Fast channel simulation via waveform over-relaxation

Abstract

This paper presents a fast simulation method for long coupled multi-chip interconnects. The channel is represented through a time-domain passive delay-rational macromodel, which is identified from tabulated scattering samples. A two-level Waveform Relaxation (WR) framework is then applied in order to perform fast transient simulation of the terminated channel via an iterative process. A new over-relaxation scheme is introduced for improving the convergence of the WR iterations.

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

“…We presented an algebraic approach to the transient simulation of high-speed channels with nonlinear terminations, which combines a Waveform Relaxation framework with a Krylovsubspace iterative scheme and an inexact Newton method. The main advantage is generality, since the limitations of previous WR implementations in terms of convergence [9] or restriction to the linear case [10] are eliminated. As in previous WR approaches involving longitudinal partitioning, channel and termination operators are not solved concurrently but only forward evaluated during the iterations.…”

“…Note that channel and termination equations are never coupled and solved concurrently, but they are just evaluated alternatively during LPTP iterations. This scheme, which corresponds to a fixed point iteration, is only characterized by conditional convergence and is not generally applicable, as discussed in [9]. However, when convergence holds, the solution is attained very fast, as far as a compact form of the nonlinear termination operator ℱ is available, e.g., as a behavioral macromodel.…”

“…Despite its promise for fast solution of mixed nonlineardistributed circuits, the WR approach may fail to converge in some cases [9]. It is clear that any numerical scheme, as fast as it can be, cannot be employed for systematic and routine channel analysis when the occurrence of instability due to lack of convergence cannot be controlled a priori.…”

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

“…Convergence holds only if P is a contraction, as discussed in [30], [36], and [49]. This condition is not always verified [50], [51], and more robust approaches are required.…”

“…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.…”