Waveform relaxation on tightly coupled systems

The waveform relaxation method applied to a system composed of two tightly coupled subsystems L₁ and L₂ is studied. The authors insert three resistors of values -R , 2R, and -R in series with the system of conductors that connect L₁ and L₂. The new system is electrically equivalent to the original system, but the authors now partition the new system so that the two partitions overlap at the 2R resistor. They call the new system the overlapped partition. They present a necessary and sufficient condition for convergence of the waveform relaxation process applied to the overlapped partition, derive sufficient conditions that are practical, and discuss the choice of R and the choice of the step size to ensure convergence. Several simple examples are given to show how with the -R+2R-R insertion, convergence is achieved, whereas without it the relaxation process diverges. They also show by examples that the condition that every node is connected to a grounded capacitor is not necessary, even in a tightly coupled system