Application of multipole expansion technique to two-dimensional nonlinear free surface flows.

The original idea by Rokhlin (1985) to rapidly solve the Laplace equation is applied here for studying nonlinear free-surface flows. The boundary integral equations for the velocity field are discretized through the Euler-McLaurin quadrature formula, which, in spite of its simplicity, displays spectral convergence properties for regular boundary data. In order to solve the discretized boundary integral equations, an iterative solver for algebraic systems is coupled to a fast summation technique based on the multipoles expansion of the influence coefficients. The resulting algorithm allows for a small size of the code (O(N)) and fast computation (O(Nlog N)) without affecting the original convergence properties. Typical long-time evolution and large-scale computations, which often arise in nonlinear free-surface flows, are discussed to show the effectiveness of the developed approach.