Parallel Evaluation of Chebyshev Approximations: Applications in Astrodynamics

Abstract

Approximations based on Chebyshev polynomials have several astrodynamic applications. The performance of these approximations can be improved by parallel implementations exploiting parallel architectures, such as OpenMP and CUDA. In this paper, we introduce the parallel implementation to two astrodynamic applications. The first is the gravitational finite element model (FEM): a piecewise Chebyshev approximation that replaces high degree and order gravitational spherical harmonic models (SHMs). Thus, much lower degree, locally valid functions can efficiently model and compute local gravity perturbations in parallel structure for efficient performance. For this model, the total gravity acceleration is split into a reference and disturbance term. The reference includes two-body plus J_2, which are relatively cheap to compute. The FEM approximates the higher-order gravity terms. It is developed from a 2D mesh grid covering a sphere of a specified radius, and a family of spherical shells is sampled using a cosine distribution in the radial direction. To reduce the required memory when seeking a specific accuracy, an adaptive version of the gravitational FEM is introduced. In addition, a parallel implementation of the FEM using OpenMP is preseneted. We show the runtime comparison for the 200 degree × 200 order EGM2008 SHM and the serial and parallel equivalent FEM algorithms. The other application is the Chebyshev-Picard method (CPM): a numerical integrator that solves an ordinary differential equation by approximating the integrand using a Chebyshev approximant and iterates over the trajectory via Picard iteration. A parallel CUDA implementation of the CPM method in conjunction with the EGM2008 SHM and the FEM is introduced. We present numerical examples for propagating four Earth-orbiting satellites considering both the 200times 200 EGM2008 SHM and the equivalent FEM representation to test the algorithm’s performance via parallel and serial computation (i.e., a single CPU thread).