Inviscid water-waves and interface modeling
Authors:
Emmanuel Dormy and Christophe Lacave
Journal:
Quart. Appl. Math.
MSC (2020):
Primary 76B15, 65M22; Secondary 35R37, 35Q31
DOI:
https://doi.org/10.1090/qam/1685
Published electronically:
January 19, 2024
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Abstract: We present a rigorous mathematical analysis of the modeling of inviscid water waves. The free-surface is described as a parametrized curve. We introduce a numerically stable algorithm which accounts for its evolution with time. The method is shown to converge using approximate solutions, such as Stokes waves and Green-Naghdi solitary waves. It is finally tested on a wave breaking problem, for which an odd-even coupling suffices to achieve numerical convergence up to the splash without the need for additional filtering.
References
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- T. Alazard, N. Burq, and C. Zuily, On the Cauchy problem for gravity water waves, Invent. Math. 198 (2014), no. 1, 71–163. MR 3260858, DOI 10.1007/s00222-014-0498-z
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- Diogo Arsénio, Emmanuel Dormy, and Christophe Lacave, The vortex method for two-dimensional ideal flows in exterior domains, SIAM J. Math. Anal. 52 (2020), no. 4, 3881–3961. MR 4137039, DOI 10.1137/19M1291947
- G. Baker, Generalized vortex methods for free-surface flows, in R. E. Meyer (ed.), Waves on Fluid Interfaces, pp. 53–81, Academic Press, 1983.
- Gregory R. Baker, Daniel I. Meiron, and Steven A. Orszag, Generalized vortex methods for free-surface flow problems, J. Fluid Mech. 123 (1982), 477–501. MR 687014, DOI 10.1017/S0022112082003164
- Gregory R. Baker and Chao Xie, Singularities in the complex physical plane for deep water waves, J. Fluid Mech. 685 (2011), 83–116. MR 2844303, DOI 10.1017/jfm.2011.283
- J. Thomas Beale, Thomas Y. Hou, and John Lowengrub, Convergence of a boundary integral method for water waves, SIAM J. Numer. Anal. 33 (1996), no. 5, 1797–1843. MR 1411850, DOI 10.1137/S0036142993245750
- Geoffrey Beck and David Lannes, Freely floating objects on a fluid governed by the Boussinesq equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 39 (2022), no. 3, 575–646. MR 4412077, DOI 10.4171/aihpc/15
- Jennifer Beichman and Sergey Denisov, 2D Euler equation on the strip: stability of a rectangular patch, Comm. Partial Differential Equations 42 (2017), no. 1, 100–120. MR 3605292, DOI 10.1080/03605302.2016.1258576
- Didier Bresch, David Lannes, and Guy Métivier, Waves interacting with a partially immersed obstacle in the Boussinesq regime, Anal. PDE 14 (2021), no. 4, 1085–1124. MR 4283690, DOI 10.2140/apde.2021.14.1085
- Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and physicists, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXVII, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954. MR 60642
- Yusong Cao, William W. Schultz, and Robert F. Beck, Three-dimensional desingularized boundary integral methods for potential problems, Internat. J. Numer. Methods Fluids 12 (1991), no. 8, 785–803. MR 1103456, DOI 10.1002/fld.1650120807
- Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, and Javier Gómez-Serrano, Finite time singularities for the free boundary incompressible Euler equations, Ann. of Math. (2) 178 (2013), no. 3, 1061–1134. MR 3092476, DOI 10.4007/annals.2013.178.3.6
- W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys. 108 (1993), no. 1, 73–83. MR 1239970, DOI 10.1006/jcph.1993.1164
- R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology, Volume 4: Integral equations and numerical methods, 2nd printing edition, Springer, Berlin, 2000.
