Abstract
With Monte Carlo simulations, we investigate the non-equilibrium critical dynamics with a domain interface, taking the two-dimensional Ising model as an example. Starting from a semi-ordered state, the domain interface grows as time evolves. Dynamic scaling behavior is carefully analyzed. The magnetization and Binder cumulant inside the domain interface are governed by critical exponents different from those outside the domain interface. The ageing phenomenon is identified. For comparison, additional simulations starting from an ordered state have been performed with a free surface and a disordered surface.