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 α^c_opt and α^d_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 α^d_opt for each quadrature. Interestingly, it is found that for the former two quadratures, the optimized parameter α^d_opt results in a much better convergence rate than α^c;_opt, while for the quadrature based on the trapezoidal rule, α^d;_opt and α^c_opt result in the same convergence rate.