A collocation method for differential equations with one‐point Taylor initial conditions

This paper presents a new multistep collocation method for nth order differential equations. The interval of interest is first divided into N subintervals. By determining interval conditions in Taylor interpolation in each subinterval, Taylor polynomials are calculated with different step lengths. Then the obtained solutions in each subinterval are used as initial conditions in the next step. Optimal error is assessed using Peano lemma, which shows that the errors decay by decreasing the step length. In particular, for fixed step length h, the error is of O(m^?m), where m is the number of collocation points in each subinterval. Meanwhile, for fixed m, the error is of O(h^m). Numerical examples demonstrate the validity and high accuracy of the proposed method even for stiff problems.