Pith. sign in

REVIEW 3 major objections 6 minor 64 references

FNPF-SEM: A parallel spectral element model in Firedrake for fully nonlinear water wave simulations

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A parallel spectral element solver for fully nonlinear potential-flow water waves is verified by convergence studies, scaling tests, and wave-structure benchmarks.

desk verdict Credible verification of a Firedrake-based spectral element FNPF solver, but the scaling claims measure only the Laplace solve and overstate end-to-end parallel performance. read the letter →

arxiv 2506.09435 v1 pith:RH44IXKU submitted 2025-06-11 math.NA cs.NA

Jens Visbech , Anders Melander , Allan Peter Engsig-Karup This is my paper · ORCID
classification math.NAcs.NA MSC 65M6065M7076B1576B0765Y05
keywords spectralelementmethodfullynonlinearpotentialflowwaterwaveswave-structureinteractionhigh-orderfiniteelementsparallelcomputingfreesurfaceFiredrake
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper presents a parallel, high-order spectral element solver for linear and fully nonlinear potential-flow water waves, built on the Firedrake finite element framework with native MPI parallelism. The authors aim to establish that the solver, FNPF-SEM, can serve as a general-purpose wave model for offshore engineering: unstructured meshes represent complex geometry, high-order polynomial bases keep numerical diffusion and dispersion low, and the parallel implementation targets large-scale, long-time simulations. Verification shows algebraic h-convergence at the expected rate and spectral p-convergence, while strong and weak scaling tests show good parallel efficiency, especially on large problems where the Laplace solve dominates. Validation against analytical solutions and experimental data for harmonic generation over a submerged bar, wave run-up on a vertical cylinder, and V-shaped breakwater diffraction supports the paper's conclusion that the framework is accurate and applicable.

What carries the argument

The load-bearing object is the Zakharov-form fully nonlinear potential flow model: the Laplace equation $\nabla^2\phi = 0$ in the fluid with Dirichlet data $\phi_\eta$ on the free surface and no-flow conditions on bottom and walls, coupled to evolution equations for the surface elevation $\eta$ and surface potential $\phi_\eta$. A mapping that extracts free-surface values from the volume mesh and a vertical mesh update tie the surface and volume discretizations together, and the Laplace solve is the dominant cost that carries the scalability analysis. Stabilization comes from vertically aligned structured elements, over-integration of the nonlinear free-surface terms, and a modal filter applied to the highest modes of free-surface quantities.

What would settle it

Run a full time-step strong-scaling experiment, including mesh update and free-surface evaluation, on a small free-surface-dominated mesh and compare total per-stage runtime with the Laplace-only scaling curve; if total efficiency drops far below the Laplace-only curve beyond 20 cores, the practical parallel performance of the whole solver is weaker than the Laplace-only numbers suggest.

Watch

Extended reading notes

Core claim

The central claim is that FNPF-SEM reproduces the expected high-order accuracy of spectral element discretizations of the Zakharov free-surface potential-flow equations while remaining parallelizable. The Laplace problem for the velocity potential is solved on an extruded, vertically moving prismatic mesh with nodal Lagrange basis functions of order p, and the free-surface evolution equations for $\eta$ and $\phi_\eta$ are integrated with explicit fourth-order Runge-Kutta, with over-integration and a mild modal filter to control aliasing instabilities. The paper reports $O(h^p)$ convergence under h-refinement, exponential convergence under p-refinement, near-ideal strong scaling for larger meshes and higher polynomial order, near-constant per-core runtime in weak scaling up to 300 cores, and agreement with the chosen analytical and experimental benchmarks. The authors conclude that these results confirm the accuracy and applicability of the proposed framework.

Load-bearing premise

The scalability claims rest on the Laplace solve remaining the dominant routine, while the paper's own profiling shows the two other main routines stagnate above about 20 cores.

