Stability of multi-rate simulation algorithms

Multi-rate simulation, in which a differential-equation model is partitioned into segments that are solved using different integration step lengths, has the potential to speed up simulations significantly. This is an important consideration especially for studies that involve many repeated simulation runs (e.g. multi-parameter, multi-objective optimizations) as well as for real-time simulation of systems with a wide dynamic range. The multi-rate approach does however raise questions of accuracy and stability arising from the methods of communicating data between segments and the effects of using different integration step lengths. A stability analysis of multi-rate integration is presented in which a general form of vector difference equation is developed that can be applied to the combination of a given system and an explicit, single-step integration algorithm. This yields stability criteria that provide information about permissible step lengths and system parameters. For the purposes of this analysis a number of simplifying assumptions are made. It is assumed that the system is divided into two regions, that the differential equations are linear and that a zero-order hold is used in communicating data between segments.