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An Efficient Two-Layer Non-hydrostatic Approach for Dispersive Water Waves

  • C. Escalante
  • E. D. Fernández-Nieto
  • T. Morales de Luna
  • M. J. Castro
Article
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Abstract

In this paper, we propose a two-layer depth-integrated non-hydrostatic system with improved dispersion relations. This improvement is obtained through three free parameters: two of them related to the representation of the pressure at the interface and a third one that controls the relative position of the interface concerning the total height. These parameters are then optimized to improve the dispersive properties of the resulting system. The optimized model shows good linear wave characteristics up to \(kH\approx 10\), that can be improved for long waves. The system is solved using an efficient formally second-order well-balanced and positive preserving hybrid finite volume/difference numerical scheme. The scheme consists of a two-step algorithm based on a projection-correction type scheme. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite-volume method. Second, the dispersive terms are solved using finite differences. The method has been applied to idealized and challenging physical situations that involve nearshore breaking. Agreement with laboratory data is excellent. This technique results in an accurate and efficient method.

Keywords

Dispersive waves Non-hydrostatic Shallow-water Finite-volume Finite-difference Breaking waves 

Notes

Acknowledgements

This research has been partially supported by the Spanish Government and FEDER through Research Project MTM2015-70490-C2-1-R and MTM2015-70490-C2-2-R, and Andalusian Government Research Project P11-FQM-8179. Funding was provided by Ministerio de Economía y Competitividad and Consejería de Economía, Innovación, Ciencia y Empleo, Junta de Andalucía.

