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arxiv: 2506.02320 · v2 · submitted 2025-06-02 · 🧮 math.NA · cs.NA· physics.comp-ph· physics.flu-dyn

Greedy recursion parameter selection for one-way spatial integration of hyperbolic equations

Michael K. Sleeman , Tim Colonius This is my paper

Pith reviewed 2026-05-19 11:30 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-phphysics.flu-dyn
keywords one-way wave equationsgreedy algorithmparameter selectionhyperbolic systemsNavier-Stokesboundary-layer flowsnumerical integration
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The pith

A greedy algorithm automatically selects recursion parameters for one-way hyperbolic wave equations, yielding faster convergence at lower cost.

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

The paper proposes a greedy algorithm to choose the recursion parameters required when building one-way approximations from full hyperbolic systems. This replaces earlier heuristic estimates that needed manual tuning and converged slowly. The algorithm is shown to produce more accurate one-way operators with fewer parameters for both linear and nonlinear disturbance evolution in boundary-layer flows. The authors also compare the projection and recursive versions of the one-way Navier-Stokes method and report that the projection version delivers better stability and convergence rates. The approach is framed for general linear first-order hyperbolic systems.

Core claim

The greedy algorithm iteratively selects each new recursion parameter to minimize the error of the resulting one-way operator relative to a reference solution, which produces faster error reduction than heuristic choices and therefore lowers the total number of operations needed to reach a target accuracy in linear and nonlinear boundary-layer simulations.

What carries the argument

The greedy recursion parameter selection procedure, which at each iteration adds the parameter value that most reduces the discrepancy between the one-way operator and a reference two-way solution.

If this is right

  • The method applies directly to any system of linear first-order hyperbolic equations.
  • The projection-based one-way method converges faster and more stably than the recursive variant.
  • Fewer recursion parameters suffice for a given accuracy, producing a net reduction in computational cost.
  • Manual trial-and-error tuning of parameters is replaced by an automatic procedure.

Where Pith is reading between the lines

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

  • The same greedy selection logic could be tested on other numerical filters used in wave propagation problems.
  • When the unidirectional assumption is only approximate, the method could be paired with an error monitor that switches back to the full system if needed.
  • Extending the tests to three-dimensional or more complex geometries would clarify how broadly the cost savings hold.

Load-bearing premise

Wave propagation must remain sufficiently unidirectional so that the recursive filter can remove left-going waves without introducing large errors into the retained right-going solution.

What would settle it

Apply the greedy method to a hyperbolic system with strong bidirectional propagation and check whether the one-way solution deviates substantially from the full two-way reference; large deviations would show the accuracy claim fails.

Figures

Figures reproduced from arXiv: 2506.02320 by Michael K. Sleeman, Tim Colonius.

Figure 1
Figure 1. Figure 1: Convergence of the objective function for greedy and heuristic parameter selec [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence of the objective function, split into upstream- and downstream [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Recursion parameters plotted against spectrum for heuristic and greedy recursion [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of OWNS spectrum with heuristic parameter selection for [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Error convergence as a function of Nβ for OWNS-P and OWNS-R for the 2D low-speed boundary-layer flow. in the polynomial approximation as max k=1,...,100 |2 QNβ j=1(ck − β j ∗ ) − QNβ j=1(ck − β j −) − QNβ j=1(ck − β j +)| | QNβ j=1(ck − β j −) − QNβ j=1(ck − β j +)| . (33) [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: For OWNS-R, the error in computing β j ∗ increases with increasing Nβ. 5.4.1. 3D low-speed boundary-layer flow We investigate the oblique-wave case studied by Joslin et al. [27] for a low￾speed isothermal flat plate boundary-layer flow with disturbance frequency and wavenumber are F = 86×10−6 and b = 0.222×10−3 , respectively, for the wall-normal domain y ∈ [0, 60] with Ny = 100 at Rex = 2.74 × 105 [PITH… view at source ↗
Figure 7
Figure 7. Figure 7: Error convergence as a function of Nβ for OWNS-P and OWNS-R for the low￾speed oblique wave case. does not. For OWNS-R only, we have excluded downstream-going eigenval￾ues with αi > 100 from the greedy selection procedure, which improves the accuracy of the approximation, since these modes decay rapidly. We also note that heuristic selection is better at evolving Mack’s second mode (MM), which is the domina… view at source ↗
Figure 8
Figure 8. Figure 8: Convergence of the error for greedy and heuristic parameter selection for 2D [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: NOWNS for oblique-wave breakdown, comparing greedy and heuristic recursion [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Temperature contours for MFD and Mack’s second mode with heuristic pa [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Temperature contours for first harmonic of Mack’s second mode with heuristic [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Temperature disturbance profile for first harmonic of Mack’s second mode at [PITH_FULL_IMAGE:figures/full_fig_p033_12.png] view at source ↗
read the original abstract

Solutions to hyperbolic systems comprise waves propagating at finite speeds. When wave propagation is predominantly unidirectional, one-way wave equations can be used to evolve only the right-going solution by removing support for left-going waves. The One-Way Navier-Stokes (OWNS) approach, which was originally developed for systems of first-order hyperbolic equations, constructs one-way approximations to the linearized Navier-Stokes equations using a recursive filter to remove left-going waves. The computational cost scales with the number of recursion parameters, which must be carefully chosen to ensure accuracy and stability of the resulting one-way equation. Previous work has chosen parameters based on heuristic estimates of key eigenvalues, which requires trial-and-error tuning while also yielding slow error convergence. We propose a greedy algorithm for automatic parameter selection, which we show yields faster convergence and a net decrease in computational cost for linear and nonlinear disturbance evolution in boundary-layer flows. We review the OWNS projection (OWNS-P) and recursive (OWNS-R) methods, comparing their convergence properties, and show through our numerical analysis and experiments that OWNS-P yields superior convergence and stability properties. Although we demonstrate the method for Navier-Stokes equations, we perform our analyses on systems of linear first-order hyperbolic equations and emphasize that the greedy algorithm is applicable to such systems.

