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

Fernando García-Alonso, José Antonio Reyes and Mónica Cortés-Molina ()
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Fernando García-Alonso: Department of Applied Mathematics, University of Alicante, 03690 Alicante, Spain
José Antonio Reyes: Department of Applied Mathematics, University of Alicante, 03690 Alicante, Spain
Mónica Cortés-Molina: Department of Applied Mathematics, University of Alicante, 03690 Alicante, Spain

Mathematics, 2024, vol. 12, issue 13, 1-22

Abstract: Based on the Ψ-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 Ψ-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 Ψ-functions and the Ψ-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 Ψ-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 J 2 . This allows us to compare the good behavior of the new code with other prestige codes.

Keywords: explicit and implicit multistep methods; predictor–corrector method; harmonic oscillator; quasi-periodic orbit; earthquake; equatorial satellite (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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