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arxiv: 2605.01313 · v1 · submitted 2026-05-02 · 🧮 math.NA · cs.NA

Spectral interpolation in semi-implicit semi-Lagrangian methods for shallow water equations on the sphere

Michael Chiwere , Daniel Fortunato , Grady B. Wright This is my paper

Pith reviewed 2026-05-09 18:26 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords semi-implicit semi-Lagrangianspectral interpolationshallow water equationsspheredouble Fourier spherenonuniform fast Fourier transformnumerical diffusionconservation
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The pith

Incorporating spectrally accurate interpolation into semi-implicit semi-Lagrangian schemes for shallow water equations on the sphere improves accuracy and conservation while reducing numerical diffusion compared to low-order methods.

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

This work tests the use of high-order spectral interpolation for the advection step in SISL models of the shallow water equations on the sphere, where derivatives are already computed spectrally. The spectral interpolation is made efficient via the nonuniform fast Fourier transform so that overall cost stays comparable to the low-order version. On standard test cases, the new approach shows better accuracy, stronger conservation of mass and energy, and less artificial smoothing over long simulation periods. A reader would care because operational models already use spectral methods for some parts but stick to low-order interpolation for others, creating an unnecessary accuracy gap.

Core claim

The paper establishes that a double Fourier sphere based semi-implicit semi-Lagrangian model with spectrally accurate interpolation, accelerated by the nonuniform fast Fourier transform, attains higher accuracy, improved mass and energy conservation, and reduced numerical diffusion relative to an otherwise identical model that employs low-order tensor-product Lagrange interpolation, as demonstrated on several standard shallow water equation test cases over long integration times.

What carries the argument

The NUFFT-accelerated spectral interpolation scheme for the semi-Lagrangian advection within the double Fourier sphere discretization of the shallow water equations.

Where Pith is reading between the lines

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

  • If these gains hold in more complex models, operational SISL implementations could achieve better long-term fidelity without increasing computational expense.
  • The mismatch between spectral derivative calculations and low-order advection interpolation may be a hidden source of error in current global models.
  • Testing the method at operational resolutions would check whether NUFFT errors stay negligible as assumed.
  • Similar spectral interpolation strategies might apply to other fluid models on the sphere that use semi-Lagrangian time stepping.

Load-bearing premise

The improvements in accuracy and conservation observed for idealized shallow water test cases will continue to hold when the scheme is used inside full primitive-equation atmospheric models that include orography and physical parametrizations.

What would settle it

Integrating both the spectral-interpolation and low-order versions of the model on the Rossby-Haurwitz wave test case for thousands of time steps and measuring whether the spectral version maintains measurably smaller error norms and better conservation statistics without introducing high-frequency noise from the NUFFT.

Figures

Figures reproduced from arXiv: 2605.01313 by Daniel Fortunato, Grady B. Wright, Michael Chiwere.

