In using multi-pass integration methods for real-time simulation of dynamic systems, it may be desirable to employ intermediate-pass state-variable calculations as real-time outputs. For example, the real-time version of 2nd-order Runge-Kutta (RTRK-2) utilizes two evaluations of each state variable per overall integration step.
However, in real-time simulations there may sometimes be a requirement for input and output data rates to be identical. This can be accomplished by using both half-frame and full frame data points as real-time outputs in the RTRK-2 simulation. When this is done, it is important to realize that the dynamic errors associated with the half-frame outputs will in general be larger than those of the full-frame outputs, which can lead to undesirable scatter in the dynamic errors of the output data points.
In this paper we introduce formulas which improve the accuracy of the intermediate output data points in multi-pass integration methods. This in turn permits the multi-pass methods to be utilized for real-time simulations with equal input and output data-point frame rates without compromising the dynamic accuracy of the simulation.