Numerical dispersion limits the accuracy of finite-difference elastic-wave modeling, particularly when relatively short spatial stencils are used. This study evaluates staggered-grid spatial-derivative coefficients obtained using a constrained simulated-annealing procedure and compares their spectral and numerical behavior primarily with Taylor-expansion coefficients within the same numerical framework. The coefficient design minimizes the maximum absolute spectral approximation error over a prescribed normalized-wavenumber interval while enforcing first-derivative consistency and an alternating, decreasing-magnitude coefficient pattern. The homogeneous and modified Marmousi examples indicate reduced visible dispersion relative to Taylor coefficients at the same stencil half-length. Least-squares coefficients are retained only as an additional qualitative reference in the homogeneous-model tests. The conclusions are limited to the tested two-dimensional isotropic models, grid spacing, time step, and source frequencies.