Editorial extensions

If this is right

  • Users can choose either mesh refinement (h) or polynomial order increase (p), with predictable algebraic or spectral error reduction.
  • On large problems the model maintains near-ideal parallel efficiency up to the tested 300 cores, making long-time simulations of engineering-scale domains practical.
  • Because the Laplace solve is the dominant cost, further improvements to its solver or preconditioner translate almost directly into whole-model speedups.
  • The validated benchmarks support the use of the solver for predicting wave run-up, harmonic generation, and diffraction around offshore structures.
  • The unstructured horizontal meshing opens the path to regional-scale wave propagation and irregular coastlines, which the paper names as ongoing work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The profiling suggests a practical ceiling for small or free-surface-dominated problems: the full time step's parallel efficiency may saturate near 20 cores because the free-surface evaluation and mesh update routines stop scaling, even though the Laplace-only scaling curves look better.
  • Since the cylinder boundary is currently approximated with affine geometry and submerged or floating bodies are not yet supported, a body-fitted high-order curvilinear boundary treatment is the natural next step for accurate near-structure predictions.
  • The solver's reliance on a code-generation finite element environment means performance portability, for example to many-core or GPU architectures, could come from swapping solver backends rather than rewriting the model equations.
  • All validated cases are non-breaking waves of moderate steepness; testing near the breaking limit would stress the stabilization and filtering machinery more severely than the presented benchmarks do.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. FNPF-SEM is a new parallel spectral element solver implemented within the Firedrake framework for linear and fully nonlinear potential-flow water waves, including wave-structure interaction. The mathematical model is the classical Zakharov-form free-surface problem: a Laplace problem in the volume plus kinematic and dynamic free-surface evolution equations, discretized with continuous nodal Lagrange elements on vertically extruded, horizontally unstructured meshes, with explicit RK4 time integration, relaxation zones for generation and absorption, and a modal filter for stabilization. The paper verifies the implementation through h-convergence (O(h^p) for the recovered vertical velocity) and p-convergence for nonlinear periodic waves over a range of kh and steepnesses, profiles the computational routines, and presents strong and weak scaling tests up to 300 cores. Validation cases are the Beji-Battjes submerged bar, linear diffraction and nonlinear run-up on a vertical cylinder, and diffraction by a V-shaped breakwater, compared with analytical, experimental, and reference numerical data. The strong and weak scaling tests are stated to focus exclusively on the Laplace solve. The authors conclude that the framework is accurate, efficient, and applicable to offshore engineering.

Significance. If the accuracy and scalability claims are fully supported, the Firedrake-based implementation is a useful engineering contribution: it brings high-order spectral element discretizations of FNPF equations to an open-source, MPI-capable framework with unstructured mesh support, and it provides a systematic benchmark suite covering convergence, scaling, and validation. The verification results are credible: the observed algebraic convergence order is consistent with the expected gradient-recovery loss, and the p-convergence in Appendix A shows exponential behavior. The validation benchmarks are well chosen and independent, with no fitted parameters. The paper also gives a clear breakdown of the computational routines, which is helpful for future optimization. If the requested evidence on end-to-end scaling and quantitative validation errors is supplied, the paper would support the advertised claims of accuracy, efficiency, and applicability.

major comments (3)
  1. [§4.1.2, Figs. 4-9, Eqs. (6)-(7)] The scaling analyses measure only the Laplace solve, not the full time step. The text states: "Consequently, the following strong- and weak-scaling analyses focus exclusively on this part of the code." Figures 6-9 therefore report LaplaceSolve-only timings and efficiencies, while the abstract and conclusion claim that the model "is scalable to full- and large-scale simulations" and exhibits "good parallel efficiency." This is load-bearing. Figure 4 shows that EvaluateRHS and LaplaceUpdate have constant runtimes beyond about 20 cores, and Figure 5 shows that the LaplaceSolve share of total time decreases as the core count grows. By Amdahl's law, the full-step speedups are strictly smaller than the Laplace-only curves, with an increasing gap at higher core counts. Please report strong and weak scaling for the complete time step (Steps 4.1-4.7), or alternatively provide Amdahl-corrected efficiency estimates using the measured cost shares.
  2. [§4.2.1-4.2.3] The validation is qualitative: the paper reports "good visual agreement" (bar test), "good agreement" (cylinder), and "good qualitative agreement" (breakwater), but no quantitative error metrics are given. The convergence tests in §4.1.1 establish discretization error only for periodic nonlinear waves, not for the inhomogeneous, structure-interaction cases in §4.2. To support the conclusion in §5 that the framework is accurate for wave-structure interaction, please add error measures such as RMS or L2 differences between numerical and experimental harmonic amplitudes in the bar test, relative errors in η_m/(2H) against MacCamy-Fuchs (TC1) and Kriebel (TC2), and profile errors against the Chang et al. reference for the breakwater. A mesh-resolution study for at least one of the structure cases is also needed to separate discretization error from physical modeling error; this is particularly important for the cylinder case, where the boundary is represented by affine geometry (§4.2.2).
  3. [§3.2] The modal filter is used for stabilization, but its effect on the solution is not quantified. The filter is taken from [23] and applied "as needed" to free-surface quantities, yet the paper does not report the filter order or strength used in the verification and validation runs, nor does it show that the filtered results are insensitive to those parameters. Since the filter modifies η, φη, or wη, the accuracy of the steep nonlinear results (e.g., ε/ε_max = 90% in Figure 3 and TC2 in Figure 12) may reflect the filter's influence as much as the spatial discretization. Please state the filter parameters and include a sensitivity study, e.g., varying the filter strength at fixed mesh resolution or comparing against an unfiltered run on a case that is stable without filtering.
minor comments (6)
  1. [§2.1, Eq. (1)] The periodic boundary condition "ϕ|Γp = ϕ|Γp" is trivially true as written; please use notation distinguishing the two periodic faces, e.g., ϕ|Γp+ = ϕ|Γp−.
  2. [§3.2] The CFL time step uses u_max, "the maximum velocity encountered in the simulation," but no a priori estimation procedure is given; for practical use, clarify how u_max is determined.
  3. [§4.1.2] The sentence "A substantial reduction in time per stage is observed; the increase is from Nc = 1 to approximately Nc = 20" is garbled; the second clause should refer to the reduction across this core range. Also, "is has been" later in the same section should read "it has been."
  4. [Figure 5] The figure caption should explicitly identify which line corresponds to Mesh 1 and which to Mesh 2; the text does this, but the figure itself is ambiguous, especially in black-and-white printing.
  5. [§4.2.3] The inequality "−L2 − 2b ≤ (x, y) ≤ 2b + L2" is not meaningful for a vector; please write it componentwise. Also, the definition of L2 as "260 [m] ≈ 2L" should be made clearer, since L2 denotes a length rather than a coordinate variable.
  6. [Title page] The title page reads "AN AR XIV PREPRINT"; the space in "AR XIV" is erroneous and should read "AN ARXIV PREPRINT."

