Comparison of Five Methods of Computing the Dirichlet–neumann Operator for the Water Wave Problem

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, realanalytic, 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 largeamplitude standing wave in deep water, and a pair of interacting traveling waves in finite depth.