The Waveform Relaxation method for systems of differential/algebraic equations

This paper reports efforts towards establishing a parallel numerical algorithm known as Waveform Relaxation (WR) for simulating large systems of differential/algebraic equations. The WR algorithm was established as a relaxation based iterative method for the numerical integration of systems of ODEs over a finite time interval. In the WR approach, the system is broken into subsystems which are solved independently, with each subsystem using the previous iterate waveform as “guesses” about the behavior of the state variables in other subsystems. Waveforms are then exchanged between subsystems, and the subsystems are then resolved repeatedly with this improved information about the other subsystems until convergence is achieved.

In this paper, a WR algorithm is introduced for the simulation of generalized high-index DAE systems. As with ODEs, DAE systems often exhibit a multirate behavior in which the states vary as differing speeds. This can be exploited by partitioning the system into subsystems as in the WR for ODEs. One additional benefit of partitioning the DAE system into subsystems is that some of the resulting subsystems may be of lower index and, therefore, do not suffer from the numerical complications that high-index systems do. These lower index subsystems may therefore be solved by less specialized simulations. This increases the efficiency of the simulation since only a portion of the problem must be solved with specially tailored code. In addition, this paper established solvability requirements and convergence theorems for varying index DAE systems for WR simulation.