Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
A modified Chebyshev pseudospectral method with an O(N–1) time step restriction
Journal of Computational Physics
Parallel and sequential methods for ordinary differential equations
Parallel and sequential methods for ordinary differential equations
NUMDIFF-7 Selected papers of the seventh conference on Numerical treatment of differential equations
Convergence of Waveform Relaxation Methods for Differential-Algebraic Systems
SIAM Journal on Numerical Analysis
Accuracy Enhancement for Higher Derivatives using Chebyshev Collocation and a Mapping Technique
SIAM Journal on Scientific Computing
Spectral collocation time-domain modeling of diffractive optical elements
Journal of Computational Physics
Applied Numerical Mathematics - Auckl numerical ordinary differential equations (ANODE 98 workshop)
Accuracy, Resolution, and Stability Properties of a Modified Chebyshev Method
SIAM Journal on Scientific Computing
On Error Estimates for Waveform Relaxation Methods for Delay-Differential Equations
SIAM Journal on Numerical Analysis
Spectral Versus Pseudospectral Solutions of the Wave Equation by Waveform Relaxation Methods
Journal of Scientific Computing
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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/dtUk(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 wave-form 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.