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Nonlinear Differential Equation Solvers via Adaptive Picard–Chebyshev Iteration: Applications in Astrodynamics

Published Online:https://doi.org/10.2514/1.G003318

An adaptive self-tuning Picard–Chebyshev numerical integration method is presented for solving initial and boundary value problems by considering high-fidelity perturbed two-body dynamics. The current adaptation technique is self-tuning and adjusts the size of the time interval segments and the number of nodes per segment automatically to achieve near-maximum efficiency. The technique also uses recent insights on local force approximations and adaptive force models that take advantage of the fixed-point nature of the Picard iteration. In addition to developing the adaptive method, an integral quasi-linearization “error feedback” term is introduced that accelerates convergence to a machine precision solution by about a of two. The integral quasi linearization can be implemented for both first- and second-order systems of ordinary differential equations. A discussion is presented regarding the subtle but significant distinction between integral quasi linearization for first-order systems, second-order systems that can be rearranged and integrated in first-order form, and second-order systems that are integrated using a kinematically consistent Picard–Chebyshev iteration in cascade form. The enhanced performance of the current algorithm is demonstrated by solving an important problem in astrodynamics: the perturbed two-body problem for near-Earth orbits. The adaptive algorithm has proven to be more efficient than an eighth-order Gauss–Jackson and a 12th/10th-order Runge–Kutta while maintaining machine precision over several weeks of propagation.

References

  • [1] Iserles A., A First Course in Numerical Analysis of Differential Equations, 2nd ed., Cambridge Univ. Press, New York, 2009. Google Scholar

  • [2] Butcher J. C., Numerical Methods for Ordinary Differential Equations, Wiley, New York, 2008. CrossrefGoogle Scholar

  • [3] Butcher J. C., “Implicit Runge–Kutta Processes,” Mathematics of Computation, Vol. 18, No. 18, 1964, pp. 50–64. CrossrefGoogle Scholar

  • [4] Dormand J. and Prince P., “A Family of Embedded Runge–Kutta Formulae,” Journal of Computational and Applied Mathematics, Vol. 6, No. 1, 1980, pp. 19–26. doi:https://doi.org/10.1016/0771-050X(80)90013-3 JCAMDI 0377-0427 CrossrefGoogle Scholar

  • [5] Dormand J. and Prince P., “A Reconsideration of Some Embedded Runge–Kutta Formulae,” Journal of Computational and Applied Mathematics, Vol. 15, No. 2, 1986, pp. 203–211. doi:https://doi.org/10.1016/0377-0427(86)90027-0 JCAMDI 0377-0427 CrossrefGoogle Scholar

  • [6] Watts H., “Starting Step Size for an ODE Solver,” Journal of Computational and Applied Mathematics, Vol. 9, No. 2, 1983, pp. 177–191. doi:https://doi.org/10.1016/0377-0427(83)90040-7 JCAMDI 0377-0427 CrossrefGoogle Scholar

  • [7] Gladwell L., Shampine L. and Brankin R., “Automatic Selection of the Initial Step Size for an ODE Solver,” Journal of Computational and Applied Mathematics, Vol. 18, No. 2, May 1987, pp. 175–192. CrossrefGoogle Scholar

  • [8] Montenbrook O. and Gill E., Satellite Orbits, Springer, New York, 2000. CrossrefGoogle Scholar

  • [9] Berry M., “A Variable-Step Double-Integration Multi-Step Integrator,” Ph.D. Dissertation, Virginia Polytechnic Inst. and State Univ., Blacksburg, VA, 2004. Google Scholar

  • [10] Berry M. and Healy L., “Implementation of Gauss–Jackson Integration for Orbit Propagation,” Journal of Astronautical Sciences, Vol. 52, No. 3, 2004, pp. 331–357. CrossrefGoogle Scholar

  • [11] Fox K., “Numerical Integration of the Equations of Motion of Celestial Mechanics,” Celestial Mechanics, Vol. 33, No. 2, 1984, pp. 127–142. doi:https://doi.org/10.1007/BF01234151 CLMCAV 0008-8714 CrossrefGoogle Scholar

