{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T23:17:43Z","timestamp":1778714263136,"version":"3.51.4"},"reference-count":72,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,10,13]],"date-time":"2018-10-13T00:00:00Z","timestamp":1539388800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00eda y Competitividad","doi-asserted-by":"publisher","award":["MTM2015-70490-C2-1-R"],"award-info":[{"award-number":["MTM2015-70490-C2-1-R"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00eda y Competitividad","doi-asserted-by":"publisher","award":["MTM2015-70490-C2-2-R"],"award-info":[{"award-number":["MTM2015-70490-C2-2-R"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002878","name":"Consejer\u00eda de Econom\u00eda, Innovaci\u00f3n, Ciencia y Empleo, Junta de Andaluc\u00eda","doi-asserted-by":"publisher","award":["P11-FQM-8179"],"award-info":[{"award-number":["P11-FQM-8179"]}],"id":[{"id":"10.13039\/501100002878","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2019,4]]},"DOI":"10.1007\/s10915-018-0849-9","type":"journal-article","created":{"date-parts":[[2018,10,13]],"date-time":"2018-10-13T11:10:03Z","timestamp":1539429003000},"page":"273-320","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["An Efficient Two-Layer Non-hydrostatic Approach for Dispersive Water Waves"],"prefix":"10.1007","volume":"79","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7602-5920","authenticated-orcid":false,"given":"C.","family":"Escalante","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6944-4019","authenticated-orcid":false,"given":"E. D.","family":"Fern\u00e1ndez-Nieto","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7162-9672","authenticated-orcid":false,"given":"T.","family":"Morales de Luna","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3164-7715","authenticated-orcid":false,"given":"M. J.","family":"Castro","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,10,13]]},"reference":[{"issue":"10","key":"849_CR1","doi-asserted-by":"publisher","first-page":"1287","DOI":"10.1061\/(ASCE)0733-9429(1984)110:10(1287)","volume":"110","author":"MB Abbott","year":"1984","unstructured":"Abbott, M.B., McCowan, A.D., Warren, I.R.: Accuracy of short wave numerical models. J. Hydraul. Eng. 110(10), 1287\u20131301 (1984)","journal-title":"J. Hydraul. Eng."},{"key":"849_CR2","doi-asserted-by":"publisher","first-page":"2759","DOI":"10.1016\/j.jcp.2009.12.015","volume":"229","author":"R Abgrall","year":"2010","unstructured":"Abgrall, R., Karni, S.: A comment on the computation of non-conservative products. J. Comput. Phys. 229, 2759\u20132763 (2010)","journal-title":"J. Comput. Phys."},{"key":"849_CR3","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1016\/j.jcp.2016.05.053","volume":"321","author":"JE Adsuara","year":"2016","unstructured":"Adsuara, J.E., Cordero-Carri\u00f3n, I., Cerd\u00e1-Dur\u00e1n, P., Aloy, M.A.: Scheduled relaxation jacobi method: Improvements and applications. J. Comput. Phys. 321, 369\u2013413 (2016)","journal-title":"J. Comput. Phys."},{"key":"849_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.coastaleng.2011.12.012","volume":"62","author":"C Ai","year":"2012","unstructured":"Ai, C., Jin, S.: A multi-layer non-hydrostatic model for wave breaking and run-up. Coast. Eng. 62, 1\u20138 (2012)","journal-title":"Coast. Eng."},{"issue":"11","key":"849_CR5","doi-asserted-by":"publisher","first-page":"1354","DOI":"10.1002\/fld.2317","volume":"66","author":"C Ai","year":"2010","unstructured":"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\u20131370 (2010)","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"849_CR6","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1007\/978-3-319-57394-6_24","volume-title":"Finite Volumes for Complex Applications VIII\u2013Hyperbolic, Elliptic and Parabolic Problems","author":"N A\u00efssiouene","year":"2017","unstructured":"A\u00efssiouene, N., Bristeau, M.-O., Godlewski, E., Mangeney, A., Par\u00e9s, C., Sainte-Marie, J.: Application of a combined finite element\u2013finite volume method to a 2D non-hydrostatic shallow water problem. In: Canc\u00e9s, C., Omnes, P. (eds.) Finite Volumes for Complex Applications VIII\u2013Hyperbolic, Elliptic and Parabolic Problems, pp. 219\u2013226. Springer, Cham (2017)"},{"key":"849_CR7","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1051\/m2an\/2010036","volume":"45","author":"E Audusse","year":"2011","unstructured":"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\u2013200 (2011). 