Circularity Check

0 steps flagged · score 1.0 of 10

No meaningful circularity: the solver is verified and validated against external stream-function, analytical, and experimental benchmarks; self-citations are methodological, not load-bearing.

full rationale

The paper's central claims are numerical verification and validation of a Firedrake implementation of the classical Zakharov FNPF equations. The h- and p-convergence studies compare the numerical vertical free-surface velocity against an external stream-function reference (Fenton 1988), and the validation cases use independent experimental and analytical data (Beji and Battjes 1994, MacCamy and Fuchs 1954, Kriebel 1992, Chang et al. 2012). No parameter is fitted to those benchmarks: CCFL = 0.95 and relaxation-zone lengths are standard modeling choices, and the harmonic decomposition in the bar test is a post-processing least-squares fit applied to both data sets rather than a model calibration. The main self-citations, notably [23] and the DTU-lineage references, supply stabilization and filtering methodology and a remark on gradient-recovery convergence loss; these are not uniqueness claims and do not define the validation targets. The statement in Section 4.1.2 that the strong- and weak-scaling analyses 'focus exclusively on this part of the code' (LaplaceSolve) is a real scoping caveat for end-to-end scalability, but it is a measurement-scope limitation, not a circular reduction: the Laplace-only efficiency numbers are computed from a well-defined component and are not assumed equal to the full-model claim. Hence no step in the derivation chain reduces by construction to its inputs.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claims depend on standard potential flow assumptions and on a few numerical choices. No entities are invented. Free parameters are solver settings and inherited filter/relaxation choices; none are fitted to the validation targets.

free parameters (4)
  • CFL coefficient = 0.95
    Set for temporal stability; standard explicit scheme choice, not fitted to accuracy targets.
  • Modal filter (strength/order) = not specified (from [23])
    Stabilizes nonlinear free surface; parameters deferred to Engsig-Karup et al. 2016 and not quantified in this paper.
  • Relaxation zone length = 2-3 wavelengths
    Chosen for wave generation/absorption; affects results but standard practice.
  • Ramp-up duration = 5 wave periods
    Gradual activation of generation zone to avoid impulsive start.
assumptions (4)
  • domain assumption Fluid is incompressible, inviscid, and irrotational
    Underlies the velocity potential representation and Laplace equation in Section 2.1.
  • domain assumption Free surface conditions in Zakharov form (Eqs. 2 and 3) accurately model nonlinear gravity waves
    Standard irrotational water wave theory; adopted without derivation.
  • ad hoc to paper Modal filter (from [23]) stabilizes without materially altering the physical solution
    Filter parameters are not specified or quantified; accuracy impact not assessed in this paper.
  • domain assumption Firedrake extruded mesh partitions coincide for surface and volume meshes (mapping m)
    Relies on Firedrake's parallel numbering for extruded meshes, Section 3.1.2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of FNPF-SEM: A parallel spectral element model in Firedrake for fully nonlinear water wave simulations." pith.science (2026). https://pith.science/paper/RH44IXKU