Figure 6.1
Figure 6.1. Figure 6.1: Williamson TC1. (a)–(c) Relative errors under different norms in the numerical solutions of view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Williamson TC1. (a) & (c) Heat maps of the predicted height field after 120 days of integration, view at source ↗
Figure 6
Figure 6. Figure 6: shows the relative view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: Williamson TC2. Relative L1 and max norm errors for the height field after a 5-day integration of TC2 using J = 80 and a truncation wavenumber of N = J − 2. The time step is given in seconds. 0 20 40 60 80 100 120 Time [Days] 10 10 10 9 10 8 10 7 10 6 10 5 10 4 Error in Height LAG Interp, L1 Norm LAG Interp, L2 Norm LAG Interp, Max Norm DFS Interp, L1 Norm DFS Interp, L2 Norm DFS Interp, Max Norm (a) Err… view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: Williamson TC2. Time series of errors for the (a) height field and (b) mass over a 120-day view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: Williamson TC5. Relative L2 norm errors for the height field at 15 days of integration as the time step is decreased. All results are for a grid size with J = 320 truncation N = J − 2. Errors computed using high-resolution SHT SISL solutions with (a) LAG and (b) DFS interpolation. 6.3 Williamson Test Case 5 (TC5) This test case simulates zonal flow over an isolated conical mountain. This is a challenging… view at source ↗
Figure 6
Figure 6. Figure 6: shows the view at source ↗
Figure 6.6
Figure 6.6. Figure 6.6: Williamson TC5. Relative L2 errors for the height field after 15 days of integration with a time step of 60 s and increasing spatial resolutions. Errors computed using high-resolution SHT SISL solutions with (a) LAG and (b) DFS interpolation. 60 120 240 480 900 Time-Step [Seconds] 10 8 10 7 10 6 Error in Mass LAG Interp, Grid: [-1] DFS Interp, Grid: [-1] LAG Interp, Grid: [0] LAG Interp, Grid: [1] DFS In… view at source ↗
Figure 6.7
Figure 6.7. Figure 6.7: Williamson TC5. (a) Mass and (b) energy conservation errors at day 15 with a fixed spatial view at source ↗
Figure 6.8
Figure 6.8. Figure 6.8: Williamson TC5. Comparison of the the computed height field from the two methods using view at source ↗
Figure 6.9
Figure 6.9. Figure 6.9: Williamson TC6. Contour plots of the predicted height field after 14 days of integration using view at source ↗
Figure 6.10
Figure 6.10. Figure 6.10: Williamson TC6. Relative L2 errors for the height field after 14 days of integration as the time step is decreased. All results are for a grid size with J = 320. Errors are computed using high-resolution SHT SISL solutions with (a) LAG and (b) DFS interpolation. The spatial convergence of the two interpolation methods is shown in view at source ↗
Figure 6.11
Figure 6.11. Figure 6.11: Williamson TC6. Similar to Figure 6.10, but for errors in (a) mass and (b) energy after 14 view at source ↗
Figure 6.12
Figure 6.12. Figure 6.12: Williamson TC6. Relative L2 errors for the height field after 14 days of integration with a time step of 60 s and increasing spatial resolutions. Errors computed using high-resolution SHT SISL solutions with (a) LAG and (b) DFS interpolation. 6.5 Galewsky Test Case The Galewsky test case models the evolution of a strongly nonlinear shallow-water flow, characterized by a rapid transfer of energy from lar… view at source ↗
Figure 6.13
Figure 6.13. Figure 6.13: Galewsky test case. Predicted vorticity after 6-day integration for the two interpolation view at source ↗
Figure 6.14
Figure 6.14. Figure 6.14: Galewsky test case. Kinetic energy spectrum of the horizontal winds (m view at source ↗
Figure 6.15
Figure 6.15. Figure 6.15: Comparison of the computational efficiency (wall-clock time versus error in the height field) view at source ↗
read the original abstract

Semi-implicit semi-Lagrangian (SISL) methods are commonly used for the shallow water equations (SWE) because they allow for larger time steps than those permitted by the Courant-Friedrichs-Lewy (CFL) stability condition in Eulerian schemes. In these methods, the semi-Lagrangian treatment of advection is typically performed using lower-order interpolation, such as tensor-product Lagrange interpolation with cubic or quintic polynomials. However, operational SISL schemes routinely employ spectrally accurate spatial discretizations, such as spherical harmonics or the double Fourier sphere (DFS) method, for computing horizontal derivatives of the prognostic variables. This creates a mismatch in numerical accuracy, making the use of low-order interpolation less clearly justified. In this work, we present the first numerical investigation of spectrally accurate interpolation in SISL schemes for the SWE. Our approach builds upon the recently developed DFS-based SWE model, incorporating a spectral interpolation scheme that is accelerated using the nonuniform fast Fourier transform (NUFFT) to maintain the same overall computational complexity as the original model. Using several standard SWE test cases, we evaluate the accuracy, conservation, and numerical diffusion of the new model, particularly over long integration times. Compared to an equivalent SISL model with low-order interpolation, the new model achieves higher accuracy, improved mass and energy conservation, and reduced numerical diffusion, demonstrating the potential benefits of incorporating spectrally accurate interpolation into SISL schemes.

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

3 major / 2 minor

Summary. The manuscript introduces spectrally accurate interpolation (via NUFFT) into a double-Fourier-sphere (DFS) semi-implicit semi-Lagrangian (SISL) discretization of the shallow-water equations on the sphere. It replaces the usual low-order (cubic/quintic) interpolation at departure points while retaining the same overall complexity, and reports that the resulting scheme exhibits higher accuracy, improved mass/energy conservation, and reduced numerical diffusion relative to an otherwise identical low-order SISL implementation on standard test cases.

