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

HPS Accelerated Spectral Solvers for Time Dependent Problems