  Nonlinear models for pattern evolution by ion beam sputtering on a material
surface present an ongoing opportunity for new numerical simulations. A
numerical analysis of the evolution of preexisting patterns is proposed to
investigate surface dynamics, based on a 2D anisotropic damped
Kuramoto-Sivashinsky equation, with periodic boundary conditions. A
finite-difference semi-implicit time splitting scheme is employed on the
discretization of the governing equation. Simulations were conducted with
realistic coefficients related to physical parameters (anisotropies, beam
orientation, diffusion). The stability of the numerical scheme is analyzed with
time step and grid spacing tests for the pattern evolution, and the Method of
Manufactured Solutions has been used to verify the proposed scheme. Ripples and
hexagonal patterns were obtained from a monomodal initial condition for certain
values of the damping coefficient, while spatiotemporal chaos appeared for
lower values. The anisotropy effects on pattern formation were studied, varying
the angle of incidence of the ion beam with respect to the irradiated surface.
Analytical discussions are based on linear and weakly nonlinear analysis.