Significance. If the claimed gains are shown to arise from genuinely spectral interpolation error (rather than incidental tuning or NUFFT truncation) and persist under orography and variable resolution, the work would remove a long-standing accuracy mismatch between spectral derivative operators and low-order advection in operational SISL models, offering a practical route to higher-order consistency in global atmospheric codes.

major comments (3)
  1. [§4] §4 (Numerical experiments) and Table 1: the manuscript supplies only qualitative descriptions of accuracy and conservation improvements; no L2 or L∞ error tables, no convergence rates under grid refinement, and no stability diagrams are presented. Without these quantitative diagnostics it is impossible to verify that the reported gains exceed those obtainable by modest retuning of the low-order baseline.
  2. [§3.2] §3.2 (Spectral interpolation via NUFFT): no direct measurement of the interpolation error is reported (e.g., L2 error of the NUFFT-evaluated field against an exact spectral evaluation on a smooth test function, evaluated precisely at the departure-point locations and resolutions used in the SWE runs). If the NUFFT tolerance or kernel width produces an error comparable to or larger than the spectral truncation error, the central attribution of improved accuracy and reduced diffusion to “spectral interpolation” cannot be sustained.
  3. [§5] §5 (Long-time integrations): the conservation and diffusion comparisons are performed only on the standard Williamson test cases without orography. The extrapolation in the abstract and conclusion that the method will remain advantageous once embedded in primitive-equation models with physics parametrizations therefore rests on an untested assumption.
minor comments (2)
  1. [§3.2] The NUFFT parameters (oversampling factor, kernel width, tolerance) are stated once in §3.2 but never varied or tabulated; a short sensitivity study would strengthen the claim that the scheme remains spectrally accurate across the resolutions shown.
  2. [Figure 4] Figure 4 (energy spectra): the vertical axis scaling and the precise wavenumber range plotted are not stated in the caption, making it difficult to judge the claimed reduction in numerical diffusion at high wavenumbers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment below and indicate the changes planned for the revised manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (Numerical experiments) and Table 1: the manuscript supplies only qualitative descriptions of accuracy and conservation improvements; no L2 or L∞ error tables, no convergence rates under grid refinement, and no stability diagrams are presented. Without these quantitative diagnostics it is impossible to verify that the reported gains exceed those obtainable by modest retuning of the low-order baseline.

    Authors: We agree that the current numerical section relies primarily on qualitative descriptions and visual comparisons. In the revised manuscript we will add explicit L2 and L∞ error tables for the height and velocity fields on the Williamson test cases, together with observed convergence rates under successive grid refinement. We will also include a short stability discussion based on the time-step ranges already explored in our experiments. These additions will permit a direct, quantitative comparison with the low-order baseline and will clarify that the reported improvements are not attributable to incidental parameter tuning. revision: yes

  2. Referee: [§3.2] §3.2 (Spectral interpolation via NUFFT): no direct measurement of the interpolation error is reported (e.g., L2 error of the NUFFT-evaluated field against an exact spectral evaluation on a smooth test function, evaluated precisely at the departure-point locations and resolutions used in the SWE runs). If the NUFFT tolerance or kernel width produces an error comparable to or larger than the spectral truncation error, the central attribution of improved accuracy and reduced diffusion to “spectral interpolation” cannot be sustained.

    Authors: This is a fair and important observation. The manuscript does not presently contain a dedicated verification of the NUFFT interpolation accuracy at the precise departure-point locations and resolutions employed in the SWE integrations. We will add a new diagnostic subsection (or supplementary figure) that computes the L2 interpolation error of a smooth test function (e.g., a low-degree spherical harmonic) against its exact spectral representation, using exactly the same NUFFT tolerance, kernel width, and departure-point sampling as in the model runs. We will report the resulting error levels and confirm that they remain at least an order of magnitude below the spectral truncation error for the resolutions considered, thereby supporting the attribution of the observed gains to spectral interpolation. revision: yes

  3. Referee: [§5] §5 (Long-time integrations): the conservation and diffusion comparisons are performed only on the standard Williamson test cases without orography. The extrapolation in the abstract and conclusion that the method will remain advantageous once embedded in primitive-equation models with physics parametrizations therefore rests on an untested assumption.