Supplementary material

References

  1. 1.
    Abbott, M.B., McCowan, A.D., Warren, I.R.: Accuracy of short wave numerical models. J. Hydraul. Eng. 110(10), 1287–1301 (1984)CrossRefGoogle Scholar
  2. 2.
    Abgrall, R., Karni, S.: A comment on the computation of non-conservative products. J. Comput. Phys. 229, 2759–2763 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Adsuara, J.E., Cordero-Carrión, I., Cerdá-Durán, P., Aloy, M.A.: Scheduled relaxation jacobi method: Improvements and applications. J. Comput. Phys. 321, 369–413 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ai, C., Jin, S.: A multi-layer non-hydrostatic model for wave breaking and run-up. Coast. Eng. 62, 1–8 (2012)CrossRefGoogle Scholar
  5. 5.
    Ai, C., Jin, S., Lv, B.: A new fully non-hydrostatic 3d free surface flow model for water wave motions. Int. J. Numer. Methods Fluids 66(11), 1354–1370 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Aïssiouene, N., Bristeau, M.-O., Godlewski, E., Mangeney, A., Parés, C., Sainte-Marie, J.: Application of a combined finite element–finite volume method to a 2D non-hydrostatic shallow water problem. In: Cancés, C., Omnes, P. (eds.) Finite Volumes for Complex Applications VIII–Hyperbolic, Elliptic and Parabolic Problems, pp. 219–226. Springer, Cham (2017)CrossRefGoogle Scholar
  7. 7.
    Audusse, E., Bristeau, M.O., Perthame, B., Sainte-Marie, J.: A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation. ESAIM Math. Model. Numer. Anal. 45, 169–200 (2011). 01MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bai, Y., Cheung, K.F.: Depth-integrated free-surface flow with a two-layer non-hydrostatic formulation. Int. J. Numer. Methods Fluids 69(2), 411–429 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bai, Y., Cheung, K.F.: Depth-integrated free-surface flow with parameterized non-hydrostatic pressure. Int. J. Numer. Methods Fluids 71(4), 403–421 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bai, Y., Cheung, K.F.: Dispersion and kinematics of multi-layer non-hydrostatic models. Ocean Model. 92, 11–27 (2015)CrossRefGoogle Scholar
  11. 11.
    Bai, Y., Cheung, K.F.: Linear shoaling of free-surface waves in multi-layer non-hydrostatic models. Ocean Model. 121, 90–104 (2018)CrossRefGoogle Scholar
  12. 12.
    Beji, S., Battjes, J.A.: Experimental investigation of wave propagation over a bar. Coast. Eng. 19, 151–162 (1993)CrossRefGoogle Scholar
  13. 13.
    Berthon, C., Coquel, F., LeFloch, P.G.: Why many theories of shock waves are necessary: kinetic functions, equivalent equations, and fourth-order models. J. Comput. Phys. 227, 4162–4189 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bonneton, P., Chazel, F., Lannes, D., Marche, F., Tissier, M.: A splitting approach for the fully nonlinear and weakly dispersive Green–Naghdi model. J. Comput. Phys. 230(4), 1479–1498 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long dun canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55–108 (1872)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Bristeau, M.-O., Mangeney, A., Sainte-Marie, J., Seguin, N.: An energy-consistent depth-averaged euler system: derivation and properties. Discrete Continuous Dyn. Syst. Ser. B 20(4), 961–988 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Castro, M.J., Fernández-Nieto, E.D.: A class of computationally fast first order finite volume solvers: PVM methods. SIAM J. Sci. Comput. 34(4), 173–196 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Castro, M.J., Ferreiro, A.M.F., García-Rodríguez, J.A., González-Vida, J.M., Macías, J., Parés, C., Vázquez-Cendón, M.E.: The numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layer systems. Math. Comput. Model. 42(3), 419–439 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Castro, M.J., Gallardo, J.M., Parés, C.: High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow water systems. Math. Comput. 75, 1103–1134 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Castro, M.J., Pardo, A., Parés, C., Toro, E.F.: On some fast well-balanced first order solvers for nonconservative systems. Math. Comput. 79(271), 1427–1472 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Castro, M.J., Morales de Luna, T., Parés, C.: Chapter 6-well-balanced schemes and path-conservative numerical methods. In: Abgrall, Rémi, Shu, C.-W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems, volume 18 of Handbook of Numerical Analysis, pp 131–175. Elsevier, Amsterdam (2017)Google Scholar
  22. 22.
    Casulli, V.: A semi-implicit finite difference method for non-hydrostatic free surface flows. Numer. Methods Fluids 30(4), 425–440 (1999)CrossRefGoogle Scholar
  23. 23.
