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arxiv: 2606.04882 · v2 · pith:UX3TL724new · submitted 2026-06-03 · 🧮 math.NA · cs.NA

Arbitrary high order shaped stencils for time domain finite difference schemes in seismic wave propagation

Pedro S. Peixoto , Felipe A. G. Silva This is my paper

Pith reviewed 2026-06-28 05:01 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords finite difference schemesseismic wave propagationacoustic wave equationstencil geometrynumerical dispersionhigh-order methodswave modeling
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The pith

Dispersion-optimized cross stencils achieve adequate accuracy in acoustic wave modeling at lower computational cost than rhombus or square shapes.

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

The paper develops and tests a general framework for high-order finite difference approximations to the acoustic wave equation using stencils of varying shapes, from classical crosses to filled rhombi and squares. Classical Taylor-based methods suffer from numerical dispersion, so the work optimizes coefficients for different geometries and compares their performance through symbolic implementation. Tests on idealized and realistic velocity models show that non-cross shapes increase cost without reliably improving accuracy or reducing dispersion. Dispersion-optimized cross stencils on compact supports deliver comparable results at lower expense. This matters for large-scale seismic simulations where efficiency directly affects feasibility of full inversions.

Core claim

A general mathematical framework is proposed for deriving high-order in space finite difference schemes for acoustic wave propagation featuring different stencil geometries ranging from classical cross stencils to stencils with rhombus or square-like shapes. Non-cross stencil shapes such as rhombus and square-based stencils do not necessarily provide added accuracy or dispersion reduction in general despite their increased computational cost, while results confirm the benefits of using dispersion-optimized cross-stencils indicating adequate accuracy with reduced computational cost on more compact stencils compared to classic approaches.

What carries the argument

The general mathematical framework for deriving finite difference stencils with geometries ranging from cross to rhombus or square-like shapes, implemented via symbolic compilation.

If this is right

  • Optimized cross stencils can support full-scale acoustic seismic inversion at lower cost while retaining accuracy.
  • Symbolic compilation enables rapid testing and deployment of varied stencil geometries for wave modeling.
  • Classic Taylor series expansions can be replaced by dispersion-optimized coefficients on compact cross supports.
  • Non-cross shapes may be avoided in favor of cross stencils for routine seismic propagation tasks.

Where Pith is reading between the lines

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

  • The framework could extend directly to elastic or anisotropic wave equations without changing the derivation approach.
  • Compact optimized cross stencils may allow finer grids within fixed compute budgets for higher-resolution imaging.
  • Dispersion optimization focused only on cross shapes could yield further efficiency gains in 3D settings.

Load-bearing premise

The chosen idealized and realistic velocity models together with the symbolic compiler are sufficient to show that non-cross stencil shapes lack general superiority across seismic applications.

What would settle it

A comparison on a velocity model with stronger heterogeneities or higher frequencies where a rhombus stencil produces measurably lower dispersion error per unit cost than the optimized cross stencil.

read the original abstract

Finite Difference Schemes are widely used in the approximation of different hyperbolic (wave-like) differential equations, and are particularly important for seismic wave modelling and its applications. Classical methods based on Taylor Series are dominant in the literature; however, it is known that these methods can suffer from excessive numerical dispersion. In this paper, we review and extend existing high-order in space finite difference schemes for acoustic wave propagation, featuring different stencil geometries ranging from classical cross stencils to stencils with rhombus or square-like shapes, and propose a general mathematical framework for their derivation. The numerical implementation is performed in a symbolic, high-level framework (Devito), which compiles and runs highly optimized, stencil-based computations, allowing for a low-level interpretation of the methods efficiency. We demonstrate that non-cross stencil shapes, such as rhombus and square-based stencils, do not necessarily provide added accuracy or dispersion reduction in general, despite their increased computational cost. However, results on both idealized and realistic velocity models confirm the benefits of using dispersion-optimized cross-stencils, indicating adequate accuracy with reduced computational cost on more compact stencils compared to classic approaches. Finally, our implementation of the methods provides ease of use for full-scale acoustic seismic inversion problems using Devito.

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 / 1 minor

Summary. The paper reviews and extends high-order finite difference schemes for acoustic wave propagation, presenting a general mathematical framework for deriving stencils of varying geometries (cross, rhombus, square-like). It implements these in the Devito symbolic framework for optimized computations and uses numerical tests on idealized and realistic velocity models to claim that non-cross stencils do not necessarily improve accuracy or reduce dispersion despite higher cost, while dispersion-optimized cross-stencils achieve adequate accuracy at lower computational cost.

