A complex-scaled boundary integral equation for time-harmonic water waves

TL;DR

This paper introduces a new boundary integral equation method for 2D time-harmonic water waves using complex-scaled Green's functions and PML, enabling efficient truncation and resonance analysis with minimal computational complexity.

Contribution

It develops a novel BIE formulation employing complex-scaled Green's functions and PML, simplifying computations and allowing resonance detection via a linear eigenvalue problem.

Findings

Exponential decay of truncation errors despite logarithmic Green's function growth

Efficient truncation of unbounded domain with simple function evaluations

Ability to compute complex resonances through a linear eigenvalue problem

Abstract

This paper presents a novel boundary integral equation (BIE) formulation for the two-dimensional time-harmonic water-waves problem. It utilizes a complex-scaled Laplace's free-space Green's function, resulting in a BIE posed on the infinite boundaries of the domain. The perfectly matched layer (PML) coordinate stretching that is used to render propagating waves exponentially decaying, allows for the effective truncation and discretization of the BIE unbounded domain. We show through a variety of numerical examples that, despite the logarithmic growth of the complex-scaled Laplace's free-space Green's function, the truncation errors are exponentially small with respect to the truncation length. Our formulation uses only simple function evaluations (e.g. complex logarithms and square roots), hence avoiding the need to compute the involved water-wave Green's function. Finally, we show that the proposed approach can also be used to find complex resonances through a \emph{linear} eigenvalue problem since the Green's function is frequency-independent.