Skip to main content
SpringerLink
Log in

A hybrid finite-volume and finite difference scheme for depth-integrated non-hydrostatic model

  • Published:
China Ocean Engineering Aims and scope Submit manuscript

Abstract

A depth-integrated, non-hydrostatic model with hybrid finite difference and finite volume numerical algorithm is proposed in this paper. By utilizing a fraction step method, the governing equations are decomposed into hydrostatic and non-hydrostatic parts. The first part is solved by using the finite volume conservative discretization method, whilst the latter is considered by solving discretized Poisson-type equations with the finite difference method. The second-order accuracy, both in time and space, of the finite volume scheme is achieved by using an explicit predictor-correction step and linear construction of variable state in cells. The fluxes across the cell faces are computed in a Godunov-based manner by using MUSTA scheme. Slope and flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purpose. Wave breaking is treated as a shock by switching off the non-hydrostatic pressure in the steep wave front locally. The model deals with moving wet/dry front in a simple way. Numerical experiments are conducted to verify the proposed model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Ai, C.F. and Jin, S., 2012. A multi-layer non-hydrostatic model for wave breaking and run-up, Coastal Engineering, 62(4), 1–8.

    Article  Google Scholar 

  • Ai, C.F., Jin, S. and Lv, B., 2011. A new fully non-hydrostatic 3D free surface flow model for water wave motions, International Journal for Numerical Methods in Fluids, 66(11), 1354–1370.

    Article  MathSciNet  MATH  Google Scholar 

  • Bai, Y.F. and Cheung, K.F., 2013. Dispersion and nonlinearity of multi-layer non-hydrostatic free-surface flow, Journal of Fluid Mechanics, 726, 226–260.

    Article  MathSciNet  MATH  Google Scholar 

  • Beji, S. and Battjes, J.A., 1993. Experimental investigation of wave propagation over a bar, Coastal Engineering, 19, 151–162.

    Article  Google Scholar 

  • Bradford, S.F., 2005. Godunov-based model for nonhydrostatic wave dynamics, Journal of the Waterway Port Coastal and Ocean Engineering, ASCE, 131(5), 226–238.

    Article  Google Scholar 

  • Bradford, S.F., 2011. Nonhydrostatic model for surf zone simulation, Journal of the Waterway Port Coastal and Ocean Engineering, ASCE, 137(4), 163–174.

    Article  Google Scholar 

  • Brocchini, M. and Peregrine, D.H., 1996. Integral flow properties of the swash zone and averaging, Journal of Fluid Mechanics, 317, 241–273.

    Article  MATH  Google Scholar 

  • Choi, D.Y. and Yuan, H.L., 2012. A horizontally curvilinear non-hydrostatic model for simulating nonlinear wave motion in curved boundaries, International Journal for Numerical Methods in Fluids, 69(12), 1923–1938.

    Article  MathSciNet  Google Scholar 

  • Fang, K.Z., Liu, Z.B. and Zou, Z.L., 2015a. Efficient computation of coastal waves using a depth-integrated. non-hydrostatic model, Coastal Engineering, 97, 21–36.

    Article  Google Scholar 

  • Fang, K.Z., Liu, Z.B. and Zou, Z.L., 2015b. Modelling coastal water wave using a depth-integrated. non-hydrostatic model with shockcapturing ability, Journal of Hydraulic Research, 53(1), 119–133.

    Article  Google Scholar 

  • Fang, K.Z., Liu, Z.B. and Zou, Z.L., 2016. Fully nonlinear modeling wave transformation over fringing reefs using shock-capturing Boussinesq model, Journal of Coastal Research, 32(1), 164–171.

    Article  Google Scholar 

  • Hansen, J.B. and Svendsen, I.A., 1979. Regular Waves in Shoaling Water: Experimental Data, Technical University of Denmark.

  • Kang, L. and Guo, X.M., 2013. Depth-integrated. non-hydrostatic model using a new alternating direction implicit scheme, Journal of Hydraulic Research, 51(4), 368–379.

    Article  Google Scholar 

  • Kennedy, A.B., Chen, Q., Kirby, J.T. and Dalrymple, R.A., 2000. Boussinesq modeling of wave transformation, breaking, end runup. I: 1D, Journal of Waterway Port Coastal and Ocean Engineering, ASCE, 126(1), 39–47.

    Google Scholar 

  • Liang, Q.H. and Borthwick, A.G.L., 2009. Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography, Computers & Fluids, 38(2), 221–234.

    Article  MathSciNet  MATH  Google Scholar 

  • Luth, H.R., Klopman, G. and Kitou, N., 1994. Project 13G: Kinematics of waves breaking partially on an offshore bar; LDV measurements for waves with and without a net onshore current, Information Bulletin on Variable Stars, 1872.

  • Lynett, P.J., 2006. Nearshore wave modeling with high-order Boussinesq-type equations, Journal of the Waterway Port Coastal and Ocean Engineering, ASCE, 132(5), 348–357.

    Article  Google Scholar 

  • Lynett, P. and Swigler, D., Son, S., Bryant, D. and Socolofsky, S., 2010. Experimental study of solitary wave evolution over a 3D shallow shelf, Proceedings of the 32nd International Conference on Coastal Engineering, Shanghai, China.