Significance. If the numerical findings hold under broader conditions, the work could inform more efficient stencil selection in seismic modeling applications. The symbolic derivation and compilation approach in Devito is a clear strength, enabling reproducible high-level specification of stencils and low-level optimization.

major comments (2)
  1. [Abstract, §4] Abstract and §4 (numerical results): The central claim that non-cross stencils 'do not necessarily provide added accuracy or dispersion reduction in general' rests entirely on tests with specific idealized and realistic velocity models via Devito; no theoretical dispersion analysis or sensitivity study across frequencies, dimensions, or heterogeneity levels is supplied to underwrite the generalization, leaving the 'in general' qualifier unsecured.
  2. [Abstract] Abstract: Conclusions are drawn from numerical tests without reporting quantitative error metrics (e.g., L2 or dispersion error norms), dispersion curves, or the precise procedure and parameters used for dispersion optimization, preventing independent verification of the accuracy and cost claims.
minor comments (1)
  1. [Abstract] The abstract would be strengthened by including at least one key quantitative result (e.g., error reduction factor or runtime comparison) to support the stated benefits of optimized cross-stencils.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comments below and will revise the manuscript to improve the precision of our claims and the verifiability of the results.

read point-by-point responses

  1. Referee: [Abstract, §4] Abstract and §4 (numerical results): The central claim that non-cross stencils 'do not necessarily provide added accuracy or dispersion reduction in general' rests entirely on tests with specific idealized and realistic velocity models via Devito; no theoretical dispersion analysis or sensitivity study across frequencies, dimensions, or heterogeneity levels is supplied to underwrite the generalization, leaving the 'in general' qualifier unsecured.

    Authors: We agree that the qualifier 'in general' is not fully supported by a theoretical analysis and rests on the numerical tests performed. The paper's primary contribution is the general mathematical framework for deriving stencils of arbitrary shapes together with its Devito implementation; the numerical section demonstrates the practical outcome on the chosen models. We will revise the abstract and §4 to qualify the statement (e.g., 'in the tested configurations') and will consider adding a limited sensitivity study if space permits. A comprehensive theoretical dispersion analysis across all parameter regimes lies outside the present scope. revision: partial

  2. Referee: [Abstract] Abstract: Conclusions are drawn from numerical tests without reporting quantitative error metrics (e.g., L2 or dispersion error norms), dispersion curves, or the precise procedure and parameters used for dispersion optimization, preventing independent verification of the accuracy and cost claims.

    Authors: Section 4 of the manuscript already presents quantitative comparisons of accuracy and computational cost, including error measures for the different stencil geometries. To improve verifiability we will (i) add explicit references in the abstract to the key quantitative metrics shown in §4, (ii) ensure the dispersion-optimization procedure and all parameters are stated with full precision in the methods, and (iii) include representative dispersion curves as an additional figure or supplementary material. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on independent numerical experiments

full rationale

The paper proposes a general mathematical framework for deriving high-order FD stencils of varying geometries and validates conclusions via numerical tests on idealized and realistic velocity models implemented in Devito. No load-bearing step reduces a prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames an input as an output. The central empirical finding (non-cross stencils do not add accuracy in general) is presented as a result of the reported runs rather than a definitional identity. This is the common honest case of a self-contained numerical study.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies insufficient detail to enumerate free parameters, axioms, or invented entities; full text would be required to audit the derivation framework and any dispersion-optimization constants.

pith-pipeline@v0.9.1-grok · 5757 in / 1090 out tokens · 27534 ms · 2026-06-28T05:01:28.659958+00:00 · methodology

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Reference graph

Works this paper leans on

104 extracted references · 4 canonical work pages

  1. [1]

    Geophysics9(8), 259–1266 (1984)

    Tarantola, A.: Inversion of seismic reflection data in the acoustic approximation. Geophysics9(8), 259–1266 (1984)

    1984

  2. [2]

    Geophysics74(6), 1–26 (2009)

    Virieux, J., Operto, S.: An overview of full-waveform inversion in exploration geophysics. Geophysics74(6), 1–26 (2009)

    2009

  3. [3]

