Disclosed are systems and methods for numerically simulating seismic-wave propagation in tilted transversely isotropic (TTI) media, using an adaptive Lebedev staggered grid. In various embodiments, the adaptive grid includes multiple horizontal zones having different associated grid spacings, which may be determined based on a vertical wave-velocity model. The numerical simulation may involve iteratively solving a set of finite-difference equations including finite-difference coefficients that vary spatially depending on the grid spacing. Additional embodiments and features are described.