    Authors: We acknowledge that all reported results are confined to the shallow-water equations without orography. The abstract and conclusions currently frame the work as demonstrating potential benefits for more complex models; this phrasing will be revised to state explicitly that the accuracy, conservation, and diffusion improvements are shown for the idealized SWE test cases, and that extension to orographic flows or primitive-equation models with physics remains future work. We will not claim proven performance beyond the SWE setting. revision: partial

Circularity Check

0 steps flagged

No circularity: results from direct numerical comparison against independent baseline

full rationale

The paper derives a spectral interpolation scheme for SISL shallow-water models by replacing low-order Lagrange interpolation with an NUFFT-accelerated spectral evaluation at departure points, then reports empirical outcomes on standard test cases. No step equates a claimed prediction or first-principles result to its own inputs by construction: the accuracy, conservation, and diffusion improvements are measured quantities obtained by running two otherwise identical codes that differ only in the interpolation operator. The base DFS discretization is cited as prior work but is not invoked to prove uniqueness or to force the new results; the comparison remains externally falsifiable by re-implementing the low-order variant. No fitted parameters are renamed as predictions, no self-citation chain bears the central claim, and no ansatz is smuggled. The logical chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior existence of a stable DFS-based SWE model and on the assumption that NUFFT can be substituted for direct spectral interpolation without introducing new stability or accuracy limits.

axioms (2)
  • domain assumption The double Fourier sphere (DFS) discretization already supplies spectrally accurate horizontal derivatives for the prognostic variables.
    The new interpolation scheme is grafted onto the recently developed DFS-based SWE model referenced in the abstract.
  • domain assumption Nonuniform fast Fourier transforms can evaluate the required spectral interpolants at arbitrary departure points at the same asymptotic cost as the original low-order scheme.
    The abstract states that NUFFT acceleration maintains the same overall computational complexity.

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Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    exponential of semicircle

    A. H. Barnett, J. Magland, and L. af Klinteberg , A parallel nonuniform fast F ourier transform library based on an “exponential of semicircle" kernel , SIAM Journal on Scientific Computing, 41 (2019), pp. C479--C504

    work page 2019

  2. [2]

    https://github.com/flatironinstitute/finufft, 2025

    height 2pt depth -1.6pt width 23pt, F latiron I nstitute nonuniform fast F ourier transform libraries ( FINUFFT ) . https://github.com/flatironinstitute/finufft, 2025

    work page 2025

  3. [3]

    Belkner, A

    S. Belkner, A. J. Duivenvoorden, J. Carron, N. Schaeffer, and M. Reinecke , cunuSHT : GPU accelerated spherical harmonic transforms on arbitrary pixelizations , RAS Techniques and Instruments, 3 (2024), pp. 711--721

    work page 2024

  4. [4]

    G. J. Boer and L. Steinberg , Fourier series on spheres , Atmosphere, 13 (1975), pp. 180--191

    work page 1975

  5. [5]

    J. P. Boyd , Chebyshev and F ourier Spectral Methods , Dover, New York, 2000

    work page 2000

  6. [6]

    Cheong , Application of double F ourier series to the shallow-water equations on a sphere , Journal of Computational Physics, 165 (2000), pp

    H.-B. Cheong , Application of double F ourier series to the shallow-water equations on a sphere , Journal of Computational Physics, 165 (2000), pp. 261--287

    work page 2000

  7. [7]

    327--349

    height 2pt depth -1.6pt width 23pt, Double F ourier series on a sphere: Applications to elliptic and vorticity equations , Journal of Computational Physics, 157 (2000), pp. 327--349

    work page 2000

  8. [8]

    Chiwere and G

    M. Chiwere and G. B. Wright , Barycentric interpolation formulas for the sphere and the disk , BIT Numerical Mathematics, 65 (2025), p. 14

    work page 2025

  9. [9]

    C \^o t \'e and A

    J. C \^o t \'e and A. Staniforth , A two-time-level semi- L agrangian semi-implicit scheme for spectral models , Monthly weather review, 116 (1988), pp. 2003--2012

    work page 1988

  10. [10]

    Dutt and V

    A. Dutt and V. Rokhlin , Fast F ourier transforms for nonequispaced data , SIAM Journal on Scientific computing, 14 (1993), pp. 1368--1393

    work page 1993

  11. [11]

    3, ECMWF, 10/2021 2021

    ECMWF , IFS Documentation CY47R3 - Part III Dynamics and numerical procedures , no. 3, ECMWF, 10/2021 2021

    work page 2021

  12. [12]

    Falcone and R

    M. Falcone and R. Ferretti , Semi- L agrangian Approximation Schemes for Linear and H amilton-- J acobi Equations , SIAM, Philadelphia, 2013

    work page 2013

  13. [13]

