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

We propose new high order accurate methods to compute the evolution of axi-symmetric interfacial Stokes flow. The velocity at a point on the interface is given by an integral over the surface. Quadrature rules to evaluate these integrals are developed using asymptotic expansions of the integrands, both locally about the point of evaluation, and about the poles, where the interface crosses the axis of symmetry. The local expansions yield methods that converge to the chosen order pointwise, for fixed evaluation point. The pole expansions yield corrections that remove maximal errors of low order, introduced by singular behaviour of the integrands as the evaluation point approaches the poles. An interesting example of roundoff error amplification due to cancellation is also addressed. The result is a uniformly accurate fifth order method. Second order, pointwise fifth order, and uniform fifth order methods are applied to compute three sample flows, each of which presents a different computational difficulty: an initially bar-belled drop that pinches in finite time, a drop in a strain flow that approaches a steady state, and a continuously extending drop. In each case, the fifth order methods significantly improve the ability to resolve the flow. The examples furthermore give insight into the effect of the corrections needed for uniformity. We determine conditions under which the pointwise method is sufficient to obtain resolved results, and others under which the corrections significantly improve the results.