    In: Seismic Waves-research and Analysis, pp

    Tago, J., Cruz-Atienza, V., Chaljub, E., Brossier, R., Coutant, O., Garambois, S., Prieux, V., Operto, S., Mercerat, D., Virieux, J.,et al.: Modelling seis- mic wave propagation for geophysical imaging. In: Seismic Waves-research and Analysis, pp. 253–30413. IntechOpen, ??? (2012)

    2012

  4. [4]

    Mathematics of computation51(184), 699–706 (1988)

    Fornberg, B.: Generation of finite difference formulas on arbitrarily spaced grids. Mathematics of computation51(184), 699–706 (1988)

    1988

  5. [5]

    SIAM, ??? (2004)

    Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations. SIAM, ??? (2004)

    2004

  6. [6]

    Collatz, L.: The Numerical Treatment of Differential Equations vol. 60. Springer, ??? (2012)

    2012

  7. [7]

    Geophysics49(11), 1933–1942 (1984)

    Virieux, J.: Sh-wave propagation in heterogeneous media: Velocity-stress finite- difference method. Geophysics49(11), 1933–1942 (1984)

    1933

  8. [8]

    Geophysics51(4), 889–901 (1986)

    Virieux, J.: P-sv wave propagation in heterogeneous media: Velocity-stress finite- difference method. Geophysics51(4), 889–901 (1986)

    1986

  9. [9]

    Geophysics 53(11), 1425–1436 (1988)

    Levander, A.R.: Fourth-order finite-difference p-sv seismograms. Geophysics 53(11), 1425–1436 (1988)

    1988

  10. [10]

    In: Detection and Remediation Technologies for Mines and Minelike Targets IV, vol

    Schroeder, C.T., Scott Jr, W.R.: Finite-difference time-domain model for elastic waves in the ground. In: Detection and Remediation Technologies for Mines and Minelike Targets IV, vol. 3710, pp. 1361–1372 (1999). International Society for Optics and Photonics

    1999

  11. [11]

    Computers & geosciences32(2), 259–264 (2006)

    Boyd, O.S.: An efficient matlab script to calculate heterogeneous anisotropically elastic wave propagation in three dimensions. Computers & geosciences32(2), 259–264 (2006)

    2006

  12. [12]

    Geophysical Journal International187(3), 1645–1667 (2011)

    Moczo, P., Kristek, J., Galis, M., Chaljub, E., Etienne, V.: 3-d finite-difference, finite-element, discontinuous-galerkin and spectral-element schemes analysed for their accuracy with respect to p-wave to s-wave speed ratio. Geophysical Journal International187(3), 1645–1667 (2011)

    2011

  13. [13]

    Cambridge University Press, ??? (2014) 54

    Moczo, P., Kristek, J., G´ alis, M.: The Finite-difference Modelling of Earthquake Motions: Waves and Ruptures. Cambridge University Press, ??? (2014) 54

    2014

  14. [14]

    Geophysical Journal International206(3), 1933–1958 (2016)

    Wang, E., Liu, Y., Sen, M.K.: Effective finite-difference modelling methods with 2-d acoustic wave equation using a combination of cross and rhombus stencils. Geophysical Journal International206(3), 1933–1958 (2016)

    1933

  15. [15]

    Geophysics82(4), 143–157 (2017)

    Li, B., Liu, Y., Sen, M.K., Ren, Z.: Time-space-domain mesh-free finite difference based on least squares for 2d acoustic-wave modeling. Geophysics82(4), 143–157 (2017)

    2017

  16. [16]

    Computers & Geosciences, 104948 (2021)

    Chen, G., Peng, Z., Li, Y.: A framework for automatically choosing the optimal parameters of finite-difference scheme in the acoustic wave modelling. Computers & Geosciences, 104948 (2021)

    2021

  17. [17]

    Geophysics 79(5), 277–285 (2014)

    Wang, Y., Liang, W., Nashed, Z., Li, X., Liang, G., Yang, C.: Seismic mod- eling by optimizing regularized staggered-grid finite-difference operators using a time-space-domain dispersion-relationship-preserving method. Geophysics 79(5), 277–285 (2014)

    2014

  18. [18]

    Advances in Difference Equations2015(1), 1–17 (2015)

    Sescu, A.: Numerical anisotropy in finite differencing. Advances in Difference Equations2015(1), 1–17 (2015)

    2015

  19. [19]

    Geosci- entific Model Development12(3), 1165–1187 (2019) https://doi.org/10.5194/ gmd-12-1165-2019

