The Application Of Waveform Relaxation Method To RC Type Circuits

As waveform relaxation method is a well parallel method, it has been widely used forlarge-scale scientific and engineering computing. In this paper, we mainly study theapplication of waveform relaxation method to RC type circuits.Firstly, for a small RC circuit, we do some improvement on the basis of the classicalJacobi waveform relaxation method. A new method is proposed, which is named theimproved waveform relaxation method. We analyze the new method about theconvergence and the speed of convergence. Compared with the classical Jacobi waveformrelaxation method, the improved waveform relaxation method converges faster. Thenumerical experiments verify the theoretical analysis.Then, based on the idea of the optimized waveform relaxation method proposed byGander and Ruehli in2003, we do some optimization on the basis of the improvedwaveform relaxation method: adding a relaxation on the original transmission condition,and get a new transmission condition. A new method is proposed, which is named theimproved and optimized waveform relaxation method. We conduct in-depth theoreticalanalysis (convergence, the speed of convergence) on the new method. Compared with theimproved waveform relaxation method and Gander’s optimized waveform relaxationmethod, the improved and optimized waveform relaxation method converges faster, whichis a more efficient waveform relaxation method. We do a lot of numerical experimentswhich verify the theoretical results.Finally, we apply the improved waveform relaxation method to solving a large RCcircuit. To the larger RC circuit, the classical Jacobi waveform relaxation method and theoptimized waveform relaxation method converge significantly slower. But the improvedwaveform relaxation method still remains a very fast speed of convergence. The speed ofits convergence is more dependent on the size of the overlapping region between thesubsystems. The results of numerical experiments are consistent with the theoreticalresults.