1. Abstract Particle tracking can serve as a useful tool in engineering analysis, visualization, and is an essential component of many Eulerian-Lagrangian (EL) transport schemes. In this work, we consider particle tracking (PT) techniques that are robust and efficient for complex, transient velocity fields on unstructured computational meshes. Several test problems from common 2-D rotational and 3-D helical examples to a remediation scenario in a large-scale, 3-D groundwater system are used to evaluate the proposed PT schemes. Introduction Transport processes are central to many of the Army’s contemporary scientific and engineering challenges. Accurate characterization of the evolution and migration of quantities of interest is essential. Inaccurate, overly diffused or oscillatory transport simulation results can be quite misleading. EL transport schemes combine a Lagrangian representation for advection with Eulerian descriptions for sources/sinks and other physical processes. When they work well, EL methods provide significantly better resolution of advection using low- order approximations in time and space. EL formulations invariably contain a core PT component. PT quality dictates much of the accuracy of the whole EL approximation as well as efficiency on serial and parallel platforms. In the past, most research has focused on how well various EL schemes solve transport with PT error avoided or minimized by using velocity fields that allow analytical tracking. Here, we consider PT techniques that are robust and efficient for complex, transient velocity fields on unstructured computational meshes. Specifically, we formulate semi-analytical approximations for relevant, low-order velocity representations and develop general purpose techniques based on adaptive, variable-order Runge-Kutta (RK) integration with error control that are suitable for generic velocity approximations.

2. Particle Tracking Governing equation: Adaptive element-by-element tracking Multistage RK: Adaptive time steps: Semi-analytical tracking RT0 velocity: Semi-analytical solution:

3. Particle Tracking Error21 x 2141 x 41 81 x 81 11 0.2660.008880.00348 ∞∞ 0.1280.03280.0152 Example 2, Semi-analytical tracking error Helical velocity field In the first example, we consider semi-analytical tracking (SA) in a helical velocity field given by on the unit cube. The maximum error associated with projecting onto RT0 was 0.0360 on a coarse 11x11x11 mesh was 0.036 Level set vortex example Next, we consider the SA tracking for level set propagation, and compare SA tracking on a 41x41 mesh with a stabilized FEM solution from a 161x161 mesh. Groundwater remediation Here, we consider efficiency of the RK schemes for an RDX cleanup scenario

4. Conclusions The semi-analytical approach can be extended to unstructured simplicial meshes in R 2,3. The resulting scheme is accurate and may be a viable computational alternative for problems where RT0 velocities are available. Increasing the approximation order from two to four in our numerical element-by-element tracking scheme improved computational efficiency significantly. The adaptive RK45 further improved efficiency and provided formal error control to meet user-defined accuracy requirements. Steady-state rotation: Accelerated vortex: