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Asymptotic Analysis for Overlap in Waveform Relaxation Methods for RC Type Circuits

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Abstract

Waveform relaxation (WR) methods are based on partitioning large circuits into sub-circuits which then are solved separately for multiple time steps in so called time windows, and an iteration is used to converge to the global circuit solution in each time window. Classical WR converges quite slowly, especially when long time windows are used. To overcome this issue, optimized WR (OWR) was introduced which is based on optimized transmission conditions that transfer information between the sub-circuits more efficiently than classical WR. We study here for the first time the influence of overlapping sub-circuits in both WR and OWR applied to RC circuits. We give a circuit interpretation of the new transmission conditions in OWR, and derive closed form asymptotic expressions for the circuit elements representing the optimization parameter in OWR. Our analysis shows that the parameter is quite different in the overlapping case, compared to the non-overlapping one. We then show numerically that our optimized choice performs well, also for cases not covered by our analysis. This paper provides a general methodology to derive optimized parameters and can be extended to other circuits or system of differential equations or space–time PDEs.

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Notes

  1. For an early approximate numerical optimization, see  [8], which followed a presentation of the first author at SciCADE 99 on why WR methods are slow, and a followup discussion. At the PDE level this problem is in the meantime much better understood, see [18] for a review.

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Funding

Funding was provided by Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (Grant No. 200020_168999 and 200020_178752).

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Authors and Affiliations

  1. Section of Mathematics, University of Geneva, 1211, Geneva, Switzerland

    Martin J. Gander & Pratik M. Kumbhar

  2. EMC Laboratory, Missouri University of Science and Technology, Rolla, MO, USA

    Albert E. Ruehli

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  1. Martin J. Gander
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  3. Albert E. Ruehli
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Correspondence to Pratik M. Kumbhar.

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Gander, M.J., Kumbhar, P.M. & Ruehli, A.E. Asymptotic Analysis for Overlap in Waveform Relaxation Methods for RC Type Circuits. J Sci Comput 84, 24 (2020). https://doi.org/10.1007/s10915-020-01270-5

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