the particle velocity data is generated from a numerical simulation; therefore, the velocity data is only available at Several techniques for the numerical integration of particle paths in steady and unsteady vector (velocity) fields are analyzed. Most some set of discrete times, t n , for n Ο 0, 1, 2, ..., N. The of the analysis applies to unsteady vector fields, however, some particle paths may be calculated in a postprocessing or results apply to steady vector field integration. Multistep, concurrent (coprocessing) manner. For postprocessing, the multistage, and some hybrid schemes are considered. It is shown time planes of velocity data are stored at some set number that due to initialization errors, many unsteady particle path integraof iterations typically determined by the amount of disk tion schemes are limited to third-order accuracy in time. Multistage schemes require at least three times more internal data storage storage available. For concurrent processing, the vector than multistep schemes of equal order. However, for timesteps field can be updated every iteration of the flow algorithm within the stability bounds, multistage schemes are generally more or after some set number of iterations. In either case, the accurate. A linearized analysis shows that the stability of these velocity field, u, is only available at some set of discrete integration algorithms are determined by the eigenvalues of the instances in time. A consequence of the temporally discrete local velocity tensor. Thus, the accuracy and stability of the methods are interpreted with concepts typically used in critical point theory. nature of the velocity data is that the timestep cannot be set This paper shows how integration schemes can lead to erroneous by the integration algorithm. This is unlike instantaneous classification of critical points when the timestep is finite and fixed. streamline integration (or steady velocity fields) for which For steady velocity fields, we demonstrate that timesteps outside the timestep can be adaptively varied to account for rapid of the relative stability region can lead to similar integration errors.
trajectory changes thereby increasing accuracy. The goal From this analysis, guidelines for accurate timestep sizing are suggested for both steady and unsteady flows. In particular, using of this paper is to determine the factors which impact the simulation data for the unsteady flow around a tapered cylinder, accuracy, efficiency, and memory requirements of particle we show that accurate particle path integration requires timesteps integration schemes. Although the majority of this paper which are at most on the order of the physical timescale of the focuses on unsteady data integration, some discussion of flow.