Fengyan Shi, Gangfeng Ma, James T. Kirby, Tian-Jian Tom Hsu


This paper describes the recent developments in a suite of coastal engineering models using Godunov-type shock-capturing schemes. The developments include a depth-integrated, wave resolving Boussinesq model, a hydrostatic, wave-averaged circulation model, and a fully 3-D non-hydrostatic model in a surface-following $\sigma$ coordinate formulation. The models implemented with the shock-capturing TVD scheme show robust performances in modeling breaking waves, nearshore circulation and coastal inundation. In this paper, we present model equations in a conservative form, MUSCLE-TVD numerical scheme and model applications. We also point out some problems caused by the TVD scheme in the recent model applications.


coastal models; TVD solver; Boussinesq wave model; non-hydrostatic model; nearshore circulation model

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Abadie S. M., Harris, J. C., Grilli, S. T., and Fabre, R., 2012, "Numerical modeling of tsunami waves generated by the flank collapse of the Cumbre Vieja Volcano (La Palma, Canary Islands): Tsunami source and near field eects", J. Geophys. Res., 117, C05030

Bradford S. F. and Sanders B. F., 2002, "Finite-volume models for unidirectional, nonlinear dispersive waves", Journal of Waterway, Port, Coastal, and Ocean Engineering, 128, 173 -182.

Bradford S.F., 2005, "Godunov-based model for nonhydrostatic wave dynamics", J. Waterway, Port, Coastal and Ocean Engineering, 131, 226-238

Bradford S.F., 2011, "Nonhydrostatic model for surf zone simulation", J. Waterway, Port, Coastal and Ocean Engineering, 137(4), 163-174

Chen, J.-L., Hsu, T.-J., Shi, F., Elgar, S., and Raubenheimer, B., 2012, "Hydrodynamics in New River Inlet, NC - a numerical investigation using NearCoM-TVD", Ocean Sciences Meeting , Salt Lake City, February, 2012.

Chen, Q., Dalrymple, R. A., Kirby, J. T., Kennedy, A. B. and Haller, M. C., 1999, "Boussinesq modelling of a rip current system", J. Geophys. Res., 104, 20,617-20,637.

Chen, Q., Kirby, J. T., Dalrymple, R. A., Shi, F. and Thornton, E. B., 2003, "Boussinesq modeling of longshore currents", J. Geophys. Res., 108(C11), 3362, doi:10.1029/2002JC001308.

Chen, Q., 2006, "Fully nonlinear Boussinesq-type equations for waves and currents over porous beds", J. Eng. Mech., 132, 220-230.

Erduran, K. S., Ilic, S., and Kutija, V., 2005, "Hybrid finite-volume finite-dierence scheme for the solution of Boussinesq equations", Int. J. Num. Meth. Fluids, 49, 1213-1232.

Geiman, J. D., Kirby, J. T., Reniers, A. J. H. M., and MacMahan, J. H., 2011, "Eects of wave averaging on estimates of fluid mixing in the surf zone", J. Geophys. Res., 116, C04006

Gottlieb, S., Shu C.-W., and Tadmore, E., 2001, "Strong stability-preserving high-order time discretization methods", SIAM Review, 43 (1), 89 - 112.

Grilli, S. T., Harris, J. C., Shi, F., Kirby, J. T., Tajalibakhsh, T. S., Estibals, E., and Tehranirad, B., 2012, "Numerical modeling of coastal tsunami impact dissipation and impact", Proc. 33d Int. Conf. Coastal Engrng., Santander.

Kennedy, A. B., Chen, Q., Kirby, J. T. and Dalrymple, R. A., 2000, "Boussinesq modeling of wave transformation, breaking and runup. I: 1D", J. Waterway, Port, Coastal and Ocean Engineering, 126, 39-47.

Kennedy, A. B., Kirby, J. T., Chen, Q. and Dalrymple, R. A., 2001, "Boussinesq-type equations with improved nonlinear performance", Wave Motion, 33, 225-243.

Kirby, J. T., Shi, F., Harris, J. C. and Grilli, S. T., 2012, "Sensitivity analysis of trans-oceanic tsunami propagation to dispersive and Coriolis eects", Ocean Modelling, under revision.

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

MacMahan, J., Brown, J. Brown, J., Thornton, E., Reniers, A., Stanton, T., Henriquez, M., Gallagher, E., Morrison, J., Austin, M. J., Scott, T. M., and Senechal, N., 2010, "Mean Lagrangian flow behavior on open coast rip-channeled beaches: New perspectives", Mar. Geol., 268, 1 - 15.

