Abstract: The motion equation is transformed into a system of the first order differential equations (ODEs); and by using the linear finite element of the Galerkin type, the explicit recurrence formula is derived with an accuracy of O(h^2). By using the element energy projection (EEP) technique, the nodal accuracy recovery approach improves the recurrence formula to yield a nodal accuracy of O(h^4). Further, the stability property and convergence orders are analyzed mathematically with a given scheme of adaptive step-size. Finally, the given numerical examples justify that the proposed approach is a simple and effective method.