Waveform relaxation: a convergence criterion for differential-algebraic equations

While waveform relaxation (also known as dynamic iteration or co-simulation) methods are known to converge for coupled systems of ordinary differential equations (ODEs), they may suffer from instabilities for coupled differential-algebraic equations (DAEs). Several convergence criteria have been developed for index-1 DAEs. We present here a convergence criterion for a coupled system of an index-2 DAE with an ODE. The analysis is motivated by the wish to combine electromagnetic field simulation with circuit simulation in a stable manner. The spatially discretized Maxwell equations in vector potential formulation with Lorenz gauging represent an ODE system. A lumped circuit model via the established modified nodal analysis is known to be a DAE system of index ≤ 2. Finally, we present sufficient network topological criteria to the coupling that are easy to check and that guarantee convergence.