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Boundary value technique for initial value problems based on Adams-type second derivative methods

In this article, we propose a family of second derivative Adams-type methods (SDAMs) of order up to 2k + 2 (k is the step number) for initial value problems. The methods are constructed through a continuous approximation of the SDAM which is obtained by multistep collocation. The continuous approximation is used to obtain initial value methods, which are simultaneously applied to generate all approximations on the entire interval. The order and the linear stability properties of the methods are discussed. Numerical experiments are performed and the results are compared with those of existing methods in the literature.