Waveform relaxation method for parallel orbital propagation

A new highly parallelizable procedure to integrate perturbed Keplerian motions numerically is presented in this study. It is based on a waveform relaxation method combined with the classic fourth-order Runge–Kutta integrator. Other parallelizable algorithms are already known in the literature, being one of the most widespread the Picard–Chebyshev method. When formulated in Cartesian variables, however, the Picard–Chebyshev method exhibits a limited convergence interval. This limitation requires sequential integration over small segments, reducing the level of parallelization. Alternatively, the equations of motion can be transformed into modified equinoctial elements. The waveform relaxation method proposed here extends both the convergence interval and rate when using Cartesian variables, carrying them to the same level as the modified equinoctial elements. Hence, this method offers an effective parallel algorithm that can be applied directly in Cartesian variables, what simplifies the formulation of the dynamical equations, the integrator structure, and the perturbation force expressions. The convergence and performance of the new waveform relaxation method was validated by performing low-Earth orbit propagations subjected to both conservative and non-conservative perturbations. The evaluation revealed a substantial enhancement with respect to a Picard–Chebyshev method, with a reduction in the Cartesian variables parallel evaluation of the perturbations of approximately 20 times, spanning from 10 to 30 orbit periods and with no significant loss of precision.