A mixed analytical-numerical time domain approach to second-order diffraction

This paper describes a mixed analytical-numerical model for solving the second-order diffraction problem on single or multiple bottom-mounted vertical cylinders. The first-order solution is given by Linton & Evans quasi-analytical frequency domain formulation [14]. We use this solution as a basis for second-order time domain diffraction calculations with advantages both in terms of accuracy and computing effort. The better accuracy comes from the forcing terms in second-order free surface conditions being evaluated quasi-analytically. Regarding the computing effort, the advantage of the proposed scheme is two-fold. Only the second-order problem has to be solved numerically, and the free surface mesh is exclusively adapted to the second-order solution. This results in a substantial reduction of the problem size, especially if the focus is on the sum-frequency problem. Applications are presented for a single cylinder, and for a square array of four cylinders in regular and bichromatic waves. Results are shown to compare favorably both with a previously developed fully-numerical time domain solution of the second-order problem, and with frequency domain semi-analytical results.