Optimized waveform relaxation solution of RLCG transmission line type circuits

Today, parallel processing is necessary for the solution of large systems of ordinary differential equations (ODEs) as they are obtained from large electronic circuits or from discretizing partial differential equations (PDEs). Using a fine mesh in the discretization of these problems also leads to large compute times and large storage requirements. The waveform relaxation (WR) technique, which is ideally suited for the use of multiple processors for problems with multiple time scales has been used to solve such problems on parallel processors for such large systems of ODEs. However, applying the so-called classical WR techniques to strongly coupled systems leads to non-uniform slow convergence over a window in time for which the equations are integrated. In this paper, we present a so-called optimizedWR algorithm applied to transmission line circuit problems based on the longitudinal partitioning into segments. This greatly improves the convergence for strongly coupled RLCG transmission line (TL) type circuits. The method can be applied to other similar circuits. The method is based on optimal parameters that lead to the optimal convergence of the iterations. Here, we present a practical optimized WR algorithm which is easy to use and is computationally inexpensive.