01","journal-title":"ESAIM Math. Model. Numer. Anal."},{"issue":"2","key":"849_CR8","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1002\/fld.2566","volume":"69","author":"Y Bai","year":"2011","unstructured":"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\u2013429 (2011)","journal-title":"Int. J. Numer. Methods Fluids"},{"issue":"4","key":"849_CR9","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1002\/fld.3664","volume":"71","author":"Y Bai","year":"2012","unstructured":"Bai, Y., Cheung, K.F.: Depth-integrated free-surface flow with parameterized non-hydrostatic pressure. Int. J. Numer. Methods Fluids 71(4), 403\u2013421 (2012)","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"849_CR10","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/j.ocemod.2015.05.005","volume":"92","author":"Y Bai","year":"2015","unstructured":"Bai, Y., Cheung, K.F.: Dispersion and kinematics of multi-layer non-hydrostatic models. Ocean Model. 92, 11\u201327 (2015)","journal-title":"Ocean Model."},{"key":"849_CR11","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1016\/j.ocemod.2017.11.005","volume":"121","author":"Y Bai","year":"2018","unstructured":"Bai, Y., Cheung, K.F.: Linear shoaling of free-surface waves in multi-layer non-hydrostatic models. Ocean Model. 121, 90\u2013104 (2018)","journal-title":"Ocean Model."},{"key":"849_CR12","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/0378-3839(93)90022-Z","volume":"19","author":"S Beji","year":"1993","unstructured":"Beji, S., Battjes, J.A.: Experimental investigation of wave propagation over a bar. Coast. Eng. 19, 151\u2013162 (1993)","journal-title":"Coast. Eng."},{"key":"849_CR13","doi-asserted-by":"publisher","first-page":"4162","DOI":"10.1016\/j.jcp.2007.12.026","volume":"227","author":"C Berthon","year":"2008","unstructured":"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\u20134189 (2008)","journal-title":"J. Comput. Phys."},{"issue":"4","key":"849_CR14","doi-asserted-by":"publisher","first-page":"1479","DOI":"10.1016\/j.jcp.2010.11.015","volume":"230","author":"P Bonneton","year":"2011","unstructured":"Bonneton, P., Chazel, F., Lannes, D., Marche, F., Tissier, M.: A splitting approach for the fully nonlinear and weakly dispersive Green\u2013Naghdi model. J. Comput. Phys. 230(4), 1479\u20131498 (2011)","journal-title":"J. Comput. Phys."},{"key":"849_CR15","first-page":"55","volume":"17","author":"J Boussinesq","year":"1872","unstructured":"Boussinesq, J.: Th\u00e9orie 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\u2013108 (1872)","journal-title":"J. Math. Pures Appl."},{"issue":"4","key":"849_CR16","doi-asserted-by":"publisher","first-page":"961","DOI":"10.3934\/dcdsb.2015.20.961","volume":"20","author":"M-O Bristeau","year":"2015","unstructured":"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\u2013988 (2015)","journal-title":"Discrete Continuous Dyn. Syst. Ser. B"},{"issue":"4","key":"849_CR17","first-page":"173","volume":"34","author":"MJ Castro","year":"2012","unstructured":"Castro, M.J., Fern\u00e1ndez-Nieto, E.D.: A class of computationally fast first order finite volume solvers: PVM methods. SIAM J. Sci. Comput. 34(4), 173\u2013196 (2012)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"849_CR18","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1016\/j.mcm.2004.01.016","volume":"42","author":"MJ Castro","year":"2005","unstructured":"Castro, M.J., Ferreiro, A.M.F., Garc\u00eda-Rodr\u00edguez, J.A., Gonz\u00e1lez-Vida, J.M., Mac\u00edas, J., Par\u00e9s, C., V\u00e1zquez-Cend\u00f3n, 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\u2013439 (2005)","journal-title":"Math. Comput. Model."},{"key":"849_CR19","doi-asserted-by":"publisher","first-page":"1103","DOI":"10.1090\/S0025-5718-06-01851-5","volume":"75","author":"MJ Castro","year":"2006","unstructured":"Castro, M.J., Gallardo, J.M., Par\u00e9s, 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\u20131134 (2006)","journal-title":"Math. Comput."},{"issue":"271","key":"849_CR20","doi-asserted-by":"publisher","first-page":"1427","DOI":"10.1090\/S0025-5718-09-02317-5","volume":"79","author":"MJ Castro","year":"2010","unstructured":"Castro, M.J., Pardo, A., Par\u00e9s, C., Toro, E.F.: On some fast well-balanced first order solvers for nonconservative systems. Math. Comput. 