Chebyshev pseudospectral spatial discretization preconditioned by the Kosloff and Tal-Ezer transformation  is applied to hyperbolic and parabolic functional equations. A Jacobi waveform relaxation method is then applied to the resulting semi-discrete functional systems, and the result is a simple system of ordinary differential equations d/dtUk+1(t) = MαUk+1(t)+f(t,U kt). Here Mα is a diagonal matrix, k is the index of waveform relaxation iterations, U kt is a functional argument computed from the previous iterate and the function f, like the matrix Mα, depends on the process of semi-discretization. This waveform relaxation splitting has the advantage of straight forward, direct application of implicit numerical methods for time integration (which allow use of large time steps than explicit methods). Another advantage of Jacobi waveform relaxation is that the resulting systems of ordinary differential equation can be efficiently integrated in a parallel computing environment. The Kosloff and Tal-Ezer transformation preconditions the matrix Mα, and this speeds up the convergence of waveform relaxation. This transformation is based on a parameter α∈ (0, 1], thus we study the relationship between this parameter and the convergence of waveform relaxation with error bounds derived here for the iteration process. We find that convergence of waveform relaxation improves as α increases, with the greatest improvement at α=1 if the spatial derivative of the solution at the boundaries is near zero. These results are confirmed by numerical experiments, and they hold for hyperbolic, parabolic and mixed hyperbolic-parabolic problems with and without delay terms.
Mead, Jodi and Zubik-Kowal, Barbara. (2005). "An Iterated Pseudospectral Method for Functional Partial Differential Equations". Applied Numerical Mathematics, 55(2), 227-250. http://dx.doi.org/10.1016/j.apnum.2005.02.010