The dynamic fracture problem under an impulsive load was analyzed using the meshless local Petrov-Galerkin method. The moving least squares (MLS) approximation was adopted to approximate displacement functions, and the penalty function method was used to impose the displacement boundary conditions. The Newmark method was applied in the time integration scheme. The stress fields near the crack tip of a double notched plate and time histories of dynamic stress intensity factors (DSIF) for mode-I and mode-II were obtained. The numerical example shows that the local Petrov-Galerkin method is feasible and effective to analyze the dynamic fracture problem under the impulsive load, and has advantages of higher precision, higher efficiency and easy implementation.