  We devise a new time-stepping algorithm for two-dimensional nonlinear
unsteady surface and interfacial waves. The algorithm uses Cauchy's integral
formula, which only requires information on the interface, to solve Laplace
equation by using iterative techniques. We derive Eulerian and mixed
Eulerian-Lagrangian descriptions by using arclength to parameterize the
interface which is updated through its inclination angle and velocity potential
at each time step. The algorithm shows broad applicability and excellent
numerical accuracy in various numerical simulations, including wave breaking,
collisions of solitary waves, vortex roll-up, etc. We especially focus on the
stability of symmetric interfacial gravity-capillary solitary waves in deep
water. Linear stability analysis is performed using a new formulation which
possesses excellent numerical efficiency and robustness. It is shown that the
depression/elevation solitary waves are linearly stable/unstable except the
portion where the monotonicity of energy curve changes firstly. These results
are supported by our fully nonlinear simulations, especially the head-on
collisions of solitary waves.
