Nonlinear Particle Tracking for High-Order Elements

The problem of calculating particle trajectories on unstructured meshes using a high-order polynomial approximation of the velocity field is addressed. The calculation of the particle trajectory is based on a Runge–Kutta integration in time. A convenient way of implementing high-order approximations is to employ an auxiliary mapping that transforms a finite element into a topologically equivalent parent element within a normalized parametric space. This presents two possible choices of space in which to perform the time integration of the particle position: the physical space or the parametric space. We present algorithms for implementing both particle tracking strategies using high-order elements and discuss their merits. The main drawback of both methods is their reliance on nonlinear procedures to calculate the particle trajectory. A novel alternative hybrid approach that advances a particle in both the physical and the parametric space without requiring nonlinear iterations is proposed. The error introduced by the alternative linearized procedures and their effect in the rate of convergence of the time integration is discussed. Finally, the performance of the different algorithms is compared using a set of analytical and computational, linear and high-order, velocity fields.