Numerical techniques on improving computational efficiency of spectral boundary integral method

Numerical techniques are suggested in this paper, in order to improve the computational efficiency of the spectral boundary integral method, initiated by Clamond & Grue [D. Clamond and J. Grue. A fast method for fully nonlinear water-wave computations. J. Fluid Mech. 2001; 447: 337-355] for simulating nonlinear water waves. This method involves dealing with the high order convolutions by using Fourier transform or inverse Fourier transform and evaluating the integrals with weakly singular integrands. A de-singularity technique is proposed here to help in efficiently evaluating the integrals with weak singularity. An anti-aliasing technique is developed in this paper to overcome the aliasing problem associated with Fourier transform or inverse Fourier transform with a limited resolution. This paper also presents a technique for determining a critical value of the free surface, under which the integrals can be neglected. Numerical tests are carried out on the numerical techniques and on the improved method equipped with the techniques. The tests will demonstrate that the improved method can significantly accelerate the computation, in particular when waves are strongly nonlinear.