  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.
