A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems

Based on the Psi-functions series method, a new numerical integration method for perturbed and damped second-order systems of differential equations is presented. This multistep method is defined for variable step and variable order (VSVO) and maintains the good properties of the Psi-functions series method. In addition, it incorporates a recurring algebraic procedure to calculate the algorithm's coefficients, which facilitates its implementation on the computer. The construction of Psi-functions and the Psi-functions series method are presented to address the construction of both explicit and implicit multistep methods and a predictor-corrector method. Three problems analogous to those solved by the Psi-functions series method are analyzed, contrasting the results obtained with the exact solution of the problem or with its first integral. The first example is the integration of a quasi-periodic orbit. The second example is a Structural Dynamics problem associated with an earthquake, and the third example studies an equatorial satellite with perturbation J2. This allows us to compare the good behavior of the new code with other prestige codes.