An efficient algorithm for high accuracy particle tracking in finite elements

Abstract

We propose an algorithm for particle tracking based on Cheng's method [Int. J. Numer. Meth. 39 (1996) 1111–1136]. Velocities in a flow field are known at a series of points and interpolated between them by finite element local functions. Tracking is performed in local coordinates, element by element, using any standard ODE solution method. The exit from an element is found using the polynomials to interpolate between the tracking points. The algorithm was tested and compared to Pollock's and Cheng's method in a series of numerical experiments, in which the Euler, Runge–Kutta 2, Runge–Kutta 5(4) and Runge–Kutta 6(4) ODE solution methods were combined with first-, second-, third- and fifth-order exit polynomials. The known velocities had a random error with standard deviation of 0%, 0.1% and 1% of the velocity. Meaningful results were obtained only when the spatial interpolation error and the error of the tracking method were calculated separately, otherwise some results were misleading. The numerical experiments confirmed that the accuracy of the exit polynomial has to be consistent with the ODE solution method. Quadratic interpolation of velocities on a coarser mesh often gives more accurate path lines and requires less computational time than linear interpolation. Pollock's method for particle tracking is viable only if input data are rather inaccurate and path lines nearly straight. Cheng's method is appropriate for moderately accurate input data, while the proposed algorithm with Runge–Kutta 5(4) or Runge–Kutta 6(4) method and fifth-order exit polynomial has excellent accuracy. Computational time is about 10 times longer than for Cheng's method while the accuracy is increased by several orders of magnitude.