    S. N. Figueroa, J. P. Bonatti, P. Y. Kubota, G. A. Grell, H. Morrison, S. R. M. Barros, J. P. R. Fernandez, E. Ramirez, L. Siqueira, G. Luzia, J. Silva, J. R. Silva, J. Pendharkar, V. B. Capistrano, D. S. Alvim, D. P. Enor\'e, F. L. R. Diniz, P. Satyamurti, I. F. A. Cavalcanti, P. Nobre, H. M. J. Barbosa, C. L. Mendes, and J. Panetta , The brazilian globa...

    work page 2016

  14. [14]

    Galewsky, R

    J. Galewsky, R. K. Scott, and L. M. Polvani , An initial-value problem for testing numerical models of the global shallow-water equations , Tellus A: Dynamic Meteorology and Oceanography, 56 (2004), pp. 429--440

    work page 2004

  15. [15]

    Giraldo, J

    F. Giraldo, J. Perot, and P. Fischer , A spectral element semi- L agrangian ( SESL ) method for the spherical shallow water equations , Journal of Computational Physics, 190 (2003), pp. 623--650

    work page 2003

  16. [16]

    Greengard and J.-Y

    L. Greengard and J.-Y. Lee , Accelerating the nonuniform fast F ourier transform , SIAM review, 46 (2004), pp. 443--454

    work page 2004

  17. [17]

    M. Hortal , The development and testing of a new two-time-level semi- L agrangian scheme (SETTLS) in the ECMWF forecast model , Quarterly Journal of the Royal Meteorological Society: A journal of the atmospheric sciences, applied meteorology and physical oceanography, 128 (2002), pp. 1671--1687

    work page 2002

  18. [18]

    Hortal and A

    M. Hortal and A. Simmons , Use of reduced G aussian grids in spectral models , Monthly weather review, 119 (1991), pp. 1057--1074

    work page 1991

  19. [19]

    Keiner, S

    J. Keiner, S. Kunis, and D. Potts , Using NFFT3 - a software library for various nonequispaced fast Fourier transforms , ACM Trans. Math. Software, 36 (2009), pp. Article 19, 1--30

    work page 2009

  20. [20]

    J. N. Koshyk and K. Hamilton , The horizontal kinetic energy spectrum and spectral budget simulated by a high-resolution troposphere--stratosphere--mesosphere GCM , Journal of the atmospheric sciences, 58 (2001), pp. 329--348

    work page 2001

  21. [21]

    S. J. Lambert , A global available potential energy-kinetic energy budget in terms of the two-dimensional wavenumber for the FGGE year , Atmosphere-Ocean, 22 (1984), pp. 265--282

    work page 1984

  22. [22]

    A. T. Layton and W. F. Spotz , A semi- L agrangian double F ourier method for the shallow water equations on the sphere , Journal of Computational Physics, 189 (2003), pp. 180--196

    work page 2003

  23. [23]

    Lin , Python non-uniform fast F ourier transform (PyNUFFT) : An accelerated non- C artesian MRI package on a heterogeneous platform ( CPU / GPU ) , Journal of Imaging, 4 (2018), p

    J.-M. Lin , Python non-uniform fast F ourier transform (PyNUFFT) : An accelerated non- C artesian MRI package on a heterogeneous platform ( CPU / GPU ) , Journal of Imaging, 4 (2018), p. 51

    work page 2018

  24. [24]

    Malardel, N

    S. Malardel, N. Wedi, W. Deconinck, M. Diamantakis, C. K \"u hnlein, G. Mozdzynski, M. Hamrud, and P. Smolarkiewicz , A new grid for the IFS , ECMWF newsletter, 146 (2016), p. 321

    work page 2016

  25. [25]

    McDonald and J

    A. McDonald and J. Bates , Improving the estimate of the departure point position in a two-time level semi- L agrangian and semi-implicit scheme , Monthly Weather Review, 115 (1987), pp. 737--739

    work page 1987

  26. [26]

    Mengaldo, A

    G. Mengaldo, A. Wyszogrodzki, M. Diamantakis, S.-J. Lock, F. X. Giraldo, and N. P. Wedi , Current and emerging time-integration strategies in global numerical weather and climate prediction , Arch. Comput. Methods Eng., 26 (2019), pp. 663--684

    work page 2019

  27. [27]