@misc{pith2026250609435,
  author       = {Pith},
  title        = {Pith review of: FNPF-SEM: A parallel spectral element model in Firedrake for fully nonlinear water wave simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RH44IXKU}},
  note         = {Machine review of arXiv:2506.09435}
}
read the original abstract

We present a new parallel spectral element solver, FNPF-SEM, for simulating linear and fully nonlinear potential flow-based water waves and their interaction with offshore structures. The tool is designed as a general-purpose wave model for offshore engineering applications. Built within the open-source framework Firedrake, the new FNPF-SEM model is designed as a computational tool capable of capturing both linear and nonlinear wave phenomena with high accuracy and efficiency, with support for high-order (spectral) finite elements. Additionally, Firedrake provides native support for MPI-based parallelism, allowing for efficient multi-CPU distributed computations needed for large-scale simulations. We demonstrate the capabilities of the high-order spectral element model through h- and p-convergence studies, and weak and strong scaling tests. Validation is performed against analytical solutions and experimental data for several benchmark cases, including nonlinear high-order harmonic generation and linear and nonlinear wave interactions with a cylinder and a breakwater. The new FNPF-SEM model offers a numerical framework for simulating wave propagation and wave-structure interactions, with the following key features: i) the ability to represent complex geometries through flexible, unstructured finite element meshes; ii) reduced numerical diffusion and dispersion by using high-order polynomial expansions; and iii) scalability to full- and large-scale simulations over long time periods through a parallel implementation.

Figures

Figures reproduced from arXiv: 2506.09435 by the authors.

Figure 1
Figure 1. Conceptual schematic of the fluid domain, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Point distribution (indicated by different colors) for a finite FNPF wave tank distributed over [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Convergence study for h-refinement for nonlinear waves in all combinations between kh = {1, 3, 6} and ε/εmax = {10%, 50%, 90%}. The p-convergence study is presented in Appendix A, in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Timings for the three dominant routines in the FNPF model for Mesh 1 (blue) and Mesh 2 (red), using a [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Percentage distribution of computational cost across the three dominant routines for Mesh 1 (blue) and Mesh [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Strong scaling efficiency test. Mesh 1 (blue), Mesh 2 (red), and Mesh 3 (orange) with polynomial orders [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: Speedup (compared to Nn = 1 as baseline) for the strong scaling test. Mesh 1 (blue), Mesh 2 (red), and Mesh 3 (orange) with polynomial orders p = {3, 4, 5}. 4.1.4 Parallel performance: Weak scaling To assess how the model scales with a constant workload per core, we pe…
Figure 8
Figure 8. Figure 8: Weak scaling timings per stage on six meshes with polynomial order [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Weak scaling efficiency on six meshes with polynomial order [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: Contour plot of the free surface elevation at [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Harmonic analysis of the numerical simulation (FNPF) and experimental data (BB94) for the four most [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: Maximum non-dimensional free surface elevation on the cylinder, [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: Free surface mesh for θ = 0◦ around the breakwater. The computational domain spans the region −L2 − 2b ≤ (x, y) ≤ 2b + L2, completely enclosing the target region defined as −2b ≤ x ≤ 2b and −L2 − 2b ≤ y ≤ 2b + L2. Here, L2 = 260 [m] ≈ 2L, with L denoting the wavelengt…
Figure 14
Figure 14. Figure 14: Breakwater solution (non-dimensional free surface variation) on the UBA and LBA for [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: Convergence study for p for nonlinear waves in all combinations between kh = {1, 3, 6} and ε/εmax = {10%, 50%, 90%}. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
Figure 16
Figure 16. Figure 16: Mesh used for the linear simulation as described in Table 5. Zoomed in around the cylinder. [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 17
Figure 17. Figure 17: Breakwater solution (normalized free surface variation) at [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]
Figure 18
Figure 18. Figure 18: Breakwater solution (normalized free surface variation) at [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]
Figure 19
Figure 19. Figure 19: Breakwater solution (normalized free surface variation) at [PITH_FULL_IMAGE:figures/full_fig_p023_19.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 64 canonical work pages

  1. [23]

    Journal of Computational Physics 318: 1–21

    Engsig-Karup AP, Eskilsson C and Bigoni D (2016) A stabilised nodal spectral element method for fully nonlinear water waves. Journal of Computational Physics 318: 1–21

    work page 2016

  2. [1]

    ACM Transactions on Mathematical Software 40(2)

