Transient analysis of high-speed channels via Newton-GMRES Waveform Relaxation

Abstract

This paper presents a technique for the numerical simulation of coupled high-speed channels terminated by arbitrary nonlinear drivers and receivers. The method builds on a number of existing techniques. A Delayed-Rational Macromodel is used to describe the channel in compact form, and a general Waveform Relaxation framework is used to cast the solution as an iterative process that refines initial estimates of transient scattering waves at the channel ports. Since a plain Waveform Relaxation approach is not able to guarantee convergence, we turn to a more general class of nonlinear algebraic solvers based on a combination of the Newton method with a Generalized Minimal Residual iteration, where the Waveform Relaxation equations act as a preconditioner. The convergence of this scheme can be proved in the general case. Numerical examples show that very few iterations are indeed required even for strongly nonlinear terminations.

Macromodel-Based Iterative Solvers for Simulation of High-Speed Links With Nonlinear Terminations

“…Unfortunately, the analysis in [49] shows that when transverse partitioning (TP) is introduced to improve simulation efficiency, the convergence of the WR scheme is only conditional and cannot be guaranteed in all cases, despite some improvements based on possibly frequency-dependent over-relaxation have been documented [50], [51]. This consideration motivated the approach in [52] and [53], where more robust iterative solvers have been suggested to replace the fixed point iteration typical of basic WR implementations. This paper extends the preliminary results of [53], by introducing a general framework for iterative solution of highspeed channels with nonlinear terminations.…”

“…This consideration motivated the approach in [52] and [53], where more robust iterative solvers have been suggested to replace the fixed point iteration typical of basic WR implementations. This paper extends the preliminary results of [53], by introducing a general framework for iterative solution of highspeed channels with nonlinear terminations. A suitable time discretization process casts the transient simulation problem as the solution of a possibly very large (but very sparse) nonlinear algebraic system.…”