  • [12] Montenbrook O., “Numerical Integration Methods for Orbital Motion,” Celestial Mechanics and Dynamical Astronomy, Vol. 53, 1992, pp. 59–69. Google Scholar

  • [13] Aristoff J., Horwood J. and Poore B., “Orbit and Uncertainty Propagation: a Comparison of Gauss-Legendre, Dormand–Prince, and Chebyshev–Picard-Based Approaches,” Celestial Mechanics and Dynamical Astronomy, Vol. 118, No. 1, 2013, pp. 13–28. CrossrefGoogle Scholar

  • [14] Wright K., “Some Relationships Between Implicit Runge–Kutta, Collocation, and Lanczos τ Methods, and Their Stability Properties,” BIT Numerical Mathematics, Vol. 10, No. 2, 1970, pp. 217–227. doi:https://doi.org/10.1007/BF01936868 CrossrefGoogle Scholar

  • [15] Jones B. and Anderson R., “A Survey of Symplectic and Collocation Methods for Orbit Propagation,” Advanced in the Astronautical Sciences, Vol. 143, 2012. Google Scholar

  • [16] Picard E., Sur l’application des methods d’approximation successives b latude de certaines bquatioiis differentielles ordinaries, Journal de Mathématiques, Vol. 9, 1893, pp. 217–271. Google Scholar

  • [17] Bai X. and Junkins J., “Modified Chebyshev–Picard Iteration Methods for Solution of Boundary Value Problems,” Advances in the Astronautical Sciences, Vol. 140, 2011, pp. 381–400. Google Scholar

  • [18] Macomber B., Probe A., Woollands R., Read J. and Junkins J., “Enhancements of Modified Chebyshev–Picard Iteration Efficiency for Perturbed Orbit Propagation,” Computational Modelling in Engineering & Sciences, Vol. 111, 2016, pp. 29–64. Google Scholar

  • [19] Macomber B., “Enhancements of Chebyshev–Picard Iteration Efficiency for Generally Perturbed Orbits and Constrained Dynamics Systems,” Ph.D. Dissertation, Texas A&M Univ., College Station, TX, 2015. Google Scholar

  • [20] Boyd J., Chebyshev and Fourier Spectral Methods, Dover, New York, 2001. CrossrefGoogle Scholar

  • [21] Clenshaw C. W., “The Numerical Solution of Linear Differential Equations in Chebyshev Series,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 53, 1957, pp. 134–149. Google Scholar

  • [22] Clenshaw C. W. and Curtis A. R., “A Method for Numerical Integration on an Automatic Computer,” Numerische Mathematik, Vol. 2, No. 1, Dec. 1960, pp. 197–205. Google Scholar

  • [23] Clenshaw C. W. and Norton H. J., “The Solution of Nonlinear Ordinary Differential Equations in Chebyshev Series,” Computer Journal, Vol. 6, No. 1, 1963, pp. 88–92. doi:https://doi.org/10.1093/comjnl/6.1.88 CrossrefGoogle Scholar

  • [24] Feagin T. W., “The Numerical Solution of Two-Point Boundary Value Problems Using Chebyshev Polynomial Series,” Ph.D. Dissertation, Univ. of Texas at Austin, Austin, TX, 1972. Google Scholar

  • [25] Feagin T. and Nacozy P., “Matrix Formulation of the Picard Method for Parallel Computation,” Celestial Mechanics and Dynamical Astronomy, Vol. 29, No. 2, 1983, pp. 107–115. doi:https://doi.org/10.1007/BF01232802 CrossrefGoogle Scholar

  • [26] Shaver J., “Formulation and Evaluation of Parallel Algorithms for the Orbit Determination Problem,” Ph.D. Dissertation, Dept. of Aeronautics and Astronautics, Massachusetts Inst. of Technology, Cambridge, MA, 1980. Google Scholar

  • [27] Fukushima T., “Vector Integration of Dynamical Motions by the Picard–Chebyshev Method,” Astronomical Journal, Vol. 113, 1997, pp. 2325–2328. doi:https://doi.org/10.1086/118443 CrossrefGoogle Scholar

