Combined nonstandard numerical methods for ODEs with polynomial right-hand sides

In this paper we formulate conditions that guarantee second-order accuracy of the one-step nonstandard finite difference methods for general first order autonomous ordinary differential equations. We also develop a new class of nonstandard finite-difference methods for differential equations of the form dx/dt=polynomial(x), based on earlier results obtained in reference [H.V. Kojouharov, B.M. Chen, Nonstandard Eulerian–Lagrangian methods for multi-dimensional reactive transport problems, Appl. Numer. Math. 49 (2) (2004) 225–243]. Error analysis and numerical examples that demonstrate the performance of the proposed new method are also provided.