Both positivity-preserving and iteration acceleration techniques are key issues when employing numerical methods to solve neutron transport equation. An algorithm combining $S_2$ synthetic acceleration method and positivity-preserving techniques to solve neutron transport equation that defined in one-dimensional planar geometry. The linear discontinuous Galerkin methods are applied to solve both the transport equation and $S_2$ equations. Two kind of simple limiters are involved. The numerical results for solving the classical Reed problem testify the efficiency of our methods.