Numerical use of exterior singularities for computation of gravity waves in shallow water

Abstract

This paper describes fully nonlinear, fully dispersive computation for two-dimensional motion of periodic gravity waves of permanent form in an ideal fluid of arbitrary uniform depth. The rate of convergence of a series solution of this motion becomes slow with decrease of the water depth-to-wavelength ratio, and a considerably large number of terms in series are required to obtain accurate numerical solutions for long waves in shallow water, even if the wave amplitude is small. Domb–Sykes plots for coefficients of the computed series solutions show that the slow rate of convergence is due to a singularity outside the flow domain in a conformally mapped complex plane. This exterior singularity can be theoretically estimated using analytic continuation of an ordinary differential equation given by the free surface condition. It is shown that a conformal mapping regularizes this singularity and enlarges the radius of convergence of a series solution. In addition, iterative use of this mapping for regularization further improves convergence. This work proposes a new method to compute long waves in shallow water using regularization of the exterior singularity. Numerical examples demonstrate that the proposed method can produce accurate enough solutions for a wide range of water depth-to-wavelength ratios. Validity of cnoidal wave solutions is also discussed using comparison with the computed results.