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A stable and high-order accurate solver for linear and nonlinear parabolicequations is presented. An additive Runge-Kutta method is used for the timestepping, which integrates the linear stiff terms by an implicit singlydiagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by anexplicit Runge-Kutta (ERK) method. In each time step, the implicit solve isperformed by the recently developed Hierarchical Poincar\'e-Steklov (HPS)method. This is a fast direct solver for elliptic equations that decomposes thespace domain into a hierarchical tree of subdomains and builds spectralcollocation solvers locally on the subdomains. These ideas are naturallycombined in the presented method since the singly diagonal coefficient inESDIRK and a fixed time-step ensures that the coefficient matrix in theimplicit solve of HPS remains the same for all time stages. This means that theprecomputed inverse can be efficiently reused, leading to a scheme withcomplexity (in two dimensions) $\mathcal{O}(N^{1.5})$ for the precomputationwhere 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 forfirst order in time and any order in space, and numerical evidencesubstantiates a claim of stability for a much broader class of timediscretization methods.Numerical experiments supporting the accuracy ofefficiency of the method in one and two dimensions are presented.

Fast and high-order approximation of parabolic equations using hierarchical direct solvers and implicit Runge-Kutta methods