Neumann-Neumann Waveform Relaxation methods for fractional RC circuits

The Waveform Relaxation (WR) methods are recognized as efficient solvers for large scale circuits and attract a lot of attention in recent years due to their favorable advantages where they are ideally suited for the use of multiple parallel processors for problems with multiple time scales. However, applying classical WR techniques to strongly coupled systems leads to non-uniform convergence. Therefore, more uniform WR methods have been developed. This paper is concerned to generalize the Neumann-Neumann waveform relaxation (NN-WR) method invented recently for time-dependent PDEs to time-fractional circuits which seems to be a promising method in circuit simulations. By choosing the RC circuit in infinite size as the model, we perform a convergence analysis for the NN-WR method and this corresponds to the analysis of this method for PDEs at the semi-discrete level. The NN-WR method contains a free parameter, namely β, which has a significant effect on the convergence rate. For PDEs, the analysis at the space-time continuous level shows β = 1 over 4, while the analysis in this paper shows that, at the semi-discrete level, i.e., for the circuit problem, we can have a better choice which leads to much faster convergence in practical computing. A comparison with the so-called Robin WR is also included.