Stable computation of the functional variation of the Dirichlet–Neumann operator

This paper presents an accurate and stable numerical scheme for computation of the first variation of the Dirichlet–Neumann operator in the context of Euler’s equations for ideal free-surface fluid flows. The Transformed Field Expansion methodology we use is not only numerically stable, but also employs a spectrally accurate Fourier/Chebyshev collocation method which delivers high-fidelity solutions. This implementation follows directly from the authors’ previous theoretical work on analyticity properties of functional variations of Dirichlet–Neumann operators. These variations can be computed in a number of ways, but we establish, via a variety of computational experiments, the superior effectiveness of our new approach as compared with another popular Boundary Perturbation algorithm (the method of Operator Expansions).