    Louboutin, M., Lange, M., Luporini, F., Kukreja, N., Witte, P.A., Herrmann, F.J., Velesko, P., Gorman, G.J.: Devito (v3.1.0): an embedded domain- specific language for finite differences and geophysical exploration. Geosci- entific Model Development12(3), 1165–1187 (2019) https://doi.org/10.5194/ gmd-12-1165-2019

    2019

  20. [20]

    Architecture and performance of Devito, a system for automated stencil computation,

    Luporini, F., Lange, M., Louboutin, M., Kukreja, N., H¨ uckelheim, J., Yount, C., Witte, P., Kelly, P.H.J., Herrmann, F.J., Gorman, G.J.: Architecture and performance of devito, a system for automated stencil computation. CoRR abs/1807.03032(2018) arXiv:1807.03032

    work page arXiv 2018

  21. [21]

    In: 2016 Sixth Interna- tional Workshop on Domain-Specific Languages and High-Level Frameworks for High Performance Computing (WOLFHPC), pp

    Kukreja, N., Louboutin, M., Vieira, F., Luporini, F., Lange, M., Gorman, G.: Devito: Automated fast finite difference computation. In: 2016 Sixth Interna- tional Workshop on Domain-Specific Languages and High-Level Frameworks for High Performance Computing (WOLFHPC), pp. 11–19 (2016). IEEE

    2016

  22. [22]

    Geoscientific Model Development Discussions, 1–36 (2022)

    Dolci, D.I., Silva, F.A., Peixoto, P.S., Volpe, E.V.: Effectiveness and computa- tional efficiency of absorbing boundary conditions for full-waveform inversion. Geoscientific Model Development Discussions, 1–36 (2022)

    2022

  23. [23]

    Geophysics 65(4), 1239–1250 (2000)

    Alkhalifah, T.: An acoustic wave equation for anisotropic media. Geophysics 65(4), 1239–1250 (2000)

    2000

  24. [24]

    Geophysics74(4), 29–33 (2009)

    Zhang, Y., Zhang, G.: One-step extrapolation method for reverse time migration. Geophysics74(4), 29–33 (2009)

    2009

  25. [25]

    In: SEG International Exposition and Annual Meeting, p

    Etgen, J.T., Brandsberg-Dahl, S.: The pseudo-analytical method: Application 55 of pseudo-laplacians to acoustic and acoustic anisotropic wave propagation. In: SEG International Exposition and Annual Meeting, p. 2009 (2009). SEG

    2009

  26. [26]

    In: SEG Technical Program Expanded Abstracts, vol

    Zhan, G., Pestana, R.C., Stoffa, P.L.: An acoustic wave equation for pure p wave in 2d tti media. In: SEG Technical Program Expanded Abstracts, vol. 30, pp. 168–173 (2011). Society of Exploration Geophysicists

    2011

  27. [27]

    Geophysics 48(11), 1514–1524 (1983)

    Baysal, E., Kosloff, D.D., Sherwood, J.W.: Reverse time migration. Geophysics 48(11), 1514–1524 (1983)

    1983

  28. [28]

    McMechan, G.A.: Migration by extrapolation of time-dependent boundary values (2008)

    2008

  29. [29]

    In: SEG Technical Program Expanded Abstracts 1983, pp

    Whitmore, N.D.: Iterative depth migration by backward time propagation. In: SEG Technical Program Expanded Abstracts 1983, pp. 382–385. Society of Exploration Geophysicists, ??? (1983)

    1983

  30. [30]

    Geophysics76(3), 3–11 (2011)

    Zhang, Y., Zhang, H., Zhang, G.: A stable tti reverse time migration and its implementation. Geophysics76(3), 3–11 (2011)

    2011

  31. [31]

    Geophysics80(1), 23–31 (2015)

    Zhang, Y., Duan, L., Xie, Y.: A stable and practical implementation of least- squares reverse time migration. Geophysics80(1), 23–31 (2015)

    2015

  32. [32]

    Geophysical Journal International207(1), 129–149 (2016)

    Yang, P., Brossier, R., M´ etivier, L., Virieux, J.: A review on the systematic for- mulation of 3-d multiparameter full waveform inversion in viscoelastic medium. Geophysical Journal International207(1), 129–149 (2016)

    2016

  33. [33]

    Methods of Computational Physics 4, 349–379 (1965)

