Waveform relaxation: a convergence criterion for differential-algebraic equations (Q2316639)

The authors are concerned with the numerical solution of initial value problems of coupled systems consisting of one ordinary differential equation (ODE) and one differential-algebraic equation (DAE) \begin{align*} \dot{u} + b(t,u) &= c_1(x), \quad u(t_0) = u_0 \in \mathbb{R}^{n_u}, \\ E(x)\dot{x} + f(x) &= q(t) + c_2(u), \quad x(t_0) = x_0 \in \mathbb{R}^{n_x}, \end{align*} where \(u\) and \(x\) are unknown vector-valued functions. The right-hand side functions \(c_1\) and \(c_2\) describe the coupling of both systems. It is well-known that, in general, this coupled system may suffer from instabilities if the considered DAE is of high index. Several convergence criteria have been developed for index-1 DAEs. The main contribution of this research is to establish a convergence criterion for a coupled system of an index-2 DAE with an ODE. Finally, the authors introduce sufficient network topological criteria to the coupling that are easy to check and that guarantee convergence. 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 \(\leq 2\).