Schwarz Waveform Relaxation for a Neutral Functional Partial Differential Equation Model of Lossless Coupled Transmission Lines

Abstract

In this paper, we investigate the convergence behavior of the Schwarz waveform relaxation (SWR) algorithm for partial neutral functional differential equations which arise from lossless coupled transmission lines. We first analyze the algorithm at continuous level with transmission conditions of Dirichlet and Robin type. We show that the convergence rate of the algorithm can be significantly improved by using the Robin transmission condition, provided the involved parameter is chosen properly. Then we discretize the model problem in space and investigate the SWR algorithm at the semidiscrete level. It would be desirable if the convergence could be further improved by considering the influence of the space discretization, since the semidiscrete SWR algorithm is closer to the fully discretized algorithm. For this point, we show that for the Dirichlet transmission condition the convergence rate of the SWR algorithm analyzed at the continuous and semidiscrete levels is equal. However, for the Robin transmission condition, using the parameter derived at the semidiscrete level can both ameliorate and deteriorate the convergence rate of the SWR algorithm in practical computations, compared to the one derived at the continuous level. In particular, when a relatively large space discretization size $\Delta x$ is used, the acceleration ability of the parameter derived at the semidiscrete level outperforms the one derived at continuous level, while if the mesh parameter $\Delta x$ is small, the latter outperforms the former. The obtained results can also predict how the convergence rate of the algorithm with Robin transmission condition behaves with respect to the time discretization size $\Delta t$ and the delay argument $\tau$. Numerical results are provided to validate our theoretical conclusions. (An erratum is attached.)