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Computing periodic solutions of linear differential-algebraic equations by waveform relaxation

Authors: Yao-Lin Jiang and Richard M. M. Chen
Journal: Math. Comp. 74 (2005), 781-804
MSC (2000): Primary 37M05, 65F10, 65L10, 65L80, 65Y05
Published electronically: July 20, 2004
MathSciNet review: 2114648
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Abstract: We propose an algorithm, which is based on the waveform relaxation (WR) approach, to compute the periodic solutions of a linear system described by differential-algebraic equations. For this kind of two-point boundary problems, we derive an analytic expression of the spectral set for the periodic WR operator. We show that the periodic WR algorithm is convergent if the supremum value of the spectral radii for a series of matrices derived from the system is less than 1. Numerical examples, where discrete waveforms are computed with a backward-difference formula, further illustrate the correctness of the theoretical work in this paper.

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  • 1. U.M. Ascher and L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-algebraic Equations, SIAM, Philadelphia, 1998. MR 99k:65052
  • 2. M. Bjørhus and A.M. Stuart, Waveform relaxation as a dynamical system, Mathematics of Computation 66:219(1997), 1101-1117. MR 97i:34059
  • 3. G.D. Gristede, A.E. Ruehli, and C.A. Zukowski, Convergence properties of waveform relaxation circuit simulation methods, IEEE Transactions on Circuits and Systems - Part I 45:7(1998), 726-738.MR 99c:94067
  • 4. C.S. Hsu, Cell-to-Cell Mapping - A Method of Global Analysis for Nonlinear Systems, Springer-Verlag, New York, 1987. MR 89d:58115
  • 5. J. Janssen and S. Vandewalle, Multigrid waveform relaxation on spatial finite element meshes: The continuous-time case, SIAM Journal on Numerical Analysis 33:2(1996), 456-474. MR 97c:65158
  • 6. Y.L. Jiang, R.M.M. Chen, and O. Wing, Improving convergence performance of relaxation-based transient analysis by matrix splitting in circuit simulation, IEEE Transactions on Circuits and Systems - Part I 48:6(2001), 769-780.
  • 7. Y.L. Jiang, W.S. Luk, and O. Wing, Convergence-theoretics of classical and Krylov waveform relaxation methods for differential-algebraic equations, IEICE Transactions on Fundamentals of Electronics, Communications, and Computer Sciences E80-A:10(1997), 1961-1972.
  • 8. Y.L. Jiang and O. Wing, Monotone waveform relaxation for systems of nonlinear differential-algebraic equations, SIAM Journal on Numerical Analysis 38:1(2000), 170-185. MR 2001i:65088
  • 9. Y.L. Jiang and O. Wing, A note on the spectra and pseudospectra of waveform relaxation operators for linear differential-algebraic equations, SIAM Journal on Numerical Analysis 38:1(2000), 186-201. MR 2001h:65089
  • 10. Y.L. Jiang, On time-domain simulation of lossless transmission lines with nonlinear terminations, to appear in SIAM Journal on Numerical Analysis, 2004.
  • 11. Y.L. Jiang and O. Wing, A note on convergence conditions of waveform relaxation algorithms for nonlinear differential-algebraic equations, Applied Numerical Mathematics 36:2-3(2001), 281-297. MR 2002e:65107
  • 12. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1984. MR 96a:47025
  • 13. U. Miekkala and O. Nevanlinna, Iterative solution of systems of linear differential equations, Acta Numerica, pp. 259-307, 1996. MR 99f:65104
  • 14. S. Vandewalle and R. Piessens, On dynamic iteration methods for solving time-periodic differential equations, SIAM Journal on Numerical Analysis 30:1(1993), 286-303. MR 94b:65125
  • 15. L. Weinberg, Network Analysis and Synthesis, McGraw-Hill Book Company, New York, 1962.
  • 16. J.K. White and A. Sangiovanni-Vincentelli, Relaxation Techniques for the Simulation of VLSI Circuits, Kluwer Academic Publishers, Boston, 1986. MR 88b:94024

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Additional Information

Yao-Lin Jiang
Affiliation: Department of Mathematical Sciences, Xi’an Jiaotong University, Xi’an, People’s Republic of China

Richard M. M. Chen
Affiliation: School of Creative Media, City University of Hong Kong, Hong Kong, People’s Republic of China

Keywords: Differential-algebraic equations, periodic solutions, waveform relaxation, spectra of linear operators, linear multistep methods, finite-difference, numerical analysis, scientific computing, circuit simulation
Received by editor(s): June 5, 2002
Received by editor(s) in revised form: August 25, 2003
Published electronically: July 20, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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