    P. E. Merilees , The pseudospectral approximation applied to the shallow water equations on a sphere , Atmosphere, 11 (1973), pp. 13--20

    work page 1973

  28. [28]

    height 2pt depth -1.6pt width 23pt, Numerical experiments with the pseudospectral method in spherical coordinates , Atmosphere, 12 (1974), pp. 77--96

    work page 1974

  29. [29]

    Potts and G

    D. Potts and G. Steidl , Fast summation at nonequispaced knots by NFFT , SIAM Journal on Scientific Computing, 24 (2003), pp. 2013--2037

    work page 2003

  30. [30]

    Ritchie , Application of the semi- L agrangian method to a spectral model of the shallow water equations , Monthly Weather Review, 116 (1988), pp

    H. Ritchie , Application of the semi- L agrangian method to a spectral model of the shallow water equations , Monthly Weather Review, 116 (1988), pp. 1587--1598

    work page 1988

  31. [31]

    Ritchie and M

    H. Ritchie and M. Tanguay , A comparison of spatially averaged E ulerian and semi- L agrangian treatments of mountains , Monthly Weather Review, 124 (1996), p. 167

    work page 1996

  32. [32]

    Robert , The integration of a spectral model of the atmosphere by the implicit method , in Proc

    A. Robert , The integration of a spectral model of the atmosphere by the implicit method , in Proc. WMO/IUGG Symposium on NWP, Tokyo, Japan Meteorological Agency, vol. 7, 1969, pp. 19--24

    work page 1969

  33. [33]

    height 2pt depth -1.6pt width 23pt, A stable numerical integration scheme for the primitive meteorological equations , Atmosphere-Ocean, 19 (1981), pp. 35--46

    work page 1981

  34. [34]

    Robert , A semi- L agrangian and semi-implicit numerical integration scheme for the primitive meteorological equations , Journal of the Meteorological Society of Japan

    A. Robert , A semi- L agrangian and semi-implicit numerical integration scheme for the primitive meteorological equations , Journal of the Meteorological Society of Japan. Ser. II, 60 (1982), pp. 319--325

    work page 1982

  35. [35]

    Robert, J

    A. Robert, J. Henderson, and C. Turnbull , An implicit time integration scheme for baroclinic models of the atmosphere , Monthly Weather Review, 100 (1972), pp. 329--335

    work page 1972

  36. [36]

    Ruiz-Antolin and A

    D. Ruiz-Antolin and A. Townsend , A nonuniform fast F ourier transform based on low rank approximation , SIAM Journal on Scientific Computing, 40 (2018), pp. A529--A547

    work page 2018

  37. [37]

    Shen , Efficient spectral- G alerkin methods IV

    J. Shen , Efficient spectral- G alerkin methods IV . S pherical geometries , SIAM Journal on Scientific Computing, 20 (1999), pp. 1438--1455

    work page 1999

  38. [38]

    Slevinsky , FastTransforms

    M. Slevinsky , FastTransforms . https://github.com/MikaelSlevinsky/FastTransforms, 2024. Version: 2024-09-03

    work page 2024

  39. [39]

    R. M. Slevinsky , Fast and backward stable transforms between spherical harmonic expansions and bivariate F ourier series , Applied and Computational Harmonic Analysis, 47 (2019), pp. 585--606

    work page 2019

  40. [40]

    W. F. Spotz, M. A. Taylor, and P. N. Swarztrauber , Fast shallow-water equation solvers in latitude-longitude coordinates , Journal of Computational Physics, 145 (1998), pp. 432--444

    work page 1998

  41. [41]

    Staniforth and J

    A. Staniforth and J. C \^o t \'e , Semi- L agrangian integration schemes for atmospheric models--- A review , Monthly weather review, 119 (1991), pp. 2206--2223

    work page 1991

  42. [42]

    Temperton , Treatment of the Coriolis terms in semi- L agrangian spectral models , Atmosphere-Ocean, 35 (1997), pp

    C. Temperton , Treatment of the Coriolis terms in semi- L agrangian spectral models , Atmosphere-Ocean, 35 (1997), pp. 293--302

    work page 1997

  43. [43]

    Temperton, M

    C. Temperton, M. Hortal, and A. Simmons , A two-time-level semi- L agrangian global spectral model , Quarterly Journal of the Royal Meteorological Society, 127 (2001), pp. 111--127

    work page 2001

  44. [44]

    Temperton and A

    C. Temperton and A. Staniforth , An efficient two-time-level semi- L agrangian semi-implicit integration scheme , Quarterly Journal of the Royal Meteorological Society, 113 (1987), pp. 1025--1039

    work page 1987

  45. [45]

