  We consider the numerical algorithm for the two-dimensional time-harmonic
elastic wave scattering by unbounded rough surfaces with Dirichlet boundary
condition. A Nystr\"{o}m method is proposed for the scattering problem based on
the integral equation method. Convergence of the Nystr\"{o}m method is
established with convergence rate depending on the smoothness of the rough
surfaces. In doing so, a crucial role is played by analyzing the singularities
of the kernels of the relevant boundary integral operators. Numerical
experiments are presented to demonstrate the effectiveness of the method.
