Based on an average-derivative method, we developed a new nine-point numerical scheme for a frequency-domain elastic-wave equation. Compared with the classic nine-point scheme, this scheme reduces the required number of grid points per wavelength for equal and unequal directional spacings. The reduction in the number of grid points increases as the Poisson’s ratio becomes larger. In particular, as the Poisson’s ratio reaches 0.5, the numerical S-wave phase velocity of this new scheme becomes zero, whereas the classical scheme produces spurious numerical S-wave phase velocity. Numerical examples demonstrate that this new scheme produces more accurate results than the classical scheme at approximately the same computational cost.