High order quadratures for the evaluation of interfacial velocities in axi-symmetric Stokes flows

Boundary integral methods have been used widely for the simulation of interfacial Stokes flows. These free surface representations reduce the problem to one defined solely on the fluid interface. Often the assumption of axi-symmetry is used to further reduce the dimensionality of the problem. However, it is difficult to obtain high order approximations to the resulting line integrals due to their singular nature and intricate structure. Existing quadrature rules for numerical integration of the interfacial velocity are, at best, of a limited second order accuracy. In this work, we analyze the problem of attaining higher order quadratures and propose new numerical approaches to overcome all the difficulties. These approaches are based on analytic and local error corrections constructed from an asymptotic analysis of the integrands. We present quadratures that achieve a uniform accuracy of up to order five and demonstrate numerically their superior accuracy and efficiency.