Second derivative two-step collocation methods for ordinary differential equations

We introduce second derivative two-step collocation methods for the numerical integration of ordinary differential equations. In these methods, the solution of the problem in each step depends on the numerical solution in some points in the two previous steps. These methods are obtained using the collocation approach by relaxing some of the collocation conditions to obtain methods with desirable stability properties. The construction technique, analysis of the order of accuracy and linear stability properties of the methods are described. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.