Fast computation of bound components of free surface gravity waves

Abstract

Abstract This paper presents a computationally fast implementation of the third-order wave interaction theory of Madsen and Fuhrman (2012) for multidirectional irregular waves. The evaluation stage is reformulated from direct summation of sinusoidal waves in the physical domain to spectral accumulation, followed by two-dimensional inverse fast Fourier transform, reducing the cost of full-grid evaluation from $${\mathcal {O}}(N_gN_{\textrm{terms}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>terms</mml:mtext> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> per time step to $${\mathcal {O}}(N_{\textrm{terms}})+{\mathcal {O}}(N_{\textrm{g}}\log N_{\textrm{g}}).$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>terms</mml:mtext> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>g</mml:mtext> </mml:msub> <mml:mo>log</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>g</mml:mtext> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Numerical verification against direct physical-space evaluation shows machine-precision agreement, with typical errors of order $$10^{-13}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>13</mml:mn> </mml:mrow> </mml:msup> </mml:math> – $$10^{-12}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> </mml:msup> </mml:math> in tests using a finite-depth, high-steepness, multidirectional crossing-sea case. Representative benchmark cases show speedup of roughly three orders of magnitude. The method provides a practical route for higher-order nonlinear surface-field evaluation in larger multidirectional bound wave calculations and related time series where repeated time step evaluations are required, and more broadly suggests spectral-domain accumulation as a scalable strategy for problems involving large-scale superposition of sinusoidal wave components.