Computing cost-effective particle trajectories in numerically calculated incompressible fluids using geometric methods

We present an novel algorithm for tracking massless solid particles in a divergence-free velocity field that is only available at discrete points in space and time such as those arising from a direct numerical simulation of Navier-Stokes. The algorithm creates a divergence-free approximation to the numerical field using matrix valued radial basis functions, which is integrated in time using a volume-preserving map. The resulting method is able to calculate accurate trajectories in a helical vortex using much larger step-sizes and a far lower number of interpolation points which results in a more efficient algorithm compared to a conventional scheme.