Optimized waveform relaxation methods for RC circuits: discrete case

¹ Sichuan University of Science and Engineering, Zigong, Sichuan 643000, P.R. China.
wushulinylp@163.com
² Department of Mathematics, Yarmouk University, 21163 Irbid, Jordan.
khaleel@yu.edu.jo
³ Department of Mathematics and Sciences, Khalifa University, 127788 Abu Dhabi, UAE.
mohammad.alkhaleel@kustar.ac.ae

Received: 17 January 2015
Revised: 28 August 2016
Accepted: 8 September 2016

Abstract

The optimized waveform relaxation (OWR) methods, benefiting from intelligent information exchange between subsystems – the so-called transmission conditions (TCs), are recognized as efficient solvers for large scale circuits and get a lot of attention in recent years. The TCs contain a free parameter, namely α, which has a significant influence on the convergence rates. So far, the analysis of finding the best parameter is merely performed at the continuous level and such an analysis does not take into account the influence of temporal discretizations. In this paper, we show that the temporal discretizations do have an important effect on the OWR methods. Precisely, for the Backward–Euler method, compared to the parameter α^c_opt from the continuous analysis, we show that the convergence rates can be further improved by using the one α^d_opt analyzed at the discrete level, while for the Trapezoidal rule, it is better to use α^c_opt. This conclusion is confirmed by numerical results.