Convergence analysis of the Neumann–Neumann waveform relaxation method for time-fractional RC circuits

Abstract

The classical waveform relaxation (WR) methods rely on decoupling the large-scale ODEs system into small-scale subsystems and then solving these subsystems in a Jacobi or Gauss–Seidel pattern. However, in general it is hard to find a clever partition and for strongly coupled systems the classical WR methods usually converge slowly and non-uniformly. On the contrary, the WR methods of longitudinal type, such as the Robin-WR method and the Neumann–Neumann waveform relaxation (NN-WR) method, possess the advantages of simple partitioning procedure and uniform convergence rate. The Robin-WR method has been extensively studied in the past few years, while the NN-WR method is just proposed very recently and does not get much attention. It was shown in our previous work that the NN-WR method converges much faster than the Robin-WR method, provided the involved parameter, namely β, is chosen properly. In this paper, we perform a convergence analysis of the NN-WR method for time-fractional RC circuits, with special attention to the optimization of the parameter β. For time-fractional PDEs, this work corresponds to the study of the NN-WR method at the semi-discrete level. We present a detailed numerical test of this method, with respect to convergence rate, CPU time and asymptotic dependence on the problem/discretization parameters, in the case of two- and multi-subcircuits.