In this paper, we introduce second derivative Multistep collocation meth-ods for the numerical integration of ordinary differential equations (ODEs). These methods combine the concepts of both Multistep methods and col-location methods, using second derivative of the solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties, that is, A-stability for ODEs. Using the second-order deriva-tives leads to high order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the r previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the second derivative of the solution in the colloca-tion methods. We present the construction of the technique and discuss the analysis of the order of accuracy and linear stability properties. Finally, some numerical results are provided to confirm the theoretical expecta-tions. A stiff system of ODEs, the Robertson chemical kinetics problem, and the two-body Pleiades problem are the case studies for comparing the efficiency of the proposed methods with existing methods.

In this paper, we study the global truncation error of the linear Multistep methods (LMM) in terms of local truncation error of the corresponding Runge-Kutta schemes. The key idea is the representation of LMM with a corresponding Runge-Kutta method. For this, we need to consider the multiple step of a linear Multistep method as a single step in the corresponding Runge-Kutta method. Therefore, the global error estimation of a LMM through the Runge-Kutta method will be provided. In this estimation, we do not take into account the effects of roundoff errors. The numerical illustrations show the accuracy and efficiency of the given estimation.

The non-local hyperbolic partial differential equations have many applications in sciences and engineering. A collocation finite element approach based on exponential cubic B-spline and quintic B-spline are presented for the numerical solution of the wave equation subject to nonlocal boundary condition. Von Neumann stability analysis is used to analyze the proposed methods. The efficiency, accuracy and stability of the methods are assessed by applying it to the test problem. The results are found to be in good agreement with known solutions and with existing collocation schemes in literature.