J. Janssen and S. Vandewalle, “Multigrid Waveform Relaxa?tion of Spatial Finite Element Meshes: The Conti?nuous-Time Case,” SIAM Journal on Numerical Analysis, Vol. 33, No. 2, 1996, pp. 456-474. doi:10.1137/0733024

has been cited by the following article:

• TITLE: A Modified Discrete-Time Jacobi Waveform Relaxation Iteration

  AUTHORS: Yong Liu, Shulin Wu

  KEYWORDS: Discrete-Time Waveform Relaxation, Convergence, Parallel Computation, Chebyshev Polynomial, Jacobi Iteration, Optimal SOR

  JOURNAL NAME: Applied Mathematics, Vol.2 No.4, March 31, 2011

  ABSTRACT: In this paper, we investigate an accelerated version of the discrete-time Jacobi waveform relaxation iteration method. Based on the well known Chebyshev polynomial theory, we show that significant speed up can be achieved by taking linear combinations of earlier iterates. The convergence and convergence speed of the new iterative method are presented and it is shown that the convergence speed of the new iterative method is sharper than that of the Jacobi method but blunter than that of the optimal SOR method. Moreover, at every iteration the new iterative method needs almost equal computation work and memory storage with the Jacobi method, and more important it can completely exploit the particular advantages of the Jacobi method in the sense of parallelism. We validate our theoretical conclusions with numerical experiments.