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 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 αoptcand αoptd). 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 αoptdfor each quadrature. Interestingly, it is found that for the former two quadratures, the optimized parameter αoptdresults in a much better convergence rate than αoptc, while for the quadrature based on the trapezoidal rule, αoptdand αoptcresult in the same convergence rate.