Optimized Waveform Relaxation Methods for RC Type Circuits

Abstract

Waveform Relaxation has been widely used in circuit theory, for the solution of large systems of ordinary differential equations and the solution of partial differential equations. In the past, clever partitioning schemes have been used for circuit theory applications to enhance convergence. However, a drawback of the classical waveform relaxation algorithm is the non-uniform convergence over the window in time for which the equations are integrated. We propose a new optimized waveform relaxation algorithm which greatly accelerates the convergence by introducing new transmission conditions. These conditions are responsible for the exchange of information between the subcircuits. We use two RC circuit examples to illustrate the theory, as well as the improved convergence behavior.

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

“…This can however be overcome with the newly developed class of optimized WR or oWR, see [9,[13][14][15]8,16], which has its roots in optimized Schwarz waveform relaxation methods, see [12,10,11,3] and also the related approach for circuits in [7,6]. In oWR, combinations of voltage and current values are transmitted at the interfaces between sub-circuits, and one can optimize the combination in order to obtain rapidly converging algorithms, which also led to the name oWR.…”

Optimized waveform relaxation solution of RLCG transmission line type circuits

“…This was an essential step for the convergence of the WR algorithm for strongly coupled circuits such as directly coupled TL circuits. They demonstrated that this new technique can be applied effectively to diffusive circuits in [15] and to TL circuits in [16], without overlapping subcircuits. These new techniques are called optimized waveform relaxation (oWR) algorithms since they are based on an optimization process.…”

Optimized Waveform Relaxation Methods for Longitudinal Partitioning of Transmission Lines

“…This is also an essential step in all of our strongly coupled circuit domain work. The first approach with the optimized transmission conditions in the circuit domain was presented for diffusive (RC) circuits in [26]. It was shown that new improved coupling between subsystems can greatly enhance the performance of WR methods.…”