Computation of water waves modelled by Nekrasov's equation

Nekrasov's integral equation, describing water waves of almost extreme form, is solved numerically. The method consists of applying a simple quadrature rule to a rearranged version of the original equation. Strongly graded meshes are used to resolve an expected boundary layer in the solution. For methods based on the trapezoidal rule, global bifurcation theory is used to prove, for fixed discretization parameter n, the existence of a continuous branch of positive numerical solutions. These are parametrized by μ a natural parameter occurring in the original integral equation. For fixed μ, collective compactness arguments then prove subsequential convergence of these solutions as the mesh is refined (i.e., as n→∞). Numerical experiments using higher-order quadrature rules are reported. These reveal that the method is capable of detecting a boundary layer and Gibbs phenomenon type oscillations of maximum height about 0.37° in a region of width O(70/μ) for μ large (typically με [10¹⁰, 10²²]). The meshes used to obtain such solutions contain some subintervals that are smaller than a typical machine epsilon.