    Alnæs MS, Logg A, Ølgaard KB, Rognes ME and Wells GN (2014) Unified form language: A domain-specific language for weak formulations of partial differential equations. ACM Transactions on Mathematical Software 40(2)

    work page 2014

  3. [2]

    Applied Ocean Research 69: 220–244

    Amini-Afshar M and Bingham HB (2017) Solving the linearized forward-speed radiation problem using a high-order finite difference method on overlapping grids. Applied Ocean Research 69: 220–244

    work page 2017

  4. [3]

    Applied Ocean Research 80: 197–219

    Amini-Afshar M and Bingham HB (2018) Pseudo-impulsive solutions of the forward-speed diffraction problem using a high-order finite-difference method. Applied Ocean Research 80: 197–219

    work page 2018

  5. [4]

    Computers & Mathematics with Applications 81: 42–74

    Anderson R, Andrej J, Barker A, Bramwell J, Camier JS, Cerveny J, Dobrev V , Dudouit Y , Fisher A, Kolev T, Pazner W, Stowell M, Tomov V , Akkerman I, Dahm J, Medina D and Zampini S (2021) MFEM: A modular finite element methods library. Computers & Mathematics with Applications 81: 42–74

    work page 2021

  6. [5]

    Computers & Mathematics with Applications 81: 407–422

    Arndt D, Bangerth W, Davydov D, Heister T, Heltai L, Kronbichler M, Maier M, Pelteret JP, Turcksin B and Wells D (2021) The deal.ii finite element library: Design, features, and insights. Computers & Mathematics with Applications 81: 407–422. Development and Application of Open-source Software for Problems with Numerical PDEs

    work page 2021

  7. [6]

    offshore wind energy

    Bailey BH, Filippelli M and Baker M (2015) Metocean data needs assessment for u.s. offshore wind energy. Technical report, AWS Truepower LLC., Albany, NY (United States)

    work page 2015

  8. [7]

    Technical Report ANL-21/39 - Revision 3.22, Argonne National Laboratory

    Balay S, Abhyankar S, Adams MF, Benson S, Brown J, Brune P, Buschelman K, Constantinescu E, Dalcin L, Dener A, Eijkhout V , Faibussowitsch J, Gropp WD, Hapla V , Isaac T, Jolivet P, Karpeev D, Kaushik D, Knepley MG, Kong F, Kruger S, May DA, McInnes LC, Mills RT, Mitchell L, Munson T, Roman JE, Rupp K, Sanan P, Sarich J, Smith BF, Suh H, Zampini S, Zhang ...

    work page 2024

Show all 64 references
  1. [8]

    In: Arge E, Bruaset AM and Langtangen HP (eds.) Modern Software Tools in Scientific Computing

    Balay S, Gropp WD, McInnes LC and Smith BF (1997) Efficient management of parallelism in object oriented numerical software libraries. In: Arge E, Bruaset AM and Langtangen HP (eds.) Modern Software Tools in Scientific Computing. Birkhäuser Press, pp. 163–202

    1997

  2. [9]

    Coastal Engineering 23(1): 1–16

    Beji S and Battjes J (1994) Numerical simulation of nonlinear wave propagation over a bar. Coastal Engineering 23(1): 1–16

    1994

  3. [10]

    Geoscientific Model Development 9(10): 3803–3815

    Bercea GT, McRae ATT, Ham DA, Mitchell L, Rathgeber F, Nardi L, Luporini F and Kelly PHJ (2016) A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in firedrake. Geoscientific Model Development 9(10): 3803–3815

    2016

  4. [11]

    Journal of Engineering Mathematics 58(1): 211–228

    Bingham HB and Zhang H (2007) On the accuracy of finite-difference solutions for nonlinear water waves. Journal of Engineering Mathematics 58(1): 211–228

    2007

  5. [12]

    Applied Ocean Research 51: 279–292

    Bitner-Gregersen EM (2015) Joint met-ocean description for design and operations of marine structures. Applied Ocean Research 51: 279–292

    2015

  6. [13]

    In: Bridges TJ, Groves MD and Nicholls DP (eds.) Lectures on the theory of water waves, LMS Lecture Note Series, volume

    Bokhove O and Kalogirou A (2016) Variational water wave modelling: from continuum to experiment. In: Bridges TJ, Groves MD and Nicholls DP (eds.) Lectures on the theory of water waves, LMS Lecture Note Series, volume

    2016

  7. [14]

    Computer Physics Communications 192: 205–219

    Cantwell C, Moxey D, Comerford A, Bolis A, Rocco G, Mengaldo G, De Grazia D, Yakovlev S, Lombard JE, Ekelschot D, Jordi B, Xu H, Mohamied Y , Eskilsson C, Nelson B, V os P, Biotto C, Kirby R and Sherwin S (2015) Nektar++: An open-source spectral/hp element framework. Computer ...