    Casulli, V., Zanolli, P.: Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems. Math. Comput. Model. 36(9), 1131–1149 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Cea, L., Stelling, G., Zijlema, M.: Non-hydrostatic 3D free surface layer-structured finite volume model for short wave propagation. Int. J. Numer. Methods Fluids 61(4), 382–410 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Chazel, F., Lannes, D., Marche, F.: Numerical simulation of strongly nonlinear and dispersive waves using a Green–Naghdi model. J. Sci. Comput. 48(1), 105–116 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
  27. 27.
    Cui, H., Pietrzak, J.D., Stelling, G.S.: Optimal dispersion with minimized poisson equations for non-hydrostatic free surface flows. Ocean Model. 81, 1–12 (2014)CrossRefGoogle Scholar
  28. 28.
    Dingemans, M.W.: Comparison of computations with Boussinesq-like models and laboratory measurements. Report H-1684.12, 32, Delft Hydraulics (1994)Google Scholar
  29. 29.
    Duran, A., Marche, F.: A discontinuous galerkin method for a new class of Green–Naghdi equations on simplicial unstructured meshes. Appl. Math. Model. 45, 840–864 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Dutykh, D., Clamond, D.: Efficient computation of steady solitary gravity waves. Wave Motion 51(1), 86–99 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Enet, F., Grilli, S.T.: Experimental study of tsunami generation by three-dimensional rigid underwater landslides. J. Waterw. Port Coast. Ocean Eng. 133(6), 442–454 (2007)CrossRefGoogle Scholar
  32. 32.
    Escalante, C., Macías, J., Castro, M. J.: Performance assessment of tsunami–HySEA model for NTHMP tsunami currents benchmarking. Part I lab data. Coast. Eng. (2017)Google Scholar
  33. 33.
    Escalante, C., Morales, T., Castro, M.J.: Non-hydrostatic pressure shallow flows: GPU implementation using finite volume and finite difference scheme. Appl. Math. Comput. 338, 631–659 (2018)MathSciNetGoogle Scholar
  34. 34.
    Fernández-Nieto, E.D., Parisot, M., Penel, Y., Sainte-Marie, J.: Layer–averaged approximation of Euler equations for free surface flows with a non-hydrostatic pressure. Commun. Math. Sci. (hal-01324012v3) (2018)Google Scholar
  35. 35.
    Fernández-Nieto, E.D., Koné, E.H., Chacón Rebollo, T.: A multilayer method for the hydrostatic Navier–Stokes equations: a particular weak solution. J. Sci. Comput. 60(2), 408–437 (2014)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Fernández-Nieto, E.D., Koné, E.H., Rebollo, T.: Chacón: A multilayer method for the hydrostatic Navier-Stokes equations: a particular weak solution. J. Sci. Comput. 60, 408–437, 08 (2014)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Gallerano, F., Cannata, G., Lasaponara, F., Petrelli, C.: A new three-dimensional finite-volume non-hydrostatic shock-capturing model for free surface flow. J. Hydrodyn. Ser. B 29(4), 552–566 (2017)CrossRefGoogle Scholar
  38. 38.
    Gobbi, M.F., Kirby, J.T., W, G.: A fully nonlinear boussinesq model for surface waves. Part 2. Extension to o(kh)4. J. Fluid Mech. 405, 181–210 (2000)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67(221), 73–85 (1998)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Green, A., Naghdi, P.: A derivation of equations for wave propagation in water of variable depth. Fluid Mech. 78, 237–246 (1976)CrossRefGoogle Scholar
  41. 41.
    Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Jeschke, A., Pedersen, G.K., Vater, S., Behrens, J.: Depth-averaged non-hydrostatic extension for shallow water equations with quadratic vertical pressure profile: equivalence to Boussinesq-type equations. Int. J. Numer. Methods Fluids 84(10), 569–583 (2017)CrossRefGoogle Scholar
  43. 43.
    Kazolea, M., Delis, A.I., Synolakis, C.E.: Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq-type equations. J. Comput. Phys. 271, 281–305 (2014)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Kurganov, A., Petrova, G.: A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5(1), 133–160, 03 (2007)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Lannes, D., Marche, F.: A new class of fully nonlinear and weakly dispersive Green–Naghdi models for efficient 2D simulations. J. Comput. Phys. 282, 238–268 (2015)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Lynett, P., Liu, P.L.F.: A two-layer approach to wave modelling. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 460(2049), 2637–2669 (2004)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Lynett, P.J., Liu, P.L.-F.: Linear analysis of the multi-layer model. Coast. Eng. 51, 439–454 (2004)CrossRefGoogle Scholar
  48. 48.
    Lynett, P.J., Wu, T.R., Liu, P.L.-F.: Modeling wave runup with depth-integrated equations. Coast. Eng. 46(2), 89–107 (2002)CrossRefGoogle Scholar
  49. 49.
