  We develop a numerical method for solving a free boundary problem which
describes shape relaxation, by surface tension, of a long and thin bubble of an
inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell. The method
of solution of the exterior Dirichlet problem employs a classical boundary
integral formulation. Our version of the numerical method is especially
advantageous for following the dynamics of a very long and thin bubble, for
which an asymptotic scaling theory has been recently developed. Because of the
very large aspect ratio of the bubble, a direct implementation of the boundary
integral algorithm would be impractical. We modify the algorithm by introducing
a new approximation of the integrals which appear in the Fredholm integral
equation and in the integral expression for the normal derivative of the
pressure at the bubble interface. The new approximation allows one to
considerably reduce the number of nodes at the almost flat part of the bubble
interface, while keeping a good accuracy. An additional benefit from the new
approximation is in that it eliminates numerical divergence of the integral for
the tangential derivative of the harmonic conjugate. The interface's position
is advanced in time by using explicit node tracking, whereas the larger node
spacing enables one to use larger time steps. The algorithm is tested on two
model problems, for which approximate analytical solutions are available.