    2015

  8. [15]

    Release 0.5.0

    Chalmers N, Karakus A, Austin AP, Swirydowicz K and Warburton T (2022) libParanumal: a performance portable high-order finite element library. Release 0.5.0

    2022

  9. [16]

    Coastal Engineering 68: 56–66

    Chang KH, Tsaur DH and Huang LH (2012) Accurate solution to diffraction around a modified v-shaped breakwater. Coastal Engineering 68: 56–66. 17 An arXiv Preprint VISBECH ET AL . (2025)

    2012

  10. [17]

    In: Proceedings of the 32nd International Conference on Ocean, Offshore and Arctic Engineering

    Christiansen TB, Bingham HB, Engsig-Karup AP, Ducrozet G and Ferrant P (2013) Efficient hybrid-spectral model for fully nonlinear numerical wave tank. In: Proceedings of the 32nd International Conference on Ocean, Offshore and Arctic Engineering

    2013

  11. [18]

    Journal of Fluid Mechanics 184: 267–288

    Dommermuth DG and Yue DKP (1987) A high-order spectral method for the study of nonlinear gravity waves. Journal of Fluid Mechanics 184: 267–288

    1987

  12. [19]

    Journal of Hydrodynamics 22(1): 225–230

    Ducrozet G, Bingham HB, Engsig-Karup AP and Ferrant P (2010) High-order finite difference solution for 3D nonlinear wave-structure interaction. Journal of Hydrodynamics 22(1): 225–230

    2010

  13. [20]

    International Journal for Numerical Methods in Fluids 74(10): 749–773

    Engsig-Karup AP (2014) Analysis of efficient preconditioned defect correction methods for nonlinear water waves. International Journal for Numerical Methods in Fluids 74(10): 749–773

    2014

  14. [21]

    Journal of Computational Physics 228(6): 2100–2118

    Engsig-Karup AP, Bingham HB and Lindberg O (2009) An efficient flexible-order model for 3D nonlinear water waves. Journal of Computational Physics 228(6): 2100–2118

    2009

  15. [22]

    International Journal of Offshore and Polar Engineering29: 141–148

    Engsig-Karup AP and Eskilsson C (2019) Spectral element fnpf simulation of focused wave groups impacting a fixed fpso-type body. International Journal of Offshore and Polar Engineering29: 141–148

    2019

  16. [24]

    International Journal for Numerical Methods in Fluids 93(9): 2823–2841

    Engsig-Karup AP and Laskowski WL (2021) An efficient p-multigrid spectral element model for fully nonlinear water waves and fixed bodies. International Journal for Numerical Methods in Fluids 93(9): 2823–2841

    2021

  17. [25]

    Water Waves1(2): 315–342

    Engsig-Karup AP, Monteserin C and Eskilsson C (2019) A Mixed Eulerian–Lagrangian Spectral Element Method for Nonlinear Wave Interaction with Fixed Structures. Water Waves1(2): 315–342

    2019

  18. [26]

    In: Proceedings of The 26th Numerical Towing Tank Symposium

    Eskilsson C and Engsig-Karup AP (2024) Water wave simulations using fully nonlinear potential flow: Spectral/hp element models implemented in nektar++. In: Proceedings of The 26th Numerical Towing Tank Symposium

    2024

  19. [27]

    In: Numerical solution of partial differential equations on parallel computers

    Falgout RD, Jones JE and Yang UM (2006) The design and implementation of hypre, a library of parallel high performance preconditioners. In: Numerical solution of partial differential equations on parallel computers . Springer, pp. 267–294

    2006

  20. [28]

    Computers & Geosciences 14(3): 357–368

    Fenton J (1988) The numerical solution of steady water wave problems. Computers & Geosciences 14(3): 357–368

    1988

  21. [29]

    the Sea 9(1): 3–25

    Fenton JD (1990) Nonlinear wave theories. the Sea 9(1): 3–25

    1990

  22. [30]

    International Journal for Numerical Methods in Engineering 79(11): 1309–1331

    Geuzaine C and Remacle JF (2009) Gmsh: A 3-D finite element mesh generator with built-in pre- and post- processing facilities. International Journal for Numerical Methods in Engineering 79(11): 1309–1331

    2009

  23. [31]

    Imperial College London and University of Oxford and Baylor University and University of Washington, first edition edition

    Ham DA, Kelly PHJ, Mitchell L, Cotter CJ, Kirby RC, Sagiyama K, Bouziani N, V orderwuelbecke S, Gregory TJ, Betteridge J, Shapero DR, Nixon-Hill RW, Ward CJ, Farrell PE, Brubeck PD, Marsden I, Gibson TH, Homolya M, Sun T, McRae ATT, Luporini F, Gregory A, Lange M, Funke SW, Ra...

