An Analysis of 3D Particle Path Integration Algorithms

Several techniques for the numerical integration of particle paths in steady and unsteady vector (velocity) elds are analyzed. Most of the analysis applies to unsteady vector elds, however, some results apply to steady vector eld integration. Multi-step, multi-stage, and some hybrid schemes are considered. It is shown that due to initialization errors, many unsteady particle path integration schemes are limited to third order accuracy in time. Multi-stage schemes require at least three times more internal data storage than multi-step schemes of equal order. However, for timesteps within the stability bounds, multi-stage schemes are generally more accurate. A linearized analysis shows that the stability of these integration algorithms are determined by the eigenvalues of the local velocity tensor. Thus, the accuracy and stability of the methods are interpreted with concepts typically used in critical point theory. This paper shows how integration schemes can lead to erroneous classi cation of critical points when the timestep is nite and xed. For steady velocity elds, we demonstrate that timesteps outside of the relative stability region can lead to similar integration errors. From this analysis, guidelines for accurate timestep sizing are suggested for both steady and unsteady ows. In particular, using simulation data for the unsteady ow around a tapered cylinder, we show that accurate particle path integration requires timesteps which are at most on the order of the physical timescale of the ow. 3