In this work, we compare different interpolation operators in the context of particle track- ing with an emphasis on situations involving velocity field with steep gradients. Since, in this case, most classical methods give rise to the Gibbs phenomenon (generation of oscil- lations near discontinuities), we present new methods for particle tracking based on sub- division schemes and especially on the Piecewise Parabolic Harmonic (PPH) scheme which has shown its advantage in image processing in presence of strong contrasts. First an ana- lytic univariate case with a discontinuous velocity field is considered in order to highlight the effect of the Gibbs phenomenon on trajectory calculation. Theoretical results are pro- vided. Then, we show, regardless of the interpolation method, the need to use a conserva- tive approach when integrating a conservative problem with a velocity field deriving from a potential. Finally, the PPH scheme is applied in a more realistic case of a time-dependent potential encountered in the edge turbulence of magnetically confined plasmas, to com- pare the propagation of density structures (turbulence bursts) with the dynamics of test particles. This study highlights the difference between particle transport and density trans- port in turbulent fields.