An iterated pseudospectral method for delay partial differential equations

Abstract

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 d d t U k ( t ) = M α U k ( t ) + f α ( t , U t k − 1 ) . Here, M α 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, U t k is a functional argument computed from the previous iterate and the function f α , like the matrix M α , depends on the process of semi-discretization. Jacobi waveform relaxation splitting has the advantage of straightforward (because M α 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.