Wavefield extrapolation is at the heart of modeling, imaging, and Full waveform inversion. Solving the wave equation using finite-difference approximations allows for fast extrapolation of the wavefield. It, however, suffers from dispersion and stability-related limitations that might hamper its efficient or proper application to high frequencies. Spectral methods are far more accurate and stable with dispersion free solutions, but that usually comes at an additional cost to the extrapolation. We use the residual formulation of the spectral implementation based on Taylor's series expansion (second order) and Shanks transform to improve the efficiency of the spectral implementation. We specifically use a velocity based perturbation approach to enhance the speed by allowing for larger time steps. The multiple smaller velocity perturbations provides a utility to decrease the residual phase operator for a large time step, and alow us to us perturbation methods. We show these advantages numerically.