A mathematical analysis of optimized waveform relaxation for a small RC circuit

Waveform relaxation techniques are an important tool for the simulation of very large scale circuits. They are based on a partition of the circuit into sub-circuits, and then use an iteration between sub-circuits to converge to the solution of the entire circuit. Their importance has increased with the wide availability of parallel computers with a large number of processors. Unfortunately classical waveform relaxation is hampered by slow convergence, but this can be addressed by better transmission conditions, which led to the new class of optimized waveform relaxation methods. In these methods, both voltage and current information is exchanged in a combination which can be optimized for the performance of the method. We prove in this paper a conjecture for the optimal combination for the particular case of a small RC circuit, and also present and analyze a transmission condition which includes a time derivative.