Parameter Optimization in Waveform Relaxation for Fractional-Order $rC$ Circuits

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The longitudinal waveform relaxation (WR) proposed by Gander and Ruehli converges faster than the classical WR method. For the former, a free parameter α is contained, which has a significant effect on the convergence rate. The optimization of this parameter is thus an important issue in practice. Here, we apply this new WR method to the fractional-order RC circuits, and optimize such a parameter at the continuous and discrete levels (this gives two parameters αcopt and αdopt). We consider three simple but widely used convolution quadrature for discretization, based on the implicit-Euler method, the two-step backward difference formula, and the trapezoidal rule, and we derive the parameter αdopt for each quadrature. Interestingly, it is found that for the former two quadratures, the optimized parameter αdopt results in a much better convergence rate than αc;opt, while for the quadrature based on the trapezoidal rule, αd;opt and αcopt result in the same convergence rate.

Original languageBritish English
Article number7888540
Pages (from-to)1781-1790
Number of pages10
JournalIEEE Transactions on Circuits and Systems I: Regular Papers
Issue number7
StatePublished - Jul 2017


  • circuit simulation
  • fractional RC circuits
  • parameter optimization
  • Waveform relaxation


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