  A stable and high-order accurate solver for linear and nonlinear parabolic
equations is presented. An additive Runge-Kutta method is used for the time
stepping, which integrates the linear stiff terms by an implicit singly
diagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by an
explicit Runge-Kutta (ERK) method. In each time step, the implicit solve is
performed by the recently developed Hierarchical Poincar\'e-Steklov (HPS)
method. This is a fast direct solver for elliptic equations that decomposes the
space domain into a hierarchical tree of subdomains and builds spectral
collocation solvers locally on the subdomains. These ideas are naturally
combined in the presented method since the singly diagonal coefficient in
ESDIRK and a fixed time-step ensures that the coefficient matrix in the
implicit solve of HPS remains the same for all time stages. This means that the
precomputed inverse can be efficiently reused, leading to a scheme with
complexity (in two dimensions) $\mathcal{O}(N^{1.5})$ for the precomputation
where the solution operator to the elliptic problems is built, and then
$\mathcal{O}(N)$ for each time step. The stability of the method is proved for
first order in time and any order in space, and numerical evidence
substantiates a claim of stability for a much broader class of time
discretization methods.Numerical experiments supporting the accuracy of
efficiency of the method in one and two dimensions are presented.

Numerical Analysis;Numerical Analysis