An Overlapping One-Step Multiderivative Hybrid Block Method for Solving Second-Order Initial Value Problems

This paper presents a one-step multiderivative hybrid block method of Order 12 that incorporates an overlapping strategy, in which intrastep points from the previous block are reused in the current step to enhance accuracy and stability when solving linear and nonlinear initial value problems. The derivation incorporates a multistep collocation and interpolation technique, using power series as the basis function for the approximate solution. Within a one-step block, three intrastep points are considered. As a foundational step, a non-overlapping one-step multiderivative scheme is first developed and expressed in matrix form. The overlapping aspect of the method is then introduced by incorporating the second-to-last intrastep point of the previous step into each integrating block. The accuracy, consistency, and stability properties of the method are analyzed. The features of the method are determined through an error analysis of the numerical solutions of linear and nonlinear second-order initial value problems. The nonlinear initial value problems are converted into linear ones using a modified Picard iteration technique. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method. The results are evaluated against other methods from the literature.