MULTIRATE INTEGRATION OF TWO-TIME-SCALE DYNAMIC SYSTEMS

Abstract

Simulation of large physical systems often leads to initial value problems in which some of the solution components contain high frequency oscillations and/or fast transients, while the remaining solution components are relatively slowly varying. Such a system is referred to as two-time-scale (TTS), which is a partial generalization of the concept of stiffness. When using conventional numerical techniques for integration of TTS systems, the rapidly varying components dictate the use of small stepsizes, with the result that the slowly varying components are integrated very inefficiently. This could mean that the computer time required for integration is excessive. To overcome this difficulty, the system is partitioned into "fast" and "slow" subsystems, containing the rapidly and slowly varying components of the solution respectively. Integration is then performed using small stepsizes for the fast subsystem and relatively large stepsizes for the slow subsystem. This is referred to as multirate integration, and it can lead to substantial savings in computer time required for integration of large systems having relatively few fast solution components. This study is devoted to multirate integration of TTS initial value problems which are partitioned into fast and slow subsystems. Techniques for partitioning are not considered here. Multirate integration algorithms based on explicit Runge-Kutta (RK) methods are developed. Such algorithms require a means for communication between the subsystems. Internally embedded RK methods are introduced to aid in computing interpolated values of the slow variables, which are supplied to the fast subsystem. The use of averaging in the fast subsystem is discussed in connection with communication from the fast to the slow subsystem. Theoretical support for this is presented in a special case. A proof of convergence is given for a multirate algorithm based on Euler's method. Absolute stability of this algorithm is also discussed. Four multirate integration routines are presented. Two of these are based on a fixed-step fourth order RK method, and one is based on the variable step Runge-Kutta-Merson scheme. The performance of these routines is compared to that of several other integration schemes, including Gear's method and Hindmarsh's EPISODE package. For this purpose, both linear and nonlinear examples are presented. It is found that multirate techniques show promise for linear systems having eigenvalues near the imaginary axis. Such systems are known to present difficulty for Gear's method and EPISODE. A nonlinear TTS model of an autopilot is presented. The variable step multirate routine is found to be substantially more efficient for this example than any other method tested. Preliminary results are also included for a pressurized water reactor model. Indications are that multirate techniques may prove fruitful for this model. Lastly, an investigation of the effects of the step-size ratio (between subsystems) is included. In addition, several suggestions for further work are given, including the possibility of using multistep methods for integration of the slow subsystem.