Optimized Waveform Relaxation Methods for Longitudinal Partitioning of Transmission Lines

Abstract

Abstract-Waveform relaxation (WR) is a technique whichcan be used to solve large systems of ordinary differential equations (ODEs). It is especially suitable for the parallel solution of ODEs with multiple time scales, and has been successfully used for solution of electronic circuits and for solving partial differential equations (PDEs). The main issue limiting the utility of waveform relaxation is the class of problems with strong subsystem to subsystem couplings and long analysis time intervals resulting in non-uniform, slow convergence. Here we consider transmission line circuits since they represent an important part of a Spice type circuit solver. For transmission lines, the coupling between different lines is relatively weak, and thus partitioning in the transverse direction leads to very fast WR algorithms. However, longitudinal partitioning of transmission lines is very challenging due to the strong coupling which results. In this paper we propose an approach with improved convergence properties for strongly coupled longitudinal partitioning of transmission lines and other similarly strongly coupled circuits.Index Terms-Waveform relaxation and transmission lines, longitudinal partitioning, convergence analysis, fast convergence.

50 Years of Time Parallel Time Integration

“…For more information, see [30,7]. Waveform relaxation methods should thus never be used with classical transmission conditions, also when applied to circuits; optimized transmission conditions have also been proposed and analyzed for circuits, see for example [1,2] and references therein. …”

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

“…Domain decomposition methods are particularly well-adapted for the parallel solution of linear systems that appear in finite difference and finite element methods. Among the various domain decomposition methods [24,28], we focus our attention here on the Classical Schwarz Waveform Relaxation (CSWR) DDM [1,13,25,26,27,28,29,30,31,32,36]. Even if this method has received much attention over the past years for many applications, to the best of the authors' knowledge, the first application to the Schrödinger equation can be found in [32].…”

Longitudinal partitioning based waveform relaxation algorithm for transient analysis of long delay transmission lines

“…However, for tightly coupled MTLs, the above algorithm may require a large number of iterations to converge. An alternative waveform relaxation algorithm based on longitudinal partitioning of the line into lumped RLGC subcircuits has also been proposed in [5]. Generally, longitudinal partitioning of the transmission line using the conventional lumped RLGC model results in strong coupling between the subcircuits, leading to slow and inefficient convergence properties of the algorithm.…”

“…Generally, longitudinal partitioning of the transmission line using the conventional lumped RLGC model results in strong coupling between the subcircuits, leading to slow and inefficient convergence properties of the algorithm. To mitigate the above problem, in [5] the convergence was accelerated by exchanging additional voltage/current waveforms between the subcircuits followed by optimization routines. However, longitudinal partitioning of the line using lumped RLGC subcircuits is still not suitable for modeling long delay lines as the number of subcircuits required to implicitly model the delay of the line can quickly become exorbitant [7].…”

“…According to the DEPACT model, the transmission line is discretized into cascaded lumped circuit elements and lossless line segments. Modeling the lossless segments using the MoC algorithm allows an elegant and simple partitioning of the line into smaller subcircuits, which by construction are relatively weakly coupled [3] compared to lumped RLGC subcircuits of [5]. Consequently, the proposed waveform relaxation algorithm can converge rapidly without exchange of additional waveforms or time consuming optimization routines.…”