  We present a hybrid asymptotic/numerical method for the accurate computation
of single and double layer heat potentials in two dimensions. It has been shown
in previous work that simple quadrature schemes suffer from a phenomenon called
"geometrically-induced stiffness," meaning that formally high-order accurate
methods require excessively small time steps before the rapid convergence rate
is observed. This can be overcome by analytic integration in time, requiring
the evaluation of a collection of spatial boundary integral operators with
non-physical, weakly singular kernels. In our hybrid scheme, we combine a local
asymptotic approximation with the evaluation of a few boundary integral
operators involving only Gaussian kernels, which are easily accelerated by a
new version of the fast Gauss transform. This new scheme is robust, avoids
geometrically-induced stiffness, and is easy to use in the presence of moving
geometries. Its extension to three dimensions is natural and straightforward,
and should permit layer heat potentials to become flexible and powerful tools
for modeling diffusion processes.
