Measuring error propagation in waveform relaxation algorithms

A new analysis tool is introduced that characterizes and measures subcircui t coupling and error attenuation in waveform relaxation (WR) circuit simula tion algorithms with full dimensionality. Unlike current methods that use h euristics to calculate scalar "coupling," this method captures all of the e ffects of error attenuation over time and space. The new method characteriz es the propagation of errors in the solution iterates by a linear time-vary ing (LTV) system model. It is shown that the LTV system model can be simpli fied by a careful discretization into an error propagation matrix which pro vides a simple and very complete characterization of the so-called "gains" in a circuit as errors propagate from one subcircuit to another, The concep t of error propagation matrices and the LTV system model are applied experi mentally and analytically to linear and nonlinear circuits to illustrate th e usefulness of these tools in understanding the convergence properties of WR methods.