79(271), 1427\u20131472 (2010)","journal-title":"Math. Comput."},{"key":"849_CR21","volume-title":"Handbook of Numerical Methods for Hyperbolic Problems, volume 18 of Handbook of Numerical Analysis, pp 131\u2013175","author":"MJ Castro","year":"2017","unstructured":"Castro, M.J., Morales de Luna, T., Par\u00e9s, C.: Chapter 6-well-balanced schemes and path-conservative numerical methods. In: Abgrall, R\u00e9mi, Shu, C.-W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems, volume 18 of Handbook of Numerical Analysis, pp 131\u2013175. Elsevier, Amsterdam (2017)"},{"issue":"4","key":"849_CR22","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1002\/(SICI)1097-0363(19990630)30:4<425::AID-FLD847>3.0.CO;2-D","volume":"30","author":"V Casulli","year":"1999","unstructured":"Casulli, V.: A semi-implicit finite difference method for non-hydrostatic free surface flows. Numer. Methods Fluids 30(4), 425\u2013440 (1999)","journal-title":"Numer. Methods Fluids"},{"issue":"9","key":"849_CR23","doi-asserted-by":"publisher","first-page":"1131","DOI":"10.1016\/S0895-7177(02)00264-9","volume":"36","author":"V Casulli","year":"2002","unstructured":"Casulli, V., Zanolli, P.: Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems. Math. Comput. Model. 36(9), 1131\u20131149 (2002)","journal-title":"Math. Comput. Model."},{"issue":"4","key":"849_CR24","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1002\/fld.1961","volume":"61","author":"L Cea","year":"2008","unstructured":"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\u2013410 (2008)","journal-title":"Int. J. Numer. Methods Fluids"},{"issue":"1","key":"849_CR25","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1007\/s10915-010-9395-9","volume":"48","author":"F Chazel","year":"2011","unstructured":"Chazel, F., Lannes, D., Marche, F.: Numerical simulation of strongly nonlinear and dispersive waves using a Green\u2013Naghdi model. J. Sci. Comput. 48(1), 105\u2013116 (2011)","journal-title":"J. Sci. Comput."},{"key":"849_CR26","unstructured":"Clamond, D., Dutykh, D. https:\/\/es.mathworks.com\/matlabcentral\/fileexchange\/39189-solitary-water-wave (2012)"},{"key":"849_CR27","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ocemod.2014.06.004","volume":"81","author":"H Cui","year":"2014","unstructured":"Cui, H., Pietrzak, J.D., Stelling, G.S.: Optimal dispersion with minimized poisson equations for non-hydrostatic free surface flows. Ocean Model. 81, 1\u201312 (2014)","journal-title":"Ocean Model."},{"key":"849_CR28","unstructured":"Dingemans, M.W.: Comparison of computations with Boussinesq-like models and laboratory measurements. Report H-1684.12, 32, Delft Hydraulics (1994)"},{"key":"849_CR29","doi-asserted-by":"publisher","first-page":"840","DOI":"10.1016\/j.apm.2017.01.030","volume":"45","author":"A Duran","year":"2017","unstructured":"Duran, A., Marche, F.: A discontinuous galerkin method for a new class of Green\u2013Naghdi equations on simplicial unstructured meshes. Appl. Math. Model. 45, 840\u2013864 (2017)","journal-title":"Appl. Math. Model."},{"issue":"1","key":"849_CR30","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/j.wavemoti.2013.06.007","volume":"51","author":"D Dutykh","year":"2014","unstructured":"Dutykh, D., Clamond, D.: Efficient computation of steady solitary gravity waves. Wave Motion 51(1), 86\u201399 (2014)","journal-title":"Wave Motion"},{"issue":"6","key":"849_CR31","doi-asserted-by":"publisher","first-page":"442","DOI":"10.1061\/(ASCE)0733-950X(2007)133:6(442)","volume":"133","author":"F Enet","year":"2007","unstructured":"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\u2013454 (2007)","journal-title":"J. Waterw. Port Coast. Ocean Eng."},{"key":"849_CR32","unstructured":"Escalante, C., Mac\u00edas, J., Castro, M. J.: Performance assessment of tsunami\u2013HySEA model for NTHMP tsunami currents benchmarking. Part I lab data. Coast. Eng. (2017)"},{"key":"849_CR33","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1016\/j.amc.2018.06.035","volume":"338","author":"C Escalante","year":"2018","unstructured":"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\u2013659 (2018)","journal-title":"Appl. Math. Comput."