[Purposes]This paper seeks a highly accurate sparse identification method for the nonlinear dynamical systems, combinating with traditional numerical analysis techniques. [Methods]Firstly, we ultilize a class of linear multistep methods to discrete nonlinear dynamical systems. Furthermore, for a noisy data set, we introduce the principle of generalized least square method for enhancing the robustness of the algorithm. Secondly, we use subspace pursuit algorithm to select a best candidate set of basis functions with the smallest coefficient error. Then, least square method is used to compute the remaining nonzero coefficients. [Findings]The proposed linear multistep subspace pursuit methods for identifying nonlinear dynamic systems possess high accuracy and robustness. [Conclusions]Numerical results are presented to demonstrate the effectiveness of the proposed methods.