An iterated pseudospectral method for delay partial differential equations

The Chebyshev pseudospectral semi-discretization preconditioned by a transformation in space is applied to delay partial differential equations. The Jacobi waveform relaxation method is then applied to the resulting semi-discrete delay systems, which gives simple systems of ordinary equations ddtUk(t)=MαUk(t)+fα(t,Utk−1). Here, Mα is a diagonal matrix, which depends on a parameter α∈[0,1], which is used in the transformation in space, k is the index of waveform relaxation iterations, Utk is a functional argument computed from the previous iterate and the function fα, like the matrix Mα, depends on the process of semi-discretization. Jacobi waveform relaxation splitting has the advantage of straightforward (because Mα is diagonal) application of implicit numerical methods for time integration. Another advantage of Jacobi waveform relaxation is that the resulting systems of ordinary differential equations can be efficiently integrated in a parallel computing environment. The spatial transformation is used to speed up the convergence of waveform relaxation by preconditioning the Chebyshev pseudospectral differentiation matrix. We study the relationship between the parameter α 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. These results are confirmed by numerical experiments for hyperbolic, parabolic and mixed hyperbolic-parabolic problems with and without delay terms.