- E. B. Fabes, M. Jodeit Jr., and N. M. Rivière, Potential techniques for boundary value problems on $C^{1}$-domains, Acta Math. 141 (1978), no. 3-4, 165–186. MR 501367, DOI 10.1007/BF02545747
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- Jonathan Goodman, Thomas Y. Hou, and John Lowengrub, Convergence of the point vortex method for the $2$-D Euler equations, Comm. Pure Appl. Math. 43 (1990), no. 3, 415–430. MR 1040146, DOI 10.1002/cpa.3160430305
- S. T. Grilli, J. Skourup, and I. Svendsen, An efficient boundary element method for nonlinear water waves, Eng. Anal. Bound. Elem. 6 (1989), no. 2, 97–107.
- P. Guyenne and S. T. Grilli, Numerical study of three-dimensional overturning waves in shallow water, J. Fluid Mech. 547 (2006), 361–388. MR 2263356, DOI 10.1017/S0022112005007317
- Tatsuo Iguchi and David Lannes, Hyperbolic free boundary problems and applications to wave-structure interactions, Indiana Univ. Math. J. 70 (2021), no. 1, 353–464. MR 4226659, DOI 10.1512/iumj.2021.70.8201
- Bo Jiang and Qinsheng Bi, Classification of traveling wave solutions to the Green-Naghdi model, Wave Motion 73 (2017), 45–56. MR 3671274, DOI 10.1016/j.wavemoti.2017.05.006
- R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. MR 1629555, DOI 10.1017/CBO9780511624056
- Oliver Dimon Kellogg, Foundations of potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR 222317
- David Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc. 18 (2005), no. 3, 605–654. MR 2138139, DOI 10.1090/S0894-0347-05-00484-4
- David Lannes, The water waves problem, Mathematical Surveys and Monographs, vol. 188, American Mathematical Society, Providence, RI, 2013. Mathematical analysis and asymptotics. MR 3060183, DOI 10.1090/surv/188
- G. Moon, Local well-posedness of the gravity-capillary water waves system in the presence of geometry and damping, arXiv:2201.04713, 2022.
- N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 355494
- Yves Pomeau, Martine Le Berre, Philippe Guyenne, and Stephan Grilli, Wave-breaking and generic singularities of nonlinear hyperbolic equations, Nonlinearity 21 (2008), no. 5, T61–T79. MR 2412317, DOI 10.1088/0951-7715/21/5/T01
- Yves Pomeau and Martine Le Berre, Topics in the theory of wave-breaking, Singularities in mechanics: formation, propagation and microscopic description, Panor. Synthèses, vol. 38, Soc. Math. France, Paris, 2012, pp. 125–162 (English, with English and French summaries). MR 3204902
- A. Riquier and E. Dormy, A numerical study of the viscous breaking water waves problem and the limit of vanishing viscosity, 2023, submitted.
- Y.-M. Scolan, Some aspects of the flip-through phenomenon: A numerical study based on the desingularized technique, J. Fluids Struct. 26 (2010), no. 6, 918–953.
- D. Scullen and E. Tuck, Nonlinear free-surface flow computations for submerged cylinders, J. Ship Res. 39 (1995), no. 3, 185–193.
- E. Tuck, Solution of free-surface problems by boundary and desingularised integral equation techniques, Computational Techniques and Applications: CTAC97, B. J. Noye, M. D. Teubner, and A. W. Gill (eds.), World Scientific Publishing, 1992.
- Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math. 130 (1997), no. 1, 39–72. MR 1471885, DOI 10.1007/s002220050177
- V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Applied Mech. Tech. Phys. 9 (1968), no. 2, 190–194.
