Abstract
We examine spectral and pseudospectral methods as well as waveform relaxation methods for the wave equation in one space dimension. Our goal is to study block Gauss–Jacobi waveform relaxation schemes which can be efficiently implemented in a parallel computing environment. These schemes are applied to semidiscrete systems written in terms of sparse or dense matrices. It is demonstrated that the spectral formulations lead to the implicit system of ordinary differential equations Wã′ = Sã + g(t) w, with sparse matrices W and S which can be effectively solved by direct application of any Runge–Kutta method. We also examine waveform relaxation iterations based on splittings W = W 1−W 2 and S = S 1 + S 2 and demonstrate that these iterations are only linearly convergent on finite time windows. Waveform relaxation methods applied to the explicit system ã′ = W −1 Sã + g(t) W −1 w are somewhat faster but less convenient to implement since the matrix W −1 S is no longer sparse. The pseudospectral methods lead to the system Ũ′ = D Ũ + g(t) w with a differentiation matrix D of order one and the corresponding waveform relaxation iterations are much faster than the iterations corresponding to the spectral cases (both implicit and explicit).
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REFERENCES
Burrage, K. (1995). Parallel and Sequential Methods for Ordinary Differential Equations, Oxford University Press, Oxford.
Burrage, K., Jackiewicz, Z., Nørsett, S. P., and Renaut, R. (1996). Preconditioning waveform relaxation iterations for differential systems, BIT 36, 54–76.
Burrage, K., Jackiewicz, Z., and Welfert, B. D. Spectral approximation of time windows in the solution of dissipative linear differential equations, submitted.
Butcher, J. C. (1987). The numerical analysis of ordinary differential equations. Runge-Kutta and General Linear Methods, Wiley, Chichester, New York.
Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1988). Spectral Methods in Fluid Mechanics, Springer-Verlag, New York, Berlin, Heidelberg.
Dekker, K., and Verwer, J. G. (1984). Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam, New York, Oxford.
Hesthaven, J. S., and Gottlieb, D. Spectral Approximation of Partial Differential Equations, manuscript.
Gottlieb, D., and Orszag, S. A. (1977). Numerical Analysis of Spectral Methods: Theory and Applications, Society for Industrial and Applied Mathematics, Philadelphia.
Jackiewicz, Z., Kwapisz, M., and Lo, E. (1997). Waveform relaxation methods for functional differential systems of neutral type, J. Math. Anal. Appl. 207, 255–285.
Jackiewicz, Z., Owren, B., and Welfert, B. D. (1998). Pseudospectral of waveform relaxation operators, Comput. Math. Appl. 36, 67–85.
Jackiewicz, Z., and Welfert, B. D. Stability of pseudospectral approximations of the one-dimensional wave equation, submitted.
Janssen, J. (1997). Acceleration of Waveform Relaxation Methods for Linear Ordinary and Partial Differential Equations, Ph.D. Thesis, Department of Computer Science, Katholieke Universiteit Leuven.
Leimkuhler, B. (1993). Estimating waveform relaxation convergence, SIAM J. Sci. Comput. 14, 872–889.
Miekkala, U., and Nevanlinna, O. (1987). Convergence of dynamic iteration methods for initial value problems, SIAM J. Sci. Stat. Comput. 8, 459–482.
Nevanlinna, O. (1989). Remarks on Picard-Lindelöf iteration, Part I, BIT 29, 328–346.
Trefethen, L. N. (1997). Pseudospectral of linear operators, SIAM Rev. 39, 383–406.
Trefethen, L. N. (2000). Spectral Methods in Matlab, Society for Industrial and Applied Mathematics, Philadelphia.
Vandewalle, S. (1993). Parallel Multigrid Waveform Relaxation for Parabolic Problems, Teubner, B. G. (ed.), Stuttgart.
White, J. K., and Sangiovanni-Vincentelli, A. (1987). Relaxation Techniques for the Simulation of VLSI Circuits, Kluwer Academic Publishers, Boston, Dordrecht, Lancaster.
Zubik-Kowal, B. (2000). Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations, Appl. Numer. Math. 34, 309–328.
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Jackiewicz, Z., Welfert, B.D. & Zubik-Kowal, B. Spectral Versus Pseudospectral Solutions of the Wave Equation by Waveform Relaxation Methods. Journal of Scientific Computing 20, 1–28 (2004). https://doi.org/10.1023/A:1025900611963
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DOI: https://doi.org/10.1023/A:1025900611963