    2023

  24. [32]

    International Journal for Numerical Methods in Fluids 94(2): 188–222

    Harris JC, Dombre E, Benoit M, Grilli ST and Kuznetsov KI (2022) Nonlinear time-domain wave-structure interaction: A parallel fast integral equation approach. International Journal for Numerical Methods in Fluids 94(2): 188–222

    2022

  25. [33]

    (1952) Methods of conjugate gradients for solving linear systems

    Hestenes MR, Stiefel E et al. (1952) Methods of conjugate gradients for solving linear systems. Journal of research of the National Bureau of Standards49(6): 409–436

    1952

  26. [34]

    Mathematics of Computation 77(263): 1425–1452

    Hesthaven JS and Kirby RM (2008) Filtering in legendre spectral methods. Mathematics of Computation 77(263): 1425–1452

    2008

  27. [35]

    Geoscientific Model Development 11(11): 4359–4382

    Kärnä T, Kramer SC, Mitchell L, Ham DA, Piggott MD and Baptista AM (2018) Thetis coastal ocean model: discontinuous galerkin discretization for the three-dimensional hydrostatic equations. Geoscientific Model Development 11(11): 4359–4382

    2018

  28. [36]

    OUP Oxford

    Karniadakis G and Sherwin S (2005) Spectral/hp element methods for computational fluid dynamics. OUP Oxford

    2005

  29. [37]

    Journal of Computational Physics 191(1): 249–264

    Kirby RM and Karniadakis GE (2003) De-aliasing on non-uniform grids: algorithms and applications. Journal of Computational Physics 191(1): 249–264

    2003

  30. [38]

    Tellus 24(3): 199–215

    Kreiss HO and Oliger J (1972) Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24(3): 199–215

    1972

  31. [39]

    part ii: Wave run-up

    Kriebel D (1992) Nonlinear wave interaction with a vertical circular cylinder. part ii: Wave run-up. Ocean Engineering 19(1): 75–99. 18 An arXiv Preprint VISBECH ET AL . (2025)

    1992

  32. [40]

    Computer Physics Communications 292: 108885

    Kurnia R and Ducrozet G (2023) Nemoh: Open-source boundary element solver for computation of first- and second-order hydrodynamic loads in the frequency domain. Computer Physics Communications 292: 108885

    2023

  33. [41]

    Larsen J and Dancy H (1983) Open boundaries in short wave simulations — a new approach.Coastal Engineering 7(3): 285–297

    1983

  34. [42]

    WAMIT, Inc42

    Lee CH and Newman JN (2006) Wamit user manual. WAMIT, Inc42

    2006

  35. [43]

    Journal of Fluid Mechanics 245: 115–136

    Liu Y , Dommermuth DG and Yue DKP (1992) A high-order spectral method for nonlinear wave–body interactions. Journal of Fluid Mechanics 245: 115–136

    1992

  36. [44]

    Liu Y , Xue M and Yue DKP (2001) Computations of fully nonlinear three-dimensional wave–wave and wave–body interactions. part 2. nonlinear waves and forces on a body. Journal of Fluid Mechanics 438: 41–66

    2001

  37. [45]

    MacCamy RC and Fuchs RA (1954) Wave forces on piles: a diffraction theory. 69. US Beach Erosion Board

    1954

  38. [46]

    SIAM Journal on Scientific Computing 38(5): S25–S47

    McRae ATT, Bercea GT, Mitchell L, Ham DA and Cotter CJ (2016) Automated generation and symbolic manipulation of tensor product finite elements. SIAM Journal on Scientific Computing 38(5): S25–S47

    2016

  39. [47]

    Water Waves 6(3): 547–570

    Melander A and Engsig-Karup AP (2024) A p-multigrid hybrid-spectral model for nonlinear water waves. Water Waves 6(3): 547–570

    2024

  40. [48]

    arXiv Preprint: 2411.14977

    Melander A, Laskowski W, Sherwin SJ and Engsig-Karup AP (2024) A p-multigrid accelerated nodal spectral element method for free-surface incompressible navier-stokes model of nonlinear water waves. arXiv Preprint: 2411.14977

    2024 arXiv

  41. [49]

    Journal of Computational Physics 54(3): 468–488

    Patera AT (1984) A spectral element method for fluid dynamics: Laminar flow in a channel expansion. Journal of Computational Physics 54(3): 468–488

    1984

  42. [50]

    (2019) A blind comparative study of focused wave interactions with a fixed fpso-like structure (ccp-wsi blind test series 1)

    Ransley E, Yan S, Brown SA, Mai T, Graham D, Ma Q, Musiedlak PH, Engsig-Karup AP, Eskilsson C, Li Q et al. (2019) A blind comparative study of focused wave interactions with a fixed fpso-like structure (ccp-wsi blind test series 1). International Journal of Offshore and Polar ...

