Direct mixed multistep block method for solving second-order differential equations

This paper presents novel mixed multistep block methods for the solution of second-order Ordinary Differential Equations (ODEs) using variable step size approach. The approach employs on the combination of Block Backward Differentiation Formulas (BBDF) and block of Adams type formulas. The theory of each method is discussed for the derivation of the mixed method. The formulas are represented in the simplest form where the integration and differentiation coefficients are stored to avoid repetitive computation of the coefficients as the step changes in the integration interval. The Newton method is used for the implementation of the BBDF method while the Adams formulas are implemented using simple iteration. Numerical examples are provided to illustrate the efficiency of the method and will be compared with ode15s in Matlab.