},{"key":"849_CR34","doi-asserted-by":"crossref","unstructured":"Fern\u00e1ndez-Nieto, E.D., Parisot, M., Penel, Y., Sainte-Marie, J.: Layer\u2013averaged approximation of Euler equations for free surface flows with a non-hydrostatic pressure. Commun. Math. Sci. (hal-01324012v3) (2018)","DOI":"10.4310\/CMS.2018.v16.n5.a1"},{"issue":"2","key":"849_CR35","doi-asserted-by":"publisher","first-page":"408","DOI":"10.1007\/s10915-013-9802-0","volume":"60","author":"ED Fern\u00e1ndez-Nieto","year":"2014","unstructured":"Fern\u00e1ndez-Nieto, E.D., Kon\u00e9, E.H., Chac\u00f3n\u00a0Rebollo, T.: A multilayer method for the hydrostatic Navier\u2013Stokes equations: a particular weak solution. J. Sci. Comput. 60(2), 408\u2013437 (2014)","journal-title":"J. Sci. Comput."},{"key":"849_CR36","doi-asserted-by":"publisher","first-page":"408","DOI":"10.1007\/s10915-013-9802-0","volume":"60","author":"ED Fern\u00e1ndez-Nieto","year":"2014","unstructured":"Fern\u00e1ndez-Nieto, E.D., Kon\u00e9, E.H., Rebollo, T.: Chac\u00f3n: A multilayer method for the hydrostatic Navier-Stokes equations: a particular weak solution. J. Sci. Comput. 60, 408\u2013437, 08 (2014)","journal-title":"J. Sci. Comput."},{"issue":"4","key":"849_CR37","doi-asserted-by":"publisher","first-page":"552","DOI":"10.1016\/S1001-6058(16)60768-0","volume":"29","author":"F Gallerano","year":"2017","unstructured":"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\u2013566 (2017)","journal-title":"J. Hydrodyn. Ser. B"},{"key":"849_CR38","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1017\/S0022112099007247","volume":"405","author":"MF Gobbi","year":"2000","unstructured":"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\u2013210 (2000)","journal-title":"J. Fluid Mech."},{"issue":"221","key":"849_CR39","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1090\/S0025-5718-98-00913-2","volume":"67","author":"S Gottlieb","year":"1998","unstructured":"Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge\u2013Kutta schemes. Math. Comput. 67(221), 73\u201385 (1998)","journal-title":"Math. Comput."},{"key":"849_CR40","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1017\/S0022112076002425","volume":"78","author":"A Green","year":"1976","unstructured":"Green, A., Naghdi, P.: A derivation of equations for wave propagation in water of variable depth. Fluid Mech. 78, 237\u2013246 (1976)","journal-title":"Fluid Mech."},{"issue":"1","key":"849_CR41","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1137\/1025002","volume":"25","author":"A Harten","year":"1983","unstructured":"Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35\u201361 (1983)","journal-title":"SIAM Rev."},{"issue":"10","key":"849_CR42","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1002\/fld.4361","volume":"84","author":"A Jeschke","year":"2017","unstructured":"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\u2013583 (2017)","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"849_CR43","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/j.jcp.2014.01.030","volume":"271","author":"M Kazolea","year":"2014","unstructured":"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\u2013305 (2014)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"849_CR44","doi-asserted-by":"publisher","first-page":"133","DOI":"10.4310\/CMS.2007.v5.n1.a6","volume":"5","author":"A Kurganov","year":"2007","unstructured":"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\u2013160, 03 (2007)","journal-title":"Commun. Math. Sci."},{"key":"849_CR45","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1016\/j.jcp.2014.11.016","volume":"282","author":"D Lannes","year":"2015","unstructured":"Lannes, D., Marche, F.: A new class of fully nonlinear and weakly dispersive Green\u2013Naghdi models for efficient 2D simulations. J. Comput. Phys. 282, 238\u2013268 (2015)","journal-title":"J. Comput. Phys."},{"issue":"2049","key":"849_CR46","doi-asserted-by":"publisher","first-page":"2637","DOI":"10.1098\/rspa.2004.1305","volume":"460","author":"P Lynett","year":"2004","unstructured":"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\u20132669 (2004)","journal-title":"Proc. R. Soc. Lond. A Math. Phys. Eng. Sci."},{"key":"849_CR47","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1016\/j.coastaleng.2004.05.004","volume":"51","author":"PJ Lynett","year":"2004","unstructured":"Lynett, P.J., Liu, P.L.-F.: Linear analysis of the multi-layer model. Coast. Eng. 51, 439\u2013454 (2004)","journal-title":"Coast. Eng."},{"issue":"2","key":"849_CR48","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/S0378-3839(02)00043-1","volume":"46","author":"PJ Lynett","year":"2002","unstructured":"Lynett, P.J., Wu, T.R., Liu, P.L.