    Google Scholar 

  • Ma, G.F., Shi, F.Y. and Kirby, J.T., 2012. Shock-capturing non-hydrostatic model for fully dispersive surface wave processes, Ocean Modelling, 43–44, 22–35.

    Article  Google Scholar 

  • Orszaghova, J., Borthwick, A.G.L. and Taylor, P.H., 2011. From the paddle to the beach-A Boussinesq shallow water numerical wave tank based on Madsen and Sørensen’s equations, Journal of Computational Physics, 231(2), 328–344.

    Article  MATH  Google Scholar 

  • Shi, F., Kirby, J.T., Harris, J.C., Geiman, J.D. and Grilli, S.T., 2012. A high-order adaptive time-stepping TVD solver for Boussinesq modeling of breaking waves and coastal inundation, Ocean Modelling, 43–44, 36–51.

    Article  Google Scholar 

  • Smit, P., Zijlema, M. and Stelling, G., 2013. Depth-induced wave breaking in a non-hydrostatic. near-shore wave model, Coastal Engineering, 76, 1–16.

    Article  Google Scholar 

  • Smit, P., Janssen, T., Holthuijsen, L. and Smith, J., 2014. Non-hydrostatic modeling of surf zone wave dynamics, Coastal Engineering, 83, 36–48.

    Article  Google Scholar 

  • Stelling, G. and Zijlema, M., 2003. An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation, International Journal for Numerical Methods in Fluids, 43(1), 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  • Synolakis, C.E., 1987. The runup of solitary waves, Journal of Fluid Mechanics, 185, 523–545.

    Article  MathSciNet  MATH  Google Scholar 

  • Tang, J., Causon, D., Mingham, C. and Qian, L., 2013. Numerical study of vegetation damping effects on solitary wave run-up using the nonlinear shallow water equations, Coastal Engineering, 75, 21–28.

    Article  Google Scholar 

  • Tonelli, M. and Petti, M., 2009. Hybrid finite volume-finite difference scheme for 2DH improved Boussinesq equations, Coastal Engineering, 56(5–6), 609–620.

    Article  Google Scholar 

  • Toro, E.F. and Titarev, V.A., 2006. MUSTA fluxes for systems of conservation laws, Journal of Computational Physics, 216(2), 403–429.

    Article  MathSciNet  MATH  Google Scholar 

  • Toro, E.F., 2009. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer, pp. 87–114.

    Book  Google Scholar 

  • van Leer, B., 1979. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, Journal of Computational Physics, 32(1), 101–136.

    Article  Google Scholar 

  • Vincent, C.L. and Briggs, M.J., 1989. Refraction-diffraction of irregular waves over a mound, Journal of the Waterway Port Coastal and Ocean Engineering, ASCE, 115(2), 269–284.

    Article  Google Scholar 

  • Vorst, H.A.V.D., 1992. Bi-CGSTAB: A fast and smoothly converging variant of bi-cg for the solution of nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing, 13(2), 631–644.

    Article  MathSciNet  MATH  Google Scholar 

  • Yamazaki, Y., Kowalik, Z. and Cheung, K.F., 2009. Depth-integrated. non-hydrostatic model for wave breaking and run-up, International Journal for Numerical Methods in Fluids, 61(5), 473–497.

    Article  MathSciNet  MATH  Google Scholar 

  • Young, C.C. and Wu, C.H., 2009. A s-coordinate non-hydrostatic model with embedded Boussinesq-type-like equations for modeling deep-water waves, International Journal for Numerical Methods in Fluids, 63(12), 1448–1470.

    MathSciNet  MATH  Google Scholar 

  • Zia, A. and Banihashemi, M., 2008. A simple efficient algorithm (SEA) for shallow flows with shock wave on dry and irregular beds, International Journal for Numerical Methods in Fluids, 56(11), 2021–2043.

    Article  MathSciNet  MATH  Google Scholar 

  • Zijlema, M., Stelling, G.S. and Smit, P., 2011. SWASH: An operational public domain code for simulating wave fields and rapidly varied flows in coastal waters, Coastal Engineering, 58(10), 992–1012.

    Article  Google Scholar 

  • Zijlema, M. and Stelling, G.S., 2008. Efficient computation of surf zone waves using the nonlinear shallow water equations with nonhydrostatic pressure, Coastal Engineering, 55(10), 780–790.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Marine Environmental Monitoring Center, Dalian, 116023, China

    Jing Yin, Jia-wen Sun & Yong-hai Yu

  2. Key State Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116023, China

    Jing Yin, Jia-wen Sun, Xing-gang Wang & Zhao-chen Sun

  3. Nanjing Hydraulic Research Institute, Nanjing, 210029, China

    Xing-gang Wang

Authors
  1. Jing Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jia-wen Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xing-gang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yong-hai Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Zhao-chen Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jia-wen Sun.

Additional information

Foundation item: The work was financially supported by the State Ocean Administration People’s Republic of China (Grant No. 201405025), and the Key Laboratory for Sea Area Management Technology (SOA) (Grant No. 201603).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yin, J., Sun, Jw., Wang, Xg. et al. A hybrid finite-volume and finite difference scheme for depth-integrated non-hydrostatic model. China Ocean Eng 31, 261–271 (2017). https://doi.org/10.1007/s13344-017-0031-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13344-017-0031-4

Key words

Search

Navigation