On computation of oscillating integrals of ship-wave theory

TL;DR

This paper develops and compares two numerical methods for evaluating oscillating integrals in ship-wave theory, improving accuracy and stability in calculating Green's functions for steady oceanic wave motion.

Contribution

It introduces a novel approach based on Levin's method with barycentric interpolation and applies steepest descent with Clenshaw--Curtis quadrature for efficient integral evaluation.

Findings

The barycentric Levin method reduces numerical instability.

Steepest descent with Clenshaw--Curtis quadrature effectively handles oscillations.

Numerical tests demonstrate the methods' accuracy across various parameters.

Abstract

Green's function of the problem describing steady forward motion of bodies in an open ocean in the framework of the linear surface wave theory (the function is often referred to as Kelvin's wave source potential) is considered. Methods for numerical evaluation of the so-called `single integral' (or, in other words, `wavelike') term, dominating in the representation of Green's function in the far field, are developed. The difficulty in the numerical evaluation is due to integration over infinite interval of the function containing two differently oscillating factors and the presence of stationary points. This work suggests two methods to approximate the integral and its derivatives. First of the methods is based on the idea suggested by D.Levin in 1982 --- evaluation of the integral is converted to finding a particular slowly oscillating solution of an ordinary differential equation. To overcome well-known numerical instability of Levin's collocation method, an alternative type of collocation is used; it is based on a barycentric Lagrange interpolation with a clustered set of nodes. The second method for evaluation of the wavelike term involves application of the steepest descent method and Clenshaw--Curtis quadrature. The methods are numerically tested and compared for a wide variety of parameters.