Fast algorithms for spectral collocation with non-periodic boundary conditions

We present a method for the numerical solution of partial differential equations using spectral collocation. By employing a structured representation of linear operators we are able to use fast algorithms without being restricted to periodic boundary conditions. The underlying ideas are introduced and developed in the context of linearly implicit methods for stiff equations. We show how different boundary conditions may be applied and illustrate the technique on the Allen–Cahn equation and the diffusion equation.