Roeber, V., Cheung, K. F., and Kobayashi, M. H., 2010, "Shock-capturing Boussinesq-type model for nearshore wave processes", Coastal Engineering, 57, 407-423.

Roeber, V., Cheung, K. F., 2012, "Boussinesq-type model for energetic breaking waves in fringing reef environments", Coast. Eng., 70, 1-20.

Shi, F. and Sun, W., 1995, "A variable boundary model of storm surge flooding in generalized curvilinear grids", International Journal for Numerical Methods in Fluids, 21 (8), 642-651.

Shi, F., Kirby, J. T., Newberger, P. and Haas, K., 2005, "NearCoM master program for nearshore community model, Version 2005.4 - Documentation and user's manual", Research Report No. CACR-05- 10, Center for Applied Coastal Research, Dept. of Civil and Environmental Engineering, Univ. of Delaware, Newark, Delaware.

Shi, F., Hanes, D. M., Kirby, J. T., Erikson, L., Barnard, P. and Eshleman, J., 2011a, "Pressure gradientdominated nearshore circulation on a beach influenced by an adjacent large inlet", J. Geophys. Res., 116, C04020, doi:10.1029/2010JC006788.

Shi, F., Kirby, J. T., Tehranirad, B. and Harris, J. C., 2011b, "FUNWAVE-TVD, documentation and users' manual", Research Report, CACR-11-03, University of Delaware, Newark, Delaware.

Shi, F., Kirby, J. T., Hsu T.-J., and Chen J.-L., 2012a, "NearCoM-TVD, documentation and users' manual", draft of Research Report, University of Delaware, Newark, Delaware.

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

Shiach, J. B. and Mingham, C. G., 2009, "A temporally second-order accurate Godunov-type scheme for solving the extended Boussinesq equations", Coast. Eng., 56, 32-45.

Smagorinsky, J., 1963, "General circulation experiments with the primitive equations. I. The basic experiment", Mon. Weather Rev, 91, 99-165.

Soulsby, R. L. ,1997, "Dynamics of Marine Sands: A Manual for Practical Applications", 248 pp., Thomas Telford, London.

Putrevu, U., and Svendsen, I. A., 1999, "Three!dimensional dispersion of momentum in wave!induced nearshore currents", Eur. J. Mech. B, 18, 83-101.

Tehranirad, B., Shi, F., Kirby, J. T., Harris, J. C. and Grilli, S., 2011, "Tsunami benchmark results for fully nonlinear Boussinesq wave model FUNWAVE-TVD, Version 1.0", Research Report No. CACR-11-02, Center for Applied Coastal Research, Univ. of Delaware, Newark, Delaware.

Tehranirad, B., Kirby, J. T., Ma, G., and Shi, F., 2012, " Tsunami benchmark results for non-hydrostatic wave model NHWAVE Version 1.1", Research Report, CACR-12-03, University of Delaware, Newark, Delaware.

Tissier, M., Bonneton, P., Marche, F. , Chazel, F., Lannes, D., 2012, "A new approach to handle wave breaking in fully non-linear Boussinesq models", Coast. Eng., 67, 54-66.

Tonelli, M. and Petti, M., 2009, "Hybrid finite volume - finite dierence scheme for 2DH improved Boussinesq equations", Coastal Eng., 56, 609-620.

Tonelli, M. and Petti, M., 2010, "Finite volume scheme for the solution of 2D extended Boussinesq equations in the surf zone", Ocean Engrng., 37, 567-582.

Tonelli, M. and Petti, M., 2012, "Shock-capturing Boussinesq model for irregular wave propagation", Coast. Eng., 61, 8-19.

Toro, E. F., 2009, Riemann solvers and numerical methods for fluid dynamics: a practical introduction, Third edition, Springer, New York.

Wei, G., Kirby, J.T., Grilli, S.T., Subramanya, R., 1995, "A fully nonlinear Boussinesq model for surface waves: Part I. Highly nonlinear unsteady waves", J. Fluid Mech., 294, 71-92.

Yamamoto, S., Kano, S. and Daiguji, H, 1998, "An ecient CFD approach for simulating unsteady hypersonic shock-shock interference flows", Computers and Fluids, 27, 571-580.

Zijlema M. and Stelling G.S., 2005, "Further experiences with computing non-hydrostatic free-surface flows involving water waves", Int. J. Numer. Meth. Fluids, 48, 169-197

Zijlema M. and Stelling G.S., 2008, "Ecient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure", Coastal Engineering, 55, 780-790