Coupled Electromagnetic Field and Electric Circuit Simulation: A Waveform Relaxation Benchmark

We consider coupled dynamical systems, arising from lumped circuit modeling coupled to distributed modeling of electromagnetic devices. The corresponding subsystems form an ordinary differential equation system, reflecting the spatially discretized electromagnetic field equations, see e.g. [33], and a system of differential-algebraic equations (DAEs), describing the circuit obtained by modified nodal analysis, see e.g. [28]. The different nature of the subsystems motivates us to solve the coupled systems by co-simulation in form of waveform relaxation methods, see e.g. [29]. It allows the use of sophisticated solvers for each subsystem. Furthermore, one can easily exploit the different structural properties of the subsystems. This becomes even more important due to the fact that the systems’ dimension may easily reach millions of unknowns. However, convergence of waveform relaxation methods is not always guaranteed as soon as DAEs are involved, see e.g. [5]. As one prototype of waveform relaxation methods we analyze the convergence behavior of the Gauss–Seidel approach. We present a criterion that guarantees convergence, supported by a numerical benchmark, and discuss the influence of three different coupling formulations onto the convergence behavior.