Abstract

We present a novel boundary integral method for the time-harmonic water-waves problem. In order to avoid the expensive computation of the water-wave Green’s function, we use only the free-space Green’s function of Laplace’s equation. The price to pay is that this leads to an integral equation set not only on the bounded scatterer but also on the infinite free surface. If this integral is truncated for numerical purposes, either roughly or even smoothly using the so-called windowed Green’s function, spurious reflections of the surface wave are generated by the truncation, and the method does not work. To overcome this difficulty, our idea is to use a perfectly matched layer (PML) coordinate-stretching in the horizontal direction. As a consequence, the outgoing surface wave becomes exponentially decaying, so that the truncation error is exponentially small with respect to the length of the PML layer.The formulation uses only simple function evaluations (e.g. complex logarithmsand square roots). Note that this PML -BIE formulation for water waves remains affine with respect to the squared frequency, which makes it very attractive for solving spectral problems like the computation of complex scattering resonances.