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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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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