    2019

  43. [51]

    ACM Transactions on Mathematical Software 43(3)

    Rathgeber F, Ham DA, Mitchell L, Lange M, Luporini F, Mcrae ATT, Bercea GT, Markall GR and Kelly PHJ (2016) Firedrake: Automating the finite element method by composing abstractions. ACM Transactions on Mathematical Software 43(3)

    2016

  44. [52]

    Journal of Computational Physics 155(1): 26–53

    Robertson I and Sherwin S (1999) Free-Surface Flow Simulation Using hp/Spectral Elements. Journal of Computational Physics 155(1): 26–53

    1999

  45. [53]

    Seixas de Medeiros J, Liu Y and Yue DK (2024) A fast high-order boundary element method for nonlinear water waves generation and propagation in large wave basins.Computer Methods in Applied Mechanics and Engineering 432: 117396

    2024

  46. [54]

    Journal of Computational Physics 274: 312–332

    Shao YL and Faltinsen OM (2014) A harmonic polynomial cell (hpc) method for 3d laplace equation with application in marine hydrodynamics. Journal of Computational Physics 274: 312–332

    2014

  47. [55]

    Physics of Fluids 36(3): 032118

    Tong C, Shao Y , Bingham HB and Hanssen FCW (2024) An adaptive harmonic polynomial cell method for three-dimensional fully nonlinear wave-structure interaction with immersed boundaries. Physics of Fluids 36(3): 032118

    2024

  48. [56]

    Visbech J, Bingham HB, Eskilsson C, Palm J and Engsig-Karup AP (2024) A High-order Accurate Spectral Element based Time-Domain Simulation of a Model-Scale Floating Offshore Wind Turbine.International Journal of Offshore and Polar Engineering34: 254–262

    2024

  49. [57]

    Computer Methods in Applied Mechanics and Engineering 423: 116871

    Visbech J, Engsig-Karup AP and Bingham HB (2024) Solving the complete pseudo-impulsive radiation and diffraction problem using a spectral element method. Computer Methods in Applied Mechanics and Engineering 423: 116871

    2024

  50. [58]

    In: Proceedings of The 39th International Workshop on Water Waves and Floating Bodies

    Visbech J, Engsig-Karup AP, Bingham HB, Amini-Afshar M and Ricchiuto M (2024) A high-order shifted boundary method for water waves and floating bodies. In: Proceedings of The 39th International Workshop on Water Waves and Floating Bodies

    2024

  51. [59]

    Applied Ocean Research 16(6): 363–372

    Wu G and Eatock Taylor R (1994) Finite element analysis of two-dimensional non-linear transient water waves. Applied Ocean Research 16(6): 363–372

    1994

  52. [60]

    Journal of Hydrodynamics 30: 1–22

    Xu H, Cantwell C, Monteserin C, Eskilsson C, Engsig-Karup AP and Sherwin S (2018) Spectral/hp element methods: Recent developments, applications, and perspectives. Journal of Hydrodynamics 30: 1–22

    2018

  53. [61]

    Xue M, Xu H, Liu Y and Yue DKP (2001) Computations of fully nonlinear three-dimensional wave–wave and wave–body interactions. part 1. dynamics of steep three-dimensional waves. Journal of Fluid Mechanics 438: 11–39. 19 An arXiv Preprint VISBECH ET AL . (2025)

    2001

  54. [62]

    Journal of Computational Physics 230(2): 402–424

    Yan H and Liu Y (2011) An efficient high-order boundary element method for nonlinear wave–wave and wave-body interactions. Journal of Computational Physics 230(2): 402–424

    2011

  55. [63]

    Journal of Applied Mechanics and Technical Physics 9: 190–194

    Zakharov VE (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics 9: 190–194. 20 An arXiv Preprint VISBECH ET AL . (2025) 6 Appendix 6.1 Appendix A: p-convergence study Figure 15: Convergence s...

    1968

  56. [426]

    Cambridge University Press, pp. 226–260

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.