Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Abstract

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

Period-Multiplying Bifurcations in the Gravitational Field of Asteroids

“…The study of orbital mechanics around an asteroid involves the computation of trajectories that a spacecraft would follow when the spacecraft is under the gravitational influence of the asteroid. Since no closed-form solutions are available, trajectories are computed by employing numerical integration schemes (see [14] for details on numerical integration schemes used in astrodynamical problems).…”