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A General Non-hydrostatic Hyperbolic Formulation for Boussinesq Dispersive Shallow Flows and Its Numerical Approximation

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Abstract

In this paper, we propose a novel first-order reformulation of the most well-known Boussinesq-type systems that are used in ocean engineering. This has the advantage of collecting in a general framework many of the well-known systems used for dispersive flows. Moreover, it avoids the use of high-order derivatives which are not easy to treat numerically, due to the large stencil usually needed. These first-order PDE dispersive systems are then approximated by a novel set of first-order hyperbolic equations. Our new hyperbolic approximation is based on a relaxed augmented system in which the divergence constraints of the velocity flow variables are coupled with the other conservation laws via an evolution equation for the depth-averaged non-hydrostatic pressures. The most important advantage of this new hyperbolic formulation is that it can be easily discretized with explicit and high-order accurate numerical schemes for hyperbolic conservation laws. There is no longer need of solving implicitly some linear system as it is usually done in many classical approaches of Boussinesq-type models. Here a third-order finite volume scheme based on a CWENO reconstruction has been used. The scheme is well-balanced and can treat correctly wet–dry areas and emerging topographies. Several numerical tests, which include idealized academic benchmarks and laboratory experiments are proposed, showing the advantage, efficiency and accuracy of the technique proposed here.

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References

  1. Abbott, M.B., McCowan, A.D., Warren, I.R.: Accuracy of short wave numerical models. J. Hydraul. Eng. 110, 1287–1301 (1984)

    Article  Google Scholar 

  2. Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un 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)

    MathSciNet  MATH  Google Scholar 

  3. Green, A.E., Naghdi, P.M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246 (1976)

    Article  MATH  Google Scholar 

  4. Madsen, P.A., Sørensen, O.R.: A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry. Coast. Eng. 18, 183–204 (1992)

    Article  Google Scholar 

  5. Nwogu, O.: Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port Coast. Ocean Eng. 119, 618–638 (1993)

    Article  Google Scholar 

  6. Peregrine, D.H.: Long waves on a beach. J. Fluid Mech. 27, 815–827 (1967)

    Article  MATH  Google Scholar 

  7. Gobbi, M.F., Kirby, J.T., Wei, G.: A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)4. J. Fluid Mech. 405, 181–210 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Whitman, G.B.: Linear and nonlinear waves. Earthq. Eng. Struct. Dyn. 4, 518 (1974)

    Google Scholar 

  9. Witting, J.M.: A unified model for the evolution of nonlinear water waves. J. Comput. Phys. 56, 203–236 (1982)

    Article  MATH  Google Scholar 

  10. Casulli, V.: A semi-implicit finite difference method for non-hydrostatic free surface flows. Numer. Methods Fluids 30, 425–440 (1999)

    Article  MATH  Google Scholar 

  11. 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, 1–23 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bristeau, M.-O., Mangeney, A., Sainte-Marie, J., Seguin, N.: An energy-consistent depth-averaged Euler system: derivation and properties. Discrete Contin. Dyn. Syst. Ser. B 20, 961–988 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yamazaki, Y., Kowalik, Z., Cheung, K.F.: Depth-integrated, non-hydrostatic model for wave breaking and run-up. Int. J. Numer. Methods Fluids 61, 473–497 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fernández-Nieto, E.D., Parisot, M., Penel, Y., Sainte-Marie, J.: A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows. Commun. Math. Sci. 16, 1169–1202 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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, 569–583 (2017)

    Article  Google Scholar 

  16. Escalante, C., Morales de Luna, 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)

    MathSciNet  MATH  Google Scholar 

  17. Escalante, C., Dumbser, M., Castro, M.J.: An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes. J. Comput. Phys. 394, 385–416 (2019)

    Article  MathSciNet  Google Scholar 

  18. Lannes, D., Bonneton, P.: Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids 21, 016601 (2009)

    Article  MATH  Google Scholar 

  19. Cienfuegos, R., Barthélemy, E., Bonneton, P.: A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I: model development and analysis. Int. J. Numer. Methods Fluids 51, 1217–1253 (2006)

    Article  MATH  Google Scholar 

  20. Roeber, V., Cheung, K.F., Kobayashi, M.H.: Shock-capturing Boussinesq-type model for nearshore wave processes. Coast. Eng. 57, 407–423 (2010)