-F.: Modeling wave runup with depth-integrated equations. Coast. Eng. 46(2), 89\u2013107 (2002)","journal-title":"Coast. Eng."},{"key":"849_CR49","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1016\/j.ocemod.2017.04.003","volume":"114","author":"PJ Lynett","year":"2017","unstructured":"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., Gonz\u00e1lez-Vida, J.M., Grilli, S.T., Heitmann, T.W., Horrillo, J., Knolu, U., Kian, R., Kirby, J.T., Li, W., Mac\u00edas, 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\u201332 (2017)","journal-title":"Ocean Model."},{"key":"849_CR50","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.ocemod.2011.12.002","volume":"43","author":"G Ma","year":"2012","unstructured":"Ma, G., Shi, F., Kirby, J.T.: Shock-capturing non-hydrostatic model for fully dispersive surface wave processes. Ocean Model. 43, 22\u201335 (2012)","journal-title":"Ocean Model."},{"key":"849_CR51","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.ocemod.2011.12.002","volume":"43\u201344","author":"G Ma","year":"2012","unstructured":"Ma, G., Shi, F., Kirby, J.T.: Shock-capturing non-hydrostatic model for fully dispersive surface wave processes. Ocean Model. 43\u201344, 22\u201335 (2012)","journal-title":"Ocean Model."},{"key":"849_CR52","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/0378-3839(92)90019-Q","volume":"18","author":"PA Madsen","year":"1992","unstructured":"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\u2013204, (1992)","journal-title":"Coast. Eng."},{"issue":"2033","key":"849_CR53","doi-asserted-by":"publisher","first-page":"1075","DOI":"10.1098\/rspa.2002.1067","volume":"459","author":"PA Madsen","year":"2003","unstructured":"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\u20131104 (2003)","journal-title":"Proc. Math. Phys. Eng. Sci."},{"key":"849_CR54","first-page":"11","volume":"59","author":"M Muoz-Ruiz","year":"2004","unstructured":"Muoz-Ruiz, M.: On a non-homogeneous bi-layer shallow-water problem: smoothness and uniqueness results. Nonlinear Anal. Theory Methods Appl. 59, 11 (2004)","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"issue":"9\u201310","key":"849_CR55","first-page":"1175","volume":"17","author":"M Muoz-Ruiz","year":"2004","unstructured":"Muoz-Ruiz, M.: On a nonhomogeneous bi-layer shallow-water problem: an existence theorem. Differ. Integral Equ. 17(9\u201310), 1175\u20131200 (2004)","journal-title":"Differ. Integral Equ."},{"key":"849_CR56","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/S1468-1218(02)00019-6","volume":"4","author":"ML Muoz-Ruiz","year":"2003","unstructured":"Muoz-Ruiz, M.L., Chatelon, F.J., Orenga, P.: On a bi-layer shallow-water problem. Nonlinear Anal. Real World Appl. 4, 139\u2013171 (2003)","journal-title":"Nonlinear Anal. Real World Appl."},{"issue":"1","key":"849_CR57","first-page":"27","volume":"43","author":"G Narbona-Reina","year":"2009","unstructured":"Narbona-Reina, G., Zabsonr\u00e9, J.D.D., Fern\u00e1ndez-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\u201371 (2009)","journal-title":"Comput. Model. Eng. Sci."},{"key":"849_CR58","doi-asserted-by":"publisher","first-page":"618","DOI":"10.1061\/(ASCE)0733-950X(1993)119:6(618)","volume":"119","author":"O Nwogu","year":"1994","unstructured":"Nwogu, O.: An alternative form of the boussinesq equations for nearshore wave propagation. Waterw. Port Coast. Ocean Eng. 119, 618\u2013638 (1994)","journal-title":"Waterw. Port Coast. Ocean Eng."},{"issue":"4","key":"849_CR59","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1017\/S0022112067002605","volume":"27","author":"DH Peregrine","year":"1967","unstructured":"Peregrine, D.H.: Long waves on a beach. Fluid Mech. 27(4), 815\u2013827 (1967)","journal-title":"Fluid Mech."},{"key":"849_CR60","doi-asserted-by":"publisher","first-page":"306","DOI":"10.1016\/j.jcp.2013.12.048","volume":"271","author":"M Ricchiuto","year":"2014","unstructured":"Ricchiuto, M., Filippini, A.G.: Upwind residual discretization of enhanced Boussinesq equations for wave propagation over complex bathymetries. J. Comput. Phys. 271, 306\u2013341 (2014)","journal-title":"J. Comput. Phys."},{"key":"849_CR61","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1016\/j.coastaleng.2009.11.007","volume":"57","author":"V Roeber","year":"2010","unstructured":"Roeber, V., Cheung, K.F., Kobayashi, M.H.: Shock-capturing Boussinesq-type model for nearshore wave processes. Coast. Eng. 57, 407\u2013423 (2010)","journal-title":"Coast. Eng."