    Lynett, P.J., Gately, K., Wilson, R., Montoya, L., Arcas, D., Aytore, B., Bai, Y., Bricker, J.D., Castro, M.J., Cheung, K.F., David, C.G., Dogan, G.G., Escalante, C., González-Vida, J.M., Grilli, S.T., Heitmann, T.W., Horrillo, J., Knolu, U., Kian, R., Kirby, J.T., Li, W., Macías, J., Nicolsky, D.J., Ortega, S., Pampell-Manis, A., Park, Y.S., Roeber, V., Sharghivand, N., Shelby, M., Shi, F., Tehranirad, B., Tolkova, E., Thio, H.K., Veliolu, D., Yalner, A.C., Yamazaki, Y., Zaytsev, A., Zhang, Y.J.: Inter-model analysis of tsunami-induced coastal currents. Ocean Model. 114, 14–32 (2017)CrossRefGoogle Scholar
  50. 50.
    Ma, G., Shi, F., Kirby, J.T.: Shock-capturing non-hydrostatic model for fully dispersive surface wave processes. Ocean Model. 43, 22–35 (2012)CrossRefGoogle Scholar
  51. 51.
    Ma, G., Shi, F., Kirby, J.T.: Shock-capturing non-hydrostatic model for fully dispersive surface wave processes. Ocean Model. 43–44, 22–35 (2012)CrossRefGoogle Scholar
  52. 52.
    Madsen, P.A., Sorensen, O.R.: A new form of the boussinesq equations with improved linear dispersion characteristics. Part 2: A slowing varying bathymetry. Coast. Eng. 18, 183–204, (1992)CrossRefGoogle Scholar
  53. 53.
    Madsen, P.A., Bingham, H.B., Schffer, H.A.: Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: Derivation and analysis. Proc. Math. Phys. Eng. Sci. 459(2033), 1075–1104 (2003)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Muoz-Ruiz, M.: On a non-homogeneous bi-layer shallow-water problem: smoothness and uniqueness results. Nonlinear Anal. Theory Methods Appl. 59, 11 (2004)MathSciNetGoogle Scholar
  55. 55.
    Muoz-Ruiz, M.: On a nonhomogeneous bi-layer shallow-water problem: an existence theorem. Differ. Integral Equ. 17(9–10), 1175–1200 (2004)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Muoz-Ruiz, M.L., Chatelon, F.J., Orenga, P.: On a bi-layer shallow-water problem. Nonlinear Anal. Real World Appl. 4, 139–171 (2003)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Narbona-Reina, G., Zabsonré, J.D.D., Fernández-Nieto, E.D., Bresch, D.: Derivation of a bilayer model for shallow water equations with viscosity. Numerical validation. Comput. Model. Eng. Sci. 43(1), 27–71 (2009)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Nwogu, O.: An alternative form of the boussinesq equations for nearshore wave propagation. Waterw. Port Coast. Ocean Eng. 119, 618–638 (1994)CrossRefGoogle Scholar
  59. 59.
    Peregrine, D.H.: Long waves on a beach. Fluid Mech. 27(4), 815–827 (1967)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Ricchiuto, M., Filippini, A.G.: Upwind residual discretization of enhanced Boussinesq equations for wave propagation over complex bathymetries. J. Comput. Phys. 271, 306–341 (2014)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Roeber, V., Cheung, K.F., Kobayashi, M.H.: Shock-capturing Boussinesq-type model for nearshore wave processes. Coast. Eng. 57, 407–423 (2010)CrossRefGoogle Scholar
  62. 62.
    Schffer, H.A., Madsen, P.A.: Further enhancements of Boussinesq-type equations. Coast. Eng. 26(1), 1–14 (1995)CrossRefGoogle Scholar
  63. 63.
    Stansby, P.K., Zhou, J.G.: Shallow-water flow solver with non-hydrostatic pressure: 2D vertical plane problems. Int. J. Numer. Methods Fluids 28(3), 541–563 (1998)CrossRefGoogle Scholar
  64. 64.
    Stelling, G., Zijlema, M.: An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation. Int. J. Numer. Methods Fluids 43(1), 1–23 (2003)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Synolakis, C.E.: The runup of solitary waves. Fluid Mech. 185, 523–545 (1987)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Van Leer, B.: Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov’s method. Comput. Phys. 32, 101–136 (1979)CrossRefGoogle Scholar
  67. 67.
    Wei, G., Kirby, J.T., Grilli, S.T., Subramanya, R.: A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. 294(–1), 71 (1995)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Whitham, G.B., Wiley, : Linear and nonlinear waves. Earthq. Eng. Struct. Dyn. 4(5), 518–518 (1976)Google Scholar
  69. 69.
    Witting, J.M.: A unified model for the evolution nonlinear water waves. J. Comput. Phys. 56(2), 203–236 (1984)CrossRefGoogle Scholar
  70. 70.
    Wu, C.H., Young, C.-C., Chen, Q., Lynett, P.J.: Efficient nonhydrostatic modeling of surface waves from deep to shallow water. J. Waterw. Port Coast. Ocean Eng. 136(2), 104–118 (2010)CrossRefGoogle Scholar
  71. 71.
    Yamazaki, Y., Kowalik, Z., Cheung, K.F.: Depth-integrated, non-hydrostatic model for wave breaking and run-up. Numer. Methods Fluids 61, 473–497 (2008)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Zijlema, M., Stelling, G.S.: Further experiences with computing non-hydrostatic free-surface flows involving water waves. Int. J. Numer. Methods Fluids 48(2), 169–197 (2005)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Universidad de MálagaMálagaSpain
  2. 2.Universidad de SevillaSevilleSpain
  3. 3.Universidad de CórdobaCórdobaSpain

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