Comparison of five methods of computing the Dirichlet-Neumann operator for the water wave problem

TL;DR

This paper compares five numerical methods for computing the Dirichlet-Neumann operator in water wave problems, analyzing their effectiveness, stability, and limitations through various examples and regularization techniques.

Contribution

It provides a comprehensive comparison of five methods, highlighting their numerical stability issues and proposing regularization strategies for improved accuracy.

Findings

AFM methods can fail with certain band-limited profiles without regularization

TFE and BIM methods are more stable and better suited for numerical computation

Regularization via over-sampling and orthogonalization improves AFM methods' robustness

Abstract

We compare the effectiveness of solving Dirichlet-Neumann problems via the Craig-Sulem (CS) expansion, the Ablowitz-Fokas-Musslimani (AFM) implicit formulation, the dual AFM formulation (AFM*), a boundary integral collocation method (BIM), and the transformed field expansion (TFE) method. The first three methods involve highly ill-conditioned intermediate calculations that we show can be overcome using multiple-precision arithmetic. The latter two methods avoid catastrophic cancellation of digits in intermediate results, and are much better suited to numerical computation. For the Craig-Sulem expansion, we explore the cancellation of terms at each order (up to 150th) for three types of wave profiles, namely band-limited, real-analytic, or smooth. For the AFM and AFM* methods, we present an example in which representing the Dirichlet or Neumann data as a series using the AFM basis functions is impossible, causing the methods to fail. The example involves band-limited wave profiles of arbitrarily small amplitude, with analytic Dirichlet data. We then show how to regularize the AFM and AFM* methods by over-sampling the basis functions and using the singular value decomposition or QR-factorization to orthogonalize them. Two additional examples are used to compare all five methods in the context of water waves, namely a large-amplitude standing wave in deep water, and a pair of interacting traveling waves in finite depth.