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 / 3 minor

Summary. The paper introduces a greedy algorithm for automatic selection of recursion parameters in one-way spatial integration methods for hyperbolic systems, focusing on the OWNS-P (projection) and OWNS-R (recursive) approaches originally developed for the linearized Navier-Stokes equations. It reviews the construction of one-way approximations that remove left-going waves, performs analysis on linear first-order hyperbolic systems, and demonstrates through numerical experiments that the greedy selection yields faster error convergence and lower net computational cost than heuristic eigenvalue-based tuning for both linear and nonlinear boundary-layer disturbance evolution. The manuscript also compares OWNS-P and OWNS-R, concluding that OWNS-P exhibits superior convergence and stability.

Significance. If the empirical results and comparisons hold, the work provides a reproducible, algorithmic procedure that reduces manual tuning in one-way wave models, which are widely used in aeroacoustics and hydrodynamic stability. The explicit demonstration of cost savings on both linear hyperbolic systems and nonlinear Navier-Stokes cases, together with the side-by-side evaluation of OWNS-P versus OWNS-R, strengthens the practical utility of one-way approximations for unidirectional wave propagation problems.

major comments (2)
  1. [§4.3] §4.3 (greedy algorithm description): the termination criterion and the precise definition of the local error indicator used to rank candidate recursion parameters are not stated with sufficient formality; without an explicit pseudocode or mathematical statement of the selection loop, it is difficult to assess reproducibility or to verify that the reported convergence improvement is independent of implementation details.
  2. [§6.2, Table 2] §6.2 and Table 2 (nonlinear boundary-layer experiments): the net computational-cost reduction is asserted after including the greedy search overhead, yet the table reports only the integration cost per time step; a breakdown showing the one-time search cost amortized over the number of runs performed would be required to substantiate the 'net decrease' claim for typical usage patterns.
minor comments (3)
  1. [§2.1] §2.1: the notation distinguishing the recursion parameters from the eigenvalues of the spatial operator is introduced without a consolidated symbol table; adding such a table would improve readability.
  2. [Figure 5] Figure 5 caption: the line styles for OWNS-P and OWNS-R are not described, making it hard to match curves to the two methods without consulting the main text.
  3. [References] Reference list: the citation to the original OWNS papers is present, but more recent works on one-way approximations for nonlinear systems (post-2020) are absent; a brief update would place the contribution in current context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have revised the manuscript to incorporate the suggested improvements for clarity and completeness.

read point-by-point responses

  1. Referee: [§4.3] §4.3 (greedy algorithm description): the termination criterion and the precise definition of the local error indicator used to rank candidate recursion parameters are not stated with sufficient formality; without an explicit pseudocode or mathematical statement of the selection loop, it is difficult to assess reproducibility or to verify that the reported convergence improvement is independent of implementation details.

    Authors: We agree that the description in §4.3 would benefit from greater formality to support reproducibility. In the revised manuscript we will add an explicit mathematical definition of the local error indicator, the procedure for ranking candidate recursion parameters, and the termination criterion for the selection loop. We will also include pseudocode for the full greedy algorithm. revision: yes

  2. Referee: [§6.2, Table 2] §6.2 and Table 2 (nonlinear boundary-layer experiments): the net computational-cost reduction is asserted after including the greedy search overhead, yet the table reports only the integration cost per time step; a breakdown showing the one-time search cost amortized over the number of runs performed would be required to substantiate the 'net decrease' claim for typical usage patterns.

    Authors: We acknowledge that Table 2 reports only the per-step integration costs. To substantiate the net-cost claim, the revised manuscript will include an explicit breakdown of the one-time greedy-search overhead together with amortization over representative numbers of runs and simulation lengths typical of boundary-layer studies. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a greedy algorithm for selecting recursion parameters in one-way spatial integration of hyperbolic systems. This is an algorithmic procedure driven by an external error indicator rather than any fitted quantity that is then re-used as a prediction. The reported improvements in convergence and cost are measured directly from numerical experiments on both linear first-order hyperbolic systems and nonlinear boundary-layer flows. The unidirectionality premise is stated as the standard modeling assumption for one-way equations and is not derived from the algorithm itself. No self-definitional steps, fitted-input predictions, or load-bearing self-citation chains appear in the derivation; the central claims remain independent of the parameter-selection procedure.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that propagation is predominantly unidirectional and on the existence of a stable recursive filter whose parameters can be chosen to control truncation error without introducing new instabilities.

free parameters (1)
  • number of recursion parameters
    The count is increased until a target accuracy is reached; the paper treats this count as the main cost driver but does not report a fixed universal value.
axioms (1)
  • domain assumption Wave propagation is predominantly unidirectional
    Invoked when constructing the one-way equation by removing support for left-going waves.

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Reference graph

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