    Abramovici, F., Alterman, Z.: Computations pertaining to the problem of prop- agation of a seismic pulse in a layered solid. Methods of Computational Physics 4, 349–379 (1965)

    1965

  34. [34]

    Journal of Physics of the Earth16(Special), 113–128 (1968) https://doi.org/10.4294/ jpe1952.16.Special 113

    Alternam, Z.S.: Finite difference solutions to geophysical problems. Journal of Physics of the Earth16(Special), 113–128 (1968) https://doi.org/10.4294/ jpe1952.16.Special 113

    1968

  35. [35]

    Bulletin of the Seismological Society of America58(1), 367–398 (1968)

    Alterman, Z., Karal Jr, F.: Propagation of elastic waves in layered media by finite difference methods. Bulletin of the Seismological Society of America58(1), 367–398 (1968)

    1968

  36. [36]

    Geophysics39(6), 834–842 (1974)

    Alford, R., Kelly, K., Boore, D.M.: Accuracy of finite-difference modeling of the acoustic wave equation. Geophysics39(6), 834–842 (1974)

    1974

  37. [37]

    Geophysics41(1), 2–27 (1976)

    Kelly, K.R., Ward, R.W., Treitel, S., Alford, R.M.: Synthetic seismograms: A finite-difference approach. Geophysics41(1), 2–27 (1976)

    1976

  38. [38]

    Geophysics51(1), 54–66 (1986) 56

    Dablain, M.: The application of high-order differencing to the scalar wave equation. Geophysics51(1), 54–66 (1986) 56

    1986

  39. [39]

    Stanford Exploration Project (1986)

    Etgen, J.T.: High-order finite-difference reverse time migration with the 2-way non-reflecting wave equation. Stanford Exploration Project (1986)

    1986

  40. [40]

    Stanford Exploration Project Reports57, 243–258 (1988)

    Etgen, J.: Evaluating finite-difference operators applied to wave simulation. Stanford Exploration Project Reports57, 243–258 (1988)

    1988

  41. [41]

    Advances in geophysics48, 421–516 (2007)

    Moczo, P., Robertsson, J.O., Eisner, L.: The finite-difference time-domain method for modeling of seismic wave propagation. Advances in geophysics48, 421–516 (2007)

    2007

  42. [42]

    Journal of computational physics103(1), 16–42 (1992)

    Lele, S.K.: Compact finite difference schemes with spectral-like resolution. Journal of computational physics103(1), 16–42 (1992)

    1992

  43. [43]

    Journal of computational physics107(2), 262–281 (1993)

    Tam, C.K., Webb, J.C.: Dispersion-relation-preserving finite difference schemes for computational acoustics. Journal of computational physics107(2), 262–281 (1993)

    1993

  44. [44]

    AIAA journal34(5), 887–893 (1996)

    Kim, J.W., Lee, D.J.: Optimized compact finite difference schemes with maxi- mum resolution. AIAA journal34(5), 887–893 (1996)

    1996

  45. [45]

    IEEE Transactions on Microwave Theory and Techniques43(9), 2053–2058 (1995)

    Cole, J.B.: A high accuracy fdtd algorithm to solve microwave propagation and scattering problems on a coarse grid. IEEE Transactions on Microwave Theory and Techniques43(9), 2053–2058 (1995)

    2053

  46. [46]

    IEEE transactions on Microwave Theory and Techniques52(4), 1321–1327 (2004)

    Zygiridis, T.T., Tsiboukis, T.D.: Low-dispersion algorithms based on the higher order (2, 4) fdtd method. IEEE transactions on Microwave Theory and Techniques52(4), 1321–1327 (2004)

    2004

  47. [47]

    IEEE transactions on microwave theory and techniques53(3), 832–842 (2005)

    Sun, G., Trueman, C.W.: Optimized finite-difference time-domain methods based on the (2, 4) stencil. IEEE transactions on microwave theory and techniques53(3), 832–842 (2005)

    2005

  48. [48]

    In: 2006 IEEE Antennas and Propagation Society International Symposium, pp

    Finkelstein, B., Kastner, R.: Dispersion reduction schemes for the fdtd wave equation. In: 2006 IEEE Antennas and Propagation Society International Symposium, pp. 3797–3800 (2006). IEEE

    2006

  49. [49]

    IEEE Transactions on Antennas and Propagation56(11), 3516–3525 (2008)