    Article  Google Scholar 

  21. Escalante, C., Fernández-Nieto, E.D., Morales de Luna, T., Castro, M.J.: An efficient two-layer non-hydrostatic approach for dispersive water waves. J. Sci. Comput. 79, 273–320 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Munz, C.D., Omnes, P., Schneider, R., Sonnendrücker, E., Voß, U.: Divergence correction techniques for Maxwell solvers based on a hyperbolic model. J. Comput. Phys. 161, 484–511 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dedner, A., Kemm, F., Kröner, D., Munz, C.D., Schnitzer, T., Wesenberg, M.: Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175, 645–673 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Manning, R.: On the flow of water in open channels and pipes. Trans. Inst. Civ. Eng. Irel. 20, 161–207 (1891)

    Google Scholar 

  25. Lynett, P., Liu, P.L.: A two-layer approach to wave modelling. Proc. R. Soc. A Math. Phys. Eng. Sci. 460, 2637–2669 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schäffer, H.A., Madsen, P.A.: Further enhancements of Boussinesq-type equations. Coast. Eng. 26, 1–14 (1995)

    Article  Google Scholar 

  28. Castro Diaz, M.J., Fernandez-Nieto, E.D.: A class of computationally fast first order finite volume solvers: PVM methods. SIAM J. Sci. Comput. 34, 173–196 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Castro, M., 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, 1103–1135 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. Am. Math. Soc. 67, 73–85 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  31. Castro, M.J., García-Rodríguez, J.A., González-Vida, J.M., Parés, C.: Solving shallow-water systems in 2D domains using finite volume methods and multimedia SSE instructions. J. Comput. Appl. Math. 221, 16–32 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Cravero, I., Semplice, M., Visconti, G.: Optimal definition of the nonlinear weights in multidimensional central Wenoz reconstructions. SIAM J. Numer. Anal. 57, 2328–2358 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Castro, M.J., Ferreiro Ferreiro, A.M., García-Rodríguez, J.A., González-Vida, J.M., MacíAs, J., Parés, C., Elena Vázquez-Cendón, M.: The numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layer systems. Math. Comput. Model. 42, 419–439 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kurganov, A., Petrova, G.: A second-order well-balanced positivity preserving central-upwind scheme for the Saint–Venant system. Commun. Math. Sci. 5, 133–160 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. 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)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Madsen, P.A., Bingham, H.B., Schäffer, H.A.: Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis. Proc. R. Soc. A Math. Phys. Eng. Sci. 459, 1075–1104 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. Macías, J., Castro, M.J., Escalante, C.: Performance assessment of the Tsunami-HySEA model for NTHMP tsunami currents benchmarking. Laboratory data. Coast. Eng. 158, 103667 (2020)

    Article  Google Scholar 

  39. Beji, S., Battjes, J.A.: Numerical simulation of nonlinear wave propagation over a bar. Coast. Eng. 23, 1–16 (1994)

    Article  Google Scholar 

  40. Synolakis, C.E.: The runup of solitary waves. J. Fluid Mech. 185, 523–545 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  41. Hammack, J.L.: A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech. 60, 769–799 (1973)

    Article  MATH  Google Scholar 

  42. Fuhrman, D.R., Madsen, P.A.: Tsunami generation, propagation, and run-up with a high-order Boussinesq model. Coast. Eng. 56, 747–758 (2009)

    Article  Google Scholar 

Download references

Acknowledgements

This research has been partially supported by the Spanish Government and FEDER through the coordinated Research Project RTI2018-096064-B-C1 and RTI2018-096064-B-C2, and the Andalusian Government Research Project UMA18-FEDERJA-161.

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  1. Departamento de Matemática Aplicada, Universidad de Málaga, Málaga, Spain

    C. Escalante

  2. Departamento de Matemáticas, Universidad de Córdoba, Córdoba, Spain

    T. Morales de Luna

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  1. C. Escalante
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Correspondence to C. Escalante.

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Escalante, C., de Luna, T.M. A General Non-hydrostatic Hyperbolic Formulation for Boussinesq Dispersive Shallow Flows and Its Numerical Approximation. J Sci Comput 83, 62 (2020). https://doi.org/10.1007/s10915-020-01244-7

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