In this paper, we use the lattice Boltzmann method with the Bhatnagar-Gross-Krook linear collision operator to study the flow physics induced by a rigid lamina undergoing moderately large harmonic oscillations in a viscous fluid. We propose a refill procedure for the hydrodynamic quantities in the lattice sites that are in the vicinity of the oscillating lamina. The numerically estimated flow field is used to compute the complex hydrodynamic function that describes the added mass and hydrodynamic damping experienced by the lamina. Results of the numerical simulations are validated against theoretical predictions for small amplitude vibrations and experimental and numerical findings for moderately large oscillations.