An Algorithm for the Numerical Integration of Perturbed and Damped Second-Order ODE Systems

A new method of numerical integration for a perturbed and damped systems of linear second-order differential equations is presented. This new method, under certain conditions, integrates, without truncation error, the IVPs (initial value problems) of the type: x ″ ( t ) + A x ′ ( t ) + C x ( t ) = ε F ( x ( t ) , t ) , x ( 0 ) = x 0 , x ′ ( 0 ) = x 0 ′ , t ∈ [ a , b ] = I , which appear in structural dynamics, astrodynamics, and other fields of physics and engineering. In this article, a succession of real functions is constructed with values in the algebra of m × m matrices. Their properties are studied and we express the solution of the proposed IVP through a serial expansion of the same, whose coefficients are calculated by means of recurrences involving the perturbation function. This expression of the solution is used for the construction of the new numerical method. Three problems are solved by means of the new series method; we contrast the results obtained with the exact solution of the problem and with its first integral. In the first problem, a quasi-periodic orbit is integrated; in the second, a problem of structural dynamics associated with an earthquake is studied; in the third, an equatorial satellite problem when the perturbation comes from zonal harmonics J 2 is solved. The good behavior of the series method is shown by comparing the results obtained against other integrators.