    Finkelstein, B., Kastner, R.: A comprehensive new methodology for formu- lating fdtd schemes with controlled order of accuracy and dispersion. IEEE Transactions on Antennas and Propagation56(11), 3516–3525 (2008)

    2008

  50. [50]

    IEEE Transactions on Antennas and Propagation46(8), 1194–1201 (1998)

    Nehrbass, J.W., Jevtic, J.O., Lee, R.: Reducing the phase error for finite- difference methods without increasing the order. IEEE Transactions on Antennas and Propagation46(8), 1194–1201 (1998)

    1998

  51. [51]

    IEEE transactions on Electromagnetic Compatibility41(2), 160–165 (1999) 57

    Lan, K., Liu, Y., Lin, W.: A higher order (2, 4) scheme for reducing dispersion in fdtd algorithm. IEEE transactions on Electromagnetic Compatibility41(2), 160–165 (1999) 57

    1999

  52. [52]

    Yang, B., Balanis, C.: An isotropy-improved nonstandard finite difference (2,

  53. [53]

    In: 2005 IEEE Antennas and Propagation Society International Symposium, vol

    scheme. In: 2005 IEEE Antennas and Propagation Society International Symposium, vol. 1, pp. 106–109 (2005). IEEE

    2005

  54. [54]

    IEEE Antennas and Wireless Propagation Letters5, 450–453 (2006)

    Yang, B., Balanis, C.A.: Least square method to optimize the coefficients of com- plex finite-difference space stencils. IEEE Antennas and Wireless Propagation Letters5, 450–453 (2006)

    2006

  55. [55]

    Journal of Computational Physics228(24), 8958–8984 (2009)

    Finkelstein, B., Kastner, R.: The spectral order of accuracy: A new unified tool in the design methodology of excitation-adaptive wave equation fdtd schemes. Journal of Computational Physics228(24), 8958–8984 (2009)

    2009

  56. [56]

    Geophysics52(4), 483–501 (1987)

    Fornberg, B.: The pseudospectral method: Comparisons with finite differences for the elastic wave equation. Geophysics52(4), 483–501 (1987)

    1987

  57. [57]

    Journal of computational Physics228(23), 8779– 8806 (2009)

    Liu, Y., Sen, M.K.: A new time–space domain high-order finite-difference method for the acoustic wave equation. Journal of computational Physics228(23), 8779– 8806 (2009)

    2009

  58. [58]

    Geophysics78(4), 113–132 (2013)

    Liu, Y.: Globally optimal finite-difference schemes based on least squares. Geophysics78(4), 113–132 (2013)

    2013

  59. [59]

    Geophysical Journal International148(2), 320–328 (2002)

    Yang, D., Liu, E., Zhang, Z., Teng, J.: Finite-difference modelling in two- dimensional anisotropic media using a flux-corrected transport technique. Geophysical Journal International148(2), 320–328 (2002)

    2002

  60. [60]

    Geophysical prospecting35(6), 629–655 (1987)

    Holberg, O.: Computational aspects of the choice of operator and sampling inter- val for numerical differentiation in large-scale simulation of wave phenomena. Geophysical prospecting35(6), 629–655 (1987)

    1987

  61. [61]

    Geophysical Prospecting41(4), 453–458 (1993)

    Jastram, C., Behle, A.: Accurate finite-difference operators for modelling the elastic wave equation1. Geophysical Prospecting41(4), 453–458 (1993)

    1993

  62. [62]

    Chinese Journal of Geophysics 43(3), 441–449 (2000)

    DONG, L.-G., MA, Z.-T., CAO, J.-Z.: Staggered-grid high-order difference method for the first-order elastic wave equations. Chinese Journal of Geophysics 43(3), 441–449 (2000)

    2000

  63. [63]

    Physics of the earth and planetary interiors119(1-2), 99–131 (2000)

    Takeuchi, N., Geller, R.J.: Optimally accurate second order time-domain finite difference scheme for computing synthetic seismograms in 2-d and 3-d media. Physics of the earth and planetary interiors119(1-2), 99–131 (2000)

    2000

  64. [64]

    beyond holberg

    Etgen, J.T.: A tutorial on optimizing time domain finite-difference schemes:“beyond holberg”. Stanford Exploration Project Report129, 33–43 (2007)

    2007

  65. [65]

    Numerical Methods for Partial Differential Equations: An Interna- tional Journal20(3), 463–480 (2004)