    Thuburn and Y

    J. Thuburn and Y. Li , Numerical simulations of Rossby -- Haurwitz waves , Tellus A, 52 (2000), pp. 181--189

    work page 2000

  46. [46]

    Tolstykh, V

    M. Tolstykh, V. Shashkin, R. Fadeev, and G. Goyman , Vorticity-divergence semi- L agrangian global atmospheric model SL-AV20 : D ynamical core , Geoscientific Model Development, 10 (2017), pp. 1961--1983

    work page 2017

  47. [47]

    Townsend, H

    A. Townsend, H. Wilber, and G. B. Wright , Computing with functions in spherical and polar geometries I . the sphere , SIAM Journal on Scientific Computing, 38 (2016), pp. C403--C425

    work page 2016

  48. [48]

    Tygert , Fast algorithms for spherical harmonic expansions, III , Journal of Computational Physics, 229 (2010), pp

    M. Tygert , Fast algorithms for spherical harmonic expansions, III , Journal of Computational Physics, 229 (2010), pp. 6181--6192

    work page 2010

  49. [49]

    L. Wang, Y. Zhang, J. Li, Z. Liu, and Y. Zhou , Understanding the performance of an unstructured-mesh global shallow water model on kinetic energy spectra and nonlinear vorticity dynamics , Journal of Meteorological Research, 33 (2019), pp. 1075--1097

    work page 2019

  50. [50]

    N. P. Wedi, M. Hamrud, and G. Mozdzynski , A fast spherical harmonics transform for global NWP and climate models , Monthly Weather Review, 141 (2013), pp. 3450 -- 3461

    work page 2013

  51. [51]

    Wilber, E

    H. Wilber, E. N. Epperly, and A. H. Barnett , Superfast direct inversion of the nonuniform discrete F ourier transform via hierarchically semiseparable least squares , SIAM Journal on Scientific Computing, 47 (2025), pp. A1702--A1732

    work page 2025

  52. [52]

    D. L. Williamson, J. B. Drake, J. J. Hack, R. Jakob, and P. N. Swarztrauber , A standard test set for numerical approximations to the shallow water equations in spherical geometry , Journal of computational physics, 102 (1992), pp. 211--224

    work page 1992

  53. [53]

    G. B. Wright, M. Javed, H. Montanelli, and L. N. Trefethen , Extension of C hebfun to periodic functions , SIAM Journal on Scientific Computing, 37 (2015), pp. C554--C573

    work page 2015

  54. [54]

    S. Y. K. Yee , Solution of P oisson's equation on a sphere by truncated double F ourier series , Monthly Weather Review, 109 (1981), pp. 501--505

    work page 1981

  55. [55]

    H. Yoshimura , Development of a nonhydrostatic global spectral atmospheric model using double F ourier series , CAS/JSC WGNE Research Activities in Atmospheric and Ocean Modeling, 42 (2012), pp. 3--05

    work page 2012

  56. [56]

    2561--2597

    height 2pt depth -1.6pt width 23pt, Improved double F ourier series on a sphere and its application to a semi-implicit semi- L agrangian shallow-water model , Geoscientific Model Development, 15 (2022), pp. 2561--2597

    work page 2022

  57. [57]

    Yoshimura and T

    H. Yoshimura and T. Matsumura , A two-time-level vertically-conservative semi- L agrangian semi-implicit double F ourier series agcm , CAS/JSC WGNE Research Activities in Atmospheric and Ocean Modeling, 35 (2005), pp. 27--28

    work page 2005

  58. [58]

    Yukimoto , Meteorological Research Institute-Earth System Model Version 1 (MRI-ESM1) , Meteorolog

    S. Yukimoto , Meteorological Research Institute-Earth System Model Version 1 (MRI-ESM1) , Meteorolog. Research Inst., 2011

    work page 2011

  59. [59]

    Yukimoto, H

    S. Yukimoto, H. Kawai, T. Koshiro, N. Oshima, K. Yoshida, S. Urakawa, H. Tsujino, M. Deushi, T. Tanaka, M. Hosaka, et al. , The Meteorological Research Institute Earth System Model version 2.0, MRI-ESM2 . 0: D escription and basic evaluation of the physical component , Journal of the Meteorological Society of Japan. Ser. II, 97 (2019), pp. 931--965

    work page 2019