 
Summary: High order quadratures for the evaluation of interfacial
velocities in axisymmetric Stokes flows
M. Nitschea
, H. D. Cenicerosb
, A. L. Karnialac
, S. Naderia
aDepartment of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131
bDepartment of Mathematics, University of California Santa Barbara, CA 93106
cBirkerod Gymnasium and International Baccalaureate School, 3460 Birkerod, Denmark
Abstract
We propose new high order accurate methods to compute the evolution of axisymmetric
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 5th order method. Second order, pointwise fifth order,