},{"issue":"1","key":"849_CR62","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0378-3839(95)00017-2","volume":"26","author":"HA Schffer","year":"1995","unstructured":"Schffer, H.A., Madsen, P.A.: Further enhancements of Boussinesq-type equations. Coast. Eng. 26(1), 1\u201314 (1995)","journal-title":"Coast. Eng."},{"issue":"3","key":"849_CR63","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1002\/(SICI)1097-0363(19980915)28:3<541::AID-FLD738>3.0.CO;2-0","volume":"28","author":"PK Stansby","year":"1998","unstructured":"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\u2013563 (1998)","journal-title":"Int. J. Numer. Methods Fluids"},{"issue":"1","key":"849_CR64","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/fld.595","volume":"43","author":"G Stelling","year":"2003","unstructured":"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\u201323 (2003)","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"849_CR65","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1017\/S002211208700329X","volume":"185","author":"CE Synolakis","year":"1987","unstructured":"Synolakis, C.E.: The runup of solitary waves. Fluid Mech. 185, 523\u2013545 (1987)","journal-title":"Fluid Mech."},{"key":"849_CR66","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/0021-9991(79)90145-1","volume":"32","author":"B Leer Van","year":"1979","unstructured":"Van Leer, B.: Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov\u2019s method. Comput. Phys. 32, 101\u2013136 (1979)","journal-title":"Comput. Phys."},{"issue":"\u20131","key":"849_CR67","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1017\/S0022112095002813","volume":"294","author":"G Wei","year":"1995","unstructured":"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(\u20131), 71 (1995)","journal-title":"J. Fluid Mech."},{"issue":"5","key":"849_CR68","first-page":"518","volume":"4","author":"GB Whitham","year":"1976","unstructured":"Whitham, G.B., Wiley, : Linear and nonlinear waves. Earthq. Eng. Struct. Dyn. 4(5), 518\u2013518 (1976)","journal-title":"Earthq. Eng. Struct. Dyn."},{"issue":"2","key":"849_CR69","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1016\/0021-9991(84)90092-5","volume":"56","author":"JM Witting","year":"1984","unstructured":"Witting, J.M.: A unified model for the evolution nonlinear water waves. J. Comput. Phys. 56(2), 203\u2013236 (1984)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"849_CR70","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1061\/(ASCE)WW.1943-5460.0000032","volume":"136","author":"CH Wu","year":"2010","unstructured":"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\u2013118 (2010)","journal-title":"J. Waterw. Port Coast. Ocean Eng."},{"key":"849_CR71","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1002\/fld.1952","volume":"61","author":"Y Yamazaki","year":"2008","unstructured":"Yamazaki, Y., Kowalik, Z., Cheung, K.F.: Depth-integrated, non-hydrostatic model for wave breaking and run-up. Numer. Methods Fluids 61, 473\u2013497 (2008)","journal-title":"Numer. Methods Fluids"},{"issue":"2","key":"849_CR72","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1002\/fld.821","volume":"48","author":"M Zijlema","year":"2005","unstructured":"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\u2013197 (2005)","journal-title":"Int. J. Numer. Methods Fluids"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0849-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-018-0849-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0849-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,3]],"date-time":"2022-09-03T23:19:57Z","timestamp":1662247197000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-018-0849-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,13]]},"references-count":72,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,4]]}},"alternative-id":["849"],"URL":"https:\/\/doi.org\/10.1007\/s10915-018-0849-9","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,10,13]]},"assertion":[{"value":"19 December 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 October 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 October 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 October 2018","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}