HPS Accelerated Spectral Solvers for Time Dependent Problems

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.