Abstract: We discuss numerical simulation of one-and two-dimensional nonlinear Schrödinger (NLS) equations (NLS).With numerical flux of diffusive generalized Riemann problem,a direct discontinuous Galerkin (DDG) method is proposed.L² stability of the DDG scheme is proved and it is shown that it is a conservative numerical scheme.The one-dimensional case indicates that the DDG scheme simulates various kinds of soliton propagations and it has excellent long-time numerical behaviors.Two-dimensional numerical results demonstrate that the method has high accuracy and is capable of capturing strong gradients.

Key words: nonlinear Schrödinger equation, direct discontinuous Galerkin method, stability