Nonlinear differential equation solvers via adaptive Picard-Chebyshev iteration: Applications in astrodynamics

We present an adaptive approach for solving initial value problems using an accelerated Picard-Chebyshev method. The new algorithm retains the large convergence domain typical of Picard iteration, and importantly, accelerated terminal convergence typical of quasi-linearization. This approach is implemented for both first order and second order systems of differential equations. Including the error feedback terms leads to about a factor of two decrease in the number of iterations required for Picard convergence to near machine precision. We discuss the subtle but significant distinction between integral quasi-linearization for systems that are naturally first order, systems that are naturally second order (but re-arranged to be integrated in first order form), and systems that are naturally second order and integrated using a kinematically consistent modified Picard-Chebyshev iteration in cascade form. The adaptation technique introduced is self-tuning and adjusts the size of time interval segments and the number of nodes per segment automatically to achieve near-maximum efficiency. The technique also utilizes recent insights on local force models and adaptive force models that take advantage of the fixed point nature of Picard iteration. We demonstrate enhanced performance by solving benchmark problems in astrodynamics, specifically gravitationally perturbed near-Earth orbits. We compare the results with those obtained using an 8^th order Gauss-Jackson integrator, a 12^th order Runge-Kutta integrator and MATLAB’s ODE45. The adaptive algorithm is more efficient than these competing methods, implemented as serial algorithms, while maintaining user prescribed accuracy tolerance ranging from engineering precision to near machine precision over at least seven weeks of orbit propagation. The method presented is well-suited for parallelization whereas the step-by-step methods are poorly suited to parallelization.