Measuring error propagation in waveform relaxation algorithms

A new analysis tool is introduced that characterizes and measures subcircuit coupling and error attenuation in waveform relaxation (WR) circuit simulation algorithms with full dimensionality. Unlike current methods that use heuristics to calculate scalar "coupling," this method captures all of the effects of error attenuation over time and space. The new method characterizes the propagation of errors in the solution iterates by a linear time-varying (LTV) system model. It is shown that the LTV system model can be simplified by a careful discretization into an error propagation matrix which provides a simple and very complete characterization of the so-called "gains" in a circuit as errors propagate from one subcircuit to another. The concept of error propagation matrices and the LTV system model are applied experimentally and analytically to linear and nonlinear circuits to illustrate the usefulness of these tools in understanding the convergence properties of WR methods.