Dommermuth, Douglas G.; Yue, Dick K. P. A high-order spectral method for the study of nonlinear gravity waves. (English) Zbl 0638.76016 J. Fluid Mech. 184, 267-288 (1987). We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number \((N=0(1000))\) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M, and the convergence with N and M is exponentially fast for waves up to approximately 80 % of Stokes limiting steepness (ka\(\sim 0.35)\). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations; calculations of long-time evolution of wavetrains using the modified fourth-order Zakharov equations; and experimental measurements of a travelling wave packet. As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies. Cited in 3 ReviewsCited in 134 Documents MSC: 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76M99 Basic methods in fluid mechanics Keywords:robust numerical method; nonlinear gravity waves; pseudospectral treatment; nonlinear free-surface conditions; convergence; efficiency; accuracy; fully nonlinear semi-Lagrangian computations; long-time evolution of wavetrains; modified fourth-order Zakharov equations; travelling wave packet PDFBibTeX XMLCite \textit{D. G. Dommermuth} and \textit{D. K. P. Yue}, J. Fluid Mech. 184, 267--288 (1987; Zbl 0638.76016) Full Text: DOI References: [1] DOI: 10.1017/S002211208200158X · Zbl 0485.76017 · doi:10.1017/S002211208200158X [2] DOI: 10.1063/1.863708 · doi:10.1063/1.863708 [3] DOI: 10.1017/S0022112085000180 · Zbl 0603.76014 · doi:10.1017/S0022112085000180 [4] Fornberg, Phil. Trans. R. Soc. Lond. 289 pp 373– (1978) [5] Dysthe, Proc. R. Soc. Lond. 369 pp 105– (1979) [6] DOI: 10.1017/S0022112081003169 · Zbl 0459.76011 · doi:10.1017/S0022112081003169 [7] DOI: 10.1063/1.861481 · Zbl 0341.76005 · doi:10.1063/1.861481 [8] Bryant, J. Austral. Math. Soc. 25 pp 2– (1983) [9] DOI: 10.1017/S0022112062001469 · Zbl 0117.43605 · doi:10.1017/S0022112062001469 [10] Stiassnie, J. Fluid Mech. 174 pp 299– (1987) [11] DOI: 10.1017/S0022112084001257 · Zbl 0551.76016 · doi:10.1017/S0022112084001257 [12] DOI: 10.1017/S0022112074000802 · Zbl 0286.76016 · doi:10.1017/S0022112074000802 [13] DOI: 10.1017/S0022112081002851 · Zbl 0494.76019 · doi:10.1017/S0022112081002851 [14] DOI: 10.1017/S0022112060001043 · Zbl 0094.41101 · doi:10.1017/S0022112060001043 [15] DOI: 10.1017/S0022112071001940 · Zbl 0229.76029 · doi:10.1017/S0022112071001940 [16] DOI: 10.1143/JPSJ.37.486 · doi:10.1143/JPSJ.37.486 [17] DOI: 10.1063/1.863075 · doi:10.1063/1.863075 [18] DOI: 10.1017/S0022112062000245 · Zbl 0105.20303 · doi:10.1017/S0022112062000245 [19] Longuet-Higgins, Proc. R. Soc. Lond. 350 pp 1– (1976) [20] Zakharov, Sov. Phys., J. Exp. Theor. Phys. 34 pp 62– (1972) [21] DOI: 10.1007/BF00913182 · doi:10.1007/BF00913182 [22] Yuen, Adv. Appl. Mech. 22 pp 67– (1982) [23] DOI: 10.1063/1.862122 · doi:10.1063/1.862122 [24] DOI: 10.1063/1.1694844 · Zbl 0294.76011 · doi:10.1063/1.1694844 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.