Optimized Waveform Relaxation Methods for Longitudinal Partitioning of Transmission Lines

Waveform relaxation (WR) is a technique that can be used to solve large systems of ordinary differential equations (ODEs). It is particularly suitable for the parallel solution of ODEs with multiple time scales and has successfully been used for the solution of electronic circuits and for solving partial differential equations. The main issue limiting the utility of WR is the class of problems with strong subsystem-to-subsystem couplings and long analysis time intervals resulting in nonuniform slow convergence. Here, we consider transmission-line (TL) circuits since they represent an important part of a Spice-type circuit solver. For TLs, 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 TLs is very challenging due to the strong coupling that results. In this paper, we propose an approach with improved convergence properties for strongly coupled longitudinal partitioning of TLs and other similarly strongly coupled circuits.