Optimized waveform relaxation solution of RLCG transmission line type circuits

Abstract

Abstract-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 optimized WR 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.

Natural Convection within Inversed T-Shaped Enclosure Filled by Nano-Enhanced Phase Change Material: Numerical Investigation

“…Various numerical methods can be exploited in order to find solutions of the differential equations whether in parallel or in sequential (see, e.g., [ 45 , 46 , 47 , 48 ]). Here, the main equations and accompanying initial boundary conditions are addressed using the Galerkin finite element method.…”

Enhancing the Melting Process of Shell-and-Tube PCM Thermal Energy Storage Unit Using Modified Tube Design

“…Roughly speaking, in engineering and applied sciences, mathematical modeling, analysis, and techniques are widely used (see for instance [ 47 , 48 , 49 , 50 , 51 ]). With that has been said, an enthalpy–porosity analysis is a well-used approach for assessing unstable heat transfer situations, such as PCM phase transitions.…”

“… Based on the above assumptions, the governing equation for the melting of NePCM in the wavy finned enclosure may be derived. The equations may be written as follows [ 47 ]. …”

Analysis of Overlap in Waveform Relaxation Methods for RC Circuits

“…Lemma 5 provides us the expressions for |ρ 0 (0, α)| and |ρ 0 (∞, α)| which are equal for α * 0 . Comparing the powers of the dominating terms of these expressions results in 1 2 − δ = δ which implies δ = 1 4 . Now, equating the coefficients of these dominating terms gives C α = √ 2 and this completes the proof.…”

“…There are also analyses at the discrete level for OWR applied to RC circuits [34]. WR and OWR were also studied for RLCG transmission lines [1,15,16,20,11]. A different approach for the interconnect circuits has been studied in [27], where the interconnect circuit is partitioned using Norton interfaces which depend upon resistors.…”