Parameter Optimization in Waveform Relaxation for Fractional-Order $RC$ Circuits.

The longitudinal waveform relaxation (WR) proposed by Gander and Ruehli converges faster than the classical WR method. For the former, a free parameter \alpha 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 \alpha ^{c}_{\mathrm{ opt}} and \alpha ^{d}_{\mathrm{ opt}} ). 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 \alpha ^d\mathrm{ opt} for each quadrature. Interestingly, it is found that for the former two quadratures, the optimized parameter \alpha ^d\mathrm{ opt} results in a much better convergence rate than \alpha ^c\mathrm{ opt} , while for the quadrature based on the trapezoidal rule, \alpha ^d\mathrm{ opt} and \alpha ^c\mathrm{ opt} result in the same convergence rate. [ABSTRACT FROM AUTHOR]