A new scheme for the calculation of spatial derivatives has been developed. The technique is based on recursive derivative operators that are generated by an fit in the spectral domain. The use of recursive operators enables us to extend acoustic and elastic wave simulations to shorter wavelengths. The method is applied to the numerical solution of the 2D acoustic wave equation and to the solution of the equations of 2D dynamic elasticity in an isotropic medium. An example of reverse-time migration of a synthetic data set shows that the numerical dispersion can be significantly reduced with respect to schemes that are based on finite differences. The method is tested for the solutions of the equations of dynamic elasticity by comparing numerical and analytic solutions to Lamb's problem.