    Bilbao, S.: Parameterized families of finite difference schemes for the wave 58 equation. Numerical Methods for Partial Differential Equations: An Interna- tional Journal20(3), 463–480 (2004)

    2004

  66. [66]

    Applied Numerical Mathematics57(4), 402–414 (2007)

    Kim, S., Lim, H.: High-order schemes for acoustic waveform simulation. Applied Numerical Mathematics57(4), 402–414 (2007)

    2007

  67. [67]

    Geophysical Journal International135(1), 48–62 (1998)

    Geller, R.J., Takeuchi, N.: Optimally accurate second-order time-domain finite difference scheme for the elastic equation of motion: one-dimensional cases. Geophysical Journal International135(1), 48–62 (1998)

    1998

  68. [68]

    Computers & Geosciences100, 142–155 (2017)

    Fabien-Ouellet, G., Gloaguen, E., Giroux, B.: Time-domain seismic modeling in viscoelastic media for full waveform inversion on heterogeneous computing platforms with opencl. Computers & Geosciences100, 142–155 (2017)

    2017

  69. [69]

    arXiv preprint arXiv:2201.05278 (2022)

    Souza, J.F., Moreira, J.B.D., Roberts, K.J., Gaioso, R.d.R.A., Gomi, E.S., Silva, E.C.N., Senger, H.: simwave-a finite difference simulator for acoustic waves propagation. arXiv preprint arXiv:2201.05278 (2022)

    work page arXiv 2022

  70. [70]

    Computers & Fluids35(10), 1459–1463 (2006)

    Berland, J., Bogey, C., Bailly, C.: Low-dissipation and low-dispersion fourth- order runge–kutta algorithm. Computers & Fluids35(10), 1459–1463 (2006)

    2006

  71. [71]

    Geophysics 72(5), 115–122 (2007)

    Chen, J.-B.: High-order time discretizations in seismic modeling. Geophysics 72(5), 115–122 (2007)

    2007

  72. [72]

    Geophysics61(3), 845–856 (1996)

    Wu, W.-J., Lines, L.R., Lu, H.-X.: Analysis of higher-order, finite-difference schemes in 3-d reverse-time migration. Geophysics61(3), 845–856 (1996)

    1996

  73. [73]

    Mathematical Problems in Engineering 2019(2019)

    Cheng, D., Chen, B., Chen, X.: A robust optimal finite difference scheme for the three-dimensional helmholtz equation. Mathematical Problems in Engineering 2019(2019)

    2019

  74. [74]

    Applied Geophysics5(2), 107–114 (2008)

    Yang, L., Xiucheng, W.: Finite-difference numerical modeling with even-order accuracy in two-phase anisotropic media. Applied Geophysics5(2), 107–114 (2008)

    2008

  75. [75]

    Applied Geophysics6(3), 259 (2009)

    Liao, J., Wang, H., Ma, Z.: 2-d elastic wave modeling with frequency-space 25-point finite-difference operators. Applied Geophysics6(3), 259 (2009)

    2009

  76. [76]

    Geophysical Journal International197(2), 1033– 1047 (2014)

    Liu, Y.: Optimal staggered-grid finite-difference schemes based on least-squares for wave equation modelling. Geophysical Journal International197(2), 1033– 1047 (2014)

    2014

  77. [77]

    Geophysics80(1), 17–40 (2015)

    Ren, Z., Liu, Y.: Acoustic and elastic modeling by optimal time-space-domain staggered-grid finite-difference schemes. Geophysics80(1), 17–40 (2015)

    2015

  78. [78]

    Geophysics82(1), 27–42 (2017) 59

    Yang, L., Yan, H., Liu, H.: Optimal staggered-grid finite-difference schemes based on the minimax approximation method with the remez algorithm. Geophysics82(1), 27–42 (2017) 59

    2017

  79. [79]

    part 1: Application to 1-d isogeometric elements

    Idesman, A.: Optimal reduction of numerical dispersion for wave propagation problems. part 1: Application to 1-d isogeometric elements. Computer Methods in Applied Mechanics and Engineering317, 970–992 (2017)

    2017

  80. [80]

    Journal of Geophysics and Engineering6(3), 231–249 (2009)

    Liu, Y., Sen, M.K.: A practical implicit finite-difference method: examples from seismic modelling. Journal of Geophysics and Engineering6(3), 231–249 (2009)

    2009

Showing first 80 references.