure 10. Figure 10: (Left) Condition number of S and S˜ versus the surface-to-grid spacing ratio for the proposed method (blue, dashed) and the prototypical continuous-forcing IB method (red, dotted) using two different grid cell sizes ∆x: 0.167 (circles) and 0.0833 (squares). (Center and right) x￾component of the forcing solution on the interior cylinder obtained with the prototypical formulation (red, dotted) and the propo… view at source ↗
read the original abstract

We introduce a refined immersed boundary (IB) methodology that is better-than-first-order accurate in practice, while preserving key properties of "continuous-forcing" IB approaches that retain a singular source term in the governing equations. Our method leverages a smoothed indicator (Heaviside) function, following ideas from multiphase flow and immersed layers formulations, to recast the IB solution as a composite of distinct interior and exterior fields. We demonstrate that, when cast through this composite-solution lens, prior continuous-forcing IB methods can be seen as neglecting terms in the governing and constraint equations that restrict the solution to first-order accuracy. We incorporate these terms to systematically improve accuracy without the need for heuristic corrections. In canonical Poisson problems, we empirically demonstrate second-order convergence, and in incompressible Navier-Stokes simulations the method achieves slightly sub-second-order performance. While our present study focuses on these cases, the framework suggests a path towards second-order accuracy or higher, with further extensions. This perspective reframes accuracy limitations typically attributed to IB schemes. Although continuous-forcing IB methods are often reported to be only first-order accurate, we show that neither smoothing nor interface interpolation inherently restricts attainable order. Moreover, we naturally incorporate this higher-order formulation into a projection-based solution process. The resulting algorithm simultaneously mitigates the spurious surface stresses produced by ill-conditioned linear systems and reduces sensitivity to geometric resolution, addressing both conditioning and accuracy concerns within a unified approach.

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 / 3 minor

Summary. The manuscript introduces a refined continuous-forcing immersed boundary (IB) method that recasts the solution as a composite of distinct interior and exterior fields separated by a smoothed Heaviside indicator function. It argues that standard continuous-forcing IB formulations neglect specific terms in the governing and constraint equations, limiting accuracy to first order; restoring these terms yields second-order convergence on canonical Poisson problems and slightly sub-second-order performance on incompressible Navier-Stokes equations, without heuristic corrections. The approach is embedded in a projection-based solver that simultaneously improves linear-system conditioning and reduces sensitivity to geometric resolution.

Significance. If the central claims hold under the reported discretizations, the work is significant for numerical analysis and computational fluid dynamics. It provides a systematic, non-heuristic route to higher-order accuracy in a class of IB methods that are widely used for their simplicity and retention of singular sources. The empirical second-order rates on Poisson problems and the unified treatment of accuracy plus conditioning constitute concrete strengths. The paper supplies reproducible numerical experiments on standard test problems, which strengthens its contribution.

major comments (1)
  1. [composite-solution formulation and modified governing equations] The central claim rests on the smoothed Heaviside permitting a clean decomposition into interior/exterior fields such that restored terms do not inject new truncation errors that cap global order (the reader's weakest assumption). While empirical rates are reported, the manuscript should supply a local truncation-error analysis at the interface (likely in the composite-field derivation section) showing that the smoothing width does not limit the scheme to first order; without this, the path to second-order or higher remains empirical rather than guaranteed.
minor comments (3)
  1. [Navier-Stokes numerical results] The abstract states 'slightly sub-second-order performance' for Navier-Stokes but does not quote the observed rates or grid sizes; a table or explicit statement of measured orders (e.g., 1.6–1.9) in the results section would allow readers to assess how close the method comes to the claimed path toward second-order accuracy.
  2. Notation for the composite fields (interior/exterior velocities, pressures, and the indicator function) should be introduced with a single, self-contained table or diagram at first use to avoid ambiguity when the restored terms are written out.
  3. [projection-based solution algorithm] The projection step is said to mitigate spurious surface stresses from ill-conditioned systems; quantitative metrics (condition-number histories or L2 norms of surface forces versus grid size) would make this benefit concrete and comparable to prior IB literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful review and the recommendation of minor revision. The single major comment is addressed below; we agree that adding a local truncation-error analysis will strengthen the manuscript and will incorporate it in the revised version.

read point-by-point responses

  1. Referee: [composite-solution formulation and modified governing equations] The central claim rests on the smoothed Heaviside permitting a clean decomposition into interior/exterior fields such that restored terms do not inject new truncation errors that cap global order (the reader's weakest assumption). While empirical rates are reported, the manuscript should supply a local truncation-error analysis at the interface (likely in the composite-field derivation section) showing that the smoothing width does not limit the scheme to first order; without this, the path to second-order or higher remains empirical rather than guaranteed.

    Authors: We agree that a local truncation-error analysis at the interface would provide a stronger theoretical guarantee and address the referee's concern directly. In the revised manuscript we will expand the composite-field derivation section to include a detailed local truncation-error analysis. Using Taylor expansions of the interior and exterior fields about the interface, we will show that the restored terms (arising from the product-rule derivatives of the smoothed indicator) are discretized consistently with the underlying second-order finite-difference operators. When the smoothing width is taken proportional to the grid spacing h, the additional interface errors remain O(h^2) locally and do not degrade the global second-order convergence, consistent with the empirical rates already reported for the Poisson problems. This analysis will be presented alongside the existing numerical experiments to make the path to second-order accuracy rigorous rather than purely empirical. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from governing equations

full rationale

The paper reframes continuous-forcing IB methods by decomposing the solution into interior/exterior fields via a smoothed Heaviside indicator drawn from multiphase literature, then identifies and restores terms omitted in the standard governing and constraint equations. This yields a composite formulation whose accuracy improvement is demonstrated empirically on Poisson and Navier-Stokes problems rather than being forced by construction. No parameter is fitted to a target quantity and then relabeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The central claim rests on re-examination of the unmodified Navier-Stokes and projection equations under the new decomposition, with external grounding in multiphase formulations and direct numerical verification of convergence rates.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on standard continuum mechanics assumptions for the governing equations and on the validity of the smoothed indicator function drawn from multiphase literature; no new free parameters, invented entities, or ad-hoc constants are introduced in the abstract.

axioms (2)
  • standard math The incompressible Navier-Stokes and Poisson equations hold separately in the interior and exterior regions and can be combined via the smoothed indicator without altering their differential form.
    Invoked when recasting the IB problem as a composite of distinct fields.
  • domain assumption The smoothed Heaviside function provides a sufficiently regular transition that permits restoration of interface terms while preserving the singular source character of continuous-forcing IB.
    Following ideas from multiphase flow and immersed layers formulations.

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Reference graph

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