  A high-order convergent numerical method for solving linear and non-linear
parabolic PDEs is presented. The time-stepping is done via an explicit, singly
diagonally implicit Runge-Kutta (ESDIRK) method of order 4 or 5, and for the
implicit solve, we use the recently developed "Hierarchial Poincare-Steklov
(HPS)" method. The HPS method combines a multidomain spectral collocation
discretization technique (a "patching method") with a nested-dissection type
direct solver. In the context under consideration, the elliptic solve required
in each time-step involves the same coefficient matrix, which makes the use of
a direct solver particularly effective. The manuscript describes the
methodology and presents numerical experiments.