References
- Thomas Alazard, Boundary observability of gravity water waves, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 3, 751–779. MR 3778651, DOI 10.1016/j.anihpc.2017.07.006
- T. Alazard, N. Burq, and C. Zuily, On the Cauchy problem for gravity water waves, Invent. Math. 198 (2014), no. 1, 71–163. MR 3260858, DOI 10.1007/s00222-014-0498-z
- David M. Ambrose, Roberto Camassa, Jeremy L. Marzuola, Richard M. McLaughlin, Quentin Robinson, and Jon Wilkening, Numerical algorithms for water waves with background flow over obstacles and topography, Adv. Comput. Math. 48 (2022), no. 4, Paper No. 46, 62. MR 4450141, DOI 10.1007/s10444-022-09957-z
- Diogo Arsénio, Emmanuel Dormy, and Christophe Lacave, The vortex method for two-dimensional ideal flows in exterior domains, SIAM J. Math. Anal. 52 (2020), no. 4, 3881–3961. MR 4137039, DOI 10.1137/19M1291947
- G. Baker, Generalized vortex methods for free-surface flows, in R. E. Meyer (ed.), Waves on Fluid Interfaces, pp. 53–81, Academic Press, 1983.
- Gregory R. Baker, Daniel I. Meiron, and Steven A. Orszag, Generalized vortex methods for free-surface flow problems, J. Fluid Mech. 123 (1982), 477–501. MR 687014, DOI 10.1017/S0022112082003164
- Gregory R. Baker and Chao Xie, Singularities in the complex physical plane for deep water waves, J. Fluid Mech. 685 (2011), 83–116. MR 2844303, DOI 10.1017/jfm.2011.283
- J. Thomas Beale, Thomas Y. Hou, and John Lowengrub, Convergence of a boundary integral method for water waves, SIAM J. Numer. Anal. 33 (1996), no. 5, 1797–1843. MR 1411850, DOI 10.1137/S0036142993245750
- Geoffrey Beck and David Lannes, Freely floating objects on a fluid governed by the Boussinesq equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 39 (2022), no. 3, 575–646. MR 4412077, DOI 10.4171/aihpc/15
- Jennifer Beichman and Sergey Denisov, 2D Euler equation on the strip: stability of a rectangular patch, Comm. Partial Differential Equations 42 (2017), no. 1, 100–120. MR 3605292, DOI 10.1080/03605302.2016.1258576
- Didier Bresch, David Lannes, and Guy Métivier, Waves interacting with a partially immersed obstacle in the Boussinesq regime, Anal. PDE 14 (2021), no. 4, 1085–1124. MR 4283690, DOI 10.2140/apde.2021.14.1085
- Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and physicists, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXVII, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954. MR 60642
- Yusong Cao, William W. Schultz, and Robert F. Beck, Three-dimensional desingularized boundary integral methods for potential problems, Internat. J. Numer. Methods Fluids 12 (1991), no. 8, 785–803. MR 1103456, DOI 10.1002/fld.1650120807
- Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, and Javier Gómez-Serrano, Finite time singularities for the free boundary incompressible Euler equations, Ann. of Math. (2) 178 (2013), no. 3, 1061–1134. MR 3092476, DOI 10.4007/annals.2013.178.3.6
- W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys. 108 (1993), no. 1, 73–83. MR 1239970, DOI 10.1006/jcph.1993.1164
- R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology, Volume 4: Integral equations and numerical methods, 2nd printing edition, Springer, Berlin, 2000.
- E. B. Fabes, M. Jodeit Jr., and N. M. Rivière, Potential techniques for boundary value problems on $C^{1}$-domains, Acta Math. 141 (1978), no. 3-4, 165–186. MR 501367, DOI 10.1007/BF02545747
- P. Germain, N. Masmoudi, and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2) 175 (2012), no. 2, 691–754. MR 2993751, DOI 10.4007/annals.2012.175.2.6
- Jonathan Goodman, Thomas Y. Hou, and John Lowengrub, Convergence of the point vortex method for the $2$-D Euler equations, Comm. Pure Appl. Math. 43 (1990), no. 3, 415–430. MR 1040146, DOI 10.1002/cpa.3160430305
- S. T. Grilli, J. Skourup, and I. Svendsen, An efficient boundary element method for nonlinear water waves, Eng. Anal. Bound. Elem. 6 (1989), no. 2, 97–107.
