||We present results showing the measured Landau damping rate using a high-order discontinuous Galerkin particle-in-cell (DG-PIC) [G. B. Jacobs and J. S. Hesthaven, J. Comput. Phys. 214, 96 (2006)] method. We show that typical damping rates measured in particle-in-cell (PIC) simulations can differ significantly from the linearized Landau damping coefficient and propose a simple numerical method to solve the plasma dispersion function exactly for moderate to high damping rates. Simulation results show a high degree of agreement between the high-order PIC results and this calculated theoretical damping rate.