Stability of Multi-Rate Simulation Algorithms

Richard Bednar and Roy E. Crosbie

Summer Computer Simulation Conference 2007 (SCSC 2007)
San Diego, California (USA), July 15-18, 2007

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Abstract

Multi-rate simulation, in which a differential-equation model is partitioned into segments that are simulated 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) and for real-time simulation of large systems with a wide dynamic range in which it may be necessary to simulate different parts of the system on different platforms using different frame rates.

The multi-rate approach does however raise questions of accuracy and stability because of delays in communicating data between segments and the complications arising from using different integration step lengths.

A stability analysis of multi-rate integration is presented in which a general form of the matrix difference equation that represents the numerical simulation process is developed. This general equation can be applied to a given combination of system differential equations and choice of explicit, single-step integration algorithm. The analysis yields stability criteria that provide information about permissible step lengths and stable ranges of 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. The analysis presented here is based on Euler integration, but the method can be generalized to other integration algorithms. Data transferred from the slower segment to the faster is assumed to be constant until updated (zero-order hold) and data from the fast segment to the slower is assumed to be the instantaneous value at the start of a slower step. Again, the method can be adapted for different data interchange approaches. A simple example illustrates the application of the method.

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