Validation of Accuracy and Efficiency of Long-Arc Orbit Propagation Using the Method of Manufactured Solutions and the Round-Trip-Closure Method
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For conservative systems, a common method for validating accuracy is that the Hamiltonian or a similar energy integral of a converged solution maintains constancy to a desired tolerance. While a Hamiltonian metric is a very useful for conservative systems, for non-conservative systems the Hamiltonian check is not applicable and other methods for validating solution accuracy must be employed. While we can utilize various ad-hoc methods for comparing the state history from a new integrator with some other well-tested code, or compare solutions using various accuracy tunings for a given method, there always remains uncertainty since rigorous convergence conclusions are difficult when comparing approximate solutions. Based on the Method of Manufactured Solutions (MMS) and the Round-Trip-Closure Method (RTC), two independent measures of solution accuracy are introduced. These metrics have the attractive property that they are both theoretically exactly zero if the integrator introduces zero error. For RTC, the convergence test is applied directly to the original differential equations and boundary conditions, whereas for MMS, a close neighbor of the unknown exact solution is established, with a known small perturbing force. Application of the solution methodology to this slightly perturbed problem permits strong conclusions on the algorithms accuracy of convergence. MMS and RTC metrics are useful for virtually any numerical process for solving differential equations. MMS and RTC are useful in evaluating the relative merits of competing algorithms; the utility of these ideas are demonstrated in an accuracy study for two numerical integrators: Modified Chebyshev Picard Iteration (MCPI) and an 8th order Gauss Jackson (GJ8) algorithm. We utilize an intermediate order spherical-harmonic gravity (40,40) model. Since this problem is conservative, we check the Hamiltonian constancy with MMS and RTC. Results demonstrate the consistency of the two metrics and high efficiency vs accuracy of MCPI relative to GJ8, for long-arc orbit propagation.