- P. Guyenne and S. T. Grilli, Numerical study of three-dimensional overturning waves in shallow water, J. Fluid Mech. 547 (2006), 361–388. MR 2263356, DOI 10.1017/S0022112005007317
- Tatsuo Iguchi and David Lannes, Hyperbolic free boundary problems and applications to wave-structure interactions, Indiana Univ. Math. J. 70 (2021), no. 1, 353–464. MR 4226659, DOI 10.1512/iumj.2021.70.8201
- Bo Jiang and Qinsheng Bi, Classification of traveling wave solutions to the Green-Naghdi model, Wave Motion 73 (2017), 45–56. MR 3671274, DOI 10.1016/j.wavemoti.2017.05.006
- R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. MR 1629555, DOI 10.1017/CBO9780511624056
- Oliver Dimon Kellogg, Foundations of potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR 222317
- David Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc. 18 (2005), no. 3, 605–654. MR 2138139, DOI 10.1090/S0894-0347-05-00484-4
- David Lannes, The water waves problem, Mathematical Surveys and Monographs, vol. 188, American Mathematical Society, Providence, RI, 2013. Mathematical analysis and asymptotics. MR 3060183, DOI 10.1090/surv/188
- G. Moon, Local well-posedness of the gravity-capillary water waves system in the presence of geometry and damping, arXiv:2201.04713, 2022.
- N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 355494
- Yves Pomeau, Martine Le Berre, Philippe Guyenne, and Stephan Grilli, Wave-breaking and generic singularities of nonlinear hyperbolic equations, Nonlinearity 21 (2008), no. 5, T61–T79. MR 2412317, DOI 10.1088/0951-7715/21/5/T01
- Yves Pomeau and Martine Le Berre, Topics in the theory of wave-breaking, Singularities in mechanics: formation, propagation and microscopic description, Panor. Synthèses, vol. 38, Soc. Math. France, Paris, 2012, pp. 125–162 (English, with English and French summaries). MR 3204902
- A. Riquier and E. Dormy, A numerical study of the viscous breaking water waves problem and the limit of vanishing viscosity, 2023, submitted.
- Y.-M. Scolan, Some aspects of the flip-through phenomenon: A numerical study based on the desingularized technique, J. Fluids Struct. 26 (2010), no. 6, 918–953.
- D. Scullen and E. Tuck, Nonlinear free-surface flow computations for submerged cylinders, J. Ship Res. 39 (1995), no. 3, 185–193.
- E. Tuck, Solution of free-surface problems by boundary and desingularised integral equation techniques, Computational Techniques and Applications: CTAC97, B. J. Noye, M. D. Teubner, and A. W. Gill (eds.), World Scientific Publishing, 1992.
- Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math. 130 (1997), no. 1, 39–72. MR 1471885, DOI 10.1007/s002220050177
- V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Applied Mech. Tech. Phys. 9 (1968), no. 2, 190–194.
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Additional Information
Emmanuel Dormy
Affiliation:
Département de Mathématiques et Applications, UMR-8553, École Normale Supérieure, CNRS, PSL University, 75005 Paris, France
MR Author ID:
622868
ORCID:
0000-0002-9683-6173
Email:
Emmanuel.Dormy@ens.fr
Christophe Lacave
Affiliation:
Université de Grenoble Alpes, CNRS, IF, 38000 Grenoble, France
Address at time of publication:
(Christophe Lacave) Université Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
MR Author ID:
878311
ORCID:
0000-0002-2488-4117
Email:
Christophe.Lacave@univ-smb.fr
Keywords:
Singular integral formulations,
vortex and dipole formulation,
overturning waves,
splash singularity
Received by editor(s):
July 5, 2023
Received by editor(s) in revised form:
September 1, 2023
Published electronically:
January 19, 2024
Additional Notes:
The authors were supported in part by the ANR project ‘SINGFLOWS’ (ANR-18-CE40-0027-01), the IMPT project ‘Ocean waves’, and the CNES-Tosca project ‘Maeva’.
Article copyright:
© Copyright 2024
Brown University