  • [28] Bai X. and Junkins J., “Modified Chebyshev–Picard Iteration Methods for Solution of Initial Value Problems,” Advances in the Astronautical Sciences, Vol. 139, 2011, pp. 345–362. Google Scholar

  • [29] Bai X., “Modified Chebyshev–Picard Iteration for Solution of Initial Value and Boundary Value Problems,” Ph.D. Dissertation, Texas A&M Univ., College Station, TX, 2010. Google Scholar

  • [30] Bani Younes A., “Orthogonal Polynomial Approximation in Higher Dimensions: Applications in Astrodynamics,” Ph.D. Dissertation, Texas A&M Univ., College Station, TX, 2013. Google Scholar

  • [31] Woollands R., Bani Younes A. and Junkins J., “New Solutions for the Perturbed Lambert Problem Using Regularization and Picard Iteration,” Journal of Guidance, Control, and Dynamics, Vol. 38, No. 9, 2015, pp. 1548–1562. doi:https://doi.org/10.2514/1.G001028 LinkGoogle Scholar

  • [32] Woollands R., “Regularization and Computational Methods for Precise Solution of Perturbed Orbit Transfer Problems,” Ph.D. Dissertation, Texas A&M Univ., College Station, TX, 2016. Google Scholar

  • [33] Wang X., Yue X., Dai H. and Atluri S., “Feedback-Accelerated Iteration for Orbit Propagation and Lambert’s Problem,” Journal of Guidance, Control, and Dynamics, Vol. 40, No. 10, 2017, pp. 2442–2451. LinkGoogle Scholar

  • [34] Hairer E., Norsett S. and Wagner G., Solving Ordinary Differential Equations I, Springer, New York, 1993. Google Scholar

  • [35] Bellman R. E. and Kalaba R. E., “Quasilinearization and Nonlinear Boundary-Value Problems,” RAND Corp., Santa Monica, CA, 1965. Google Scholar

  • [36] Swenson T., Woollands R., Junkins J. and Lo M., “Application of Modified Chebyshev–Picard Iteration to Differential Correction for Improved Robustness and Computation Time,” Journal of Astronautical Sciences, No. 3, 2017, pp. 267–284. CrossrefGoogle Scholar

  • [37] Sundman K. F., “Memoire sur le Probleme Retreint des Trois Corps,” Acta Mathematica, Vol. 30, 1912, pp. 105–179. ACMAA8 0001-5962 Google Scholar

  • [38] Aristoff J. and Poore A., “Implicit Runge–Kutta Method for Orbit Propagation,” AIAA/AAS Astrodynamics Specialist Conference, AIAA Paper 2012-4880, 2012. LinkGoogle Scholar

  • [39] Junkins J., “Investigation of Finite-Element Representations of the Geopotential,” AIAA Journal, Vol. 14, No. 6, 1976, pp. 803–808. doi:https://doi.org/10.2514/3.61420 AIAJAH 0001-1452 LinkGoogle Scholar

  • [40] Engles R. and Junkins J., “Local Representation of the Geopotential by Weighted Orthogonal Polynomials,” Journal of Guidance and Control, Vol. 3, No. 1, 1980, pp. 55–61. doi:https://doi.org/10.2514/3.55947 LinkGoogle Scholar

  • [41] Arora N., Vittaldev V. and Russell R., “Parallel Computation of Trajectories Using Graphics Processing Units and Interpolated Gravity Models,” Journal of Guidance, Control, and Dynamics, Vol. 38, No. 8, 2015, pp. 1345–1355. LinkGoogle Scholar

  • [42] Bradley B., Jones B., Beylkin G., Sandberg K. and Axelrad P., “Bandlimited Implicit Runge–Kutta Integration for Astrodynamics,” Celestial Mechanics and Dynamical Astronomy, Vol. 119, No. 2, 2014, pp. 143–168. doi:https://doi.org/10.1007/s10569-014-9551-x CrossrefGoogle Scholar

  • [43] Probe A., Macomber B., Kim D., Woollands R. and Junkins J., “Terminal Convergence Approximation Modified Chebyshev–Picard Iteration for Efficient Numerical Integration of Orbital Trajectories,” Advanced Maui Optical Space Surveillance Technologies Conference, Maui, HI, 2014. Google Scholar