  Two fundamental difficulties are encountered in the numerical evaluation of
time-dependent layer potentials. One is the quadratic cost of history
dependence, which has been successfully addressed by splitting the potentials
into two parts - a local part that contains the most recent contributions and a
history part that contains the contributions from all earlier times. The
history part is smooth, easily discretized using high-order quadratures, and
straightforward to compute using a variety of fast algorithms. The local part,
however, involves complicated singularities in the underlying Green's function.
Existing methods, based on exchanging the order of integration in space and
time, are able to achieve high order accuracy, but are limited to the case of
stationary boundaries. Here, we present a new quadrature method that leaves the
order of integration unchanged, making use of a change of variables that
converts the singular integrals with respect to time into smooth ones. We have
also derived asymptotic formulas for the local part that lead to fast and
accurate hybrid schemes, extending earlier work for scalar heat potentials and
applicable to moving boundaries. The performance of the overall scheme is
demonstrated via numerical examples.
