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Numerical solution of ordinary differential equations in geodetic science using adaptive Gauss numerical integration method

Numerical solution of ordinary differential equations in geosciences: applications in satellite geodesy

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Abstract

In this paper a new method of numerically solving ordinary differential equations is presented. This method is based on the Gaussian numerical integration of different orders. Using two different orders for numerical integration, an adaptive method is derived. Any other numerical solver for ordinary differential equations can be used alongside this method. For instance, Runge–Kutta and Adams–Bashforth–Moulton methods are used together with this new adaptive method. This method is fast, stable, consistent, and suitable for very high accuracies. The accuracy of this method is always higher than the method used alongside with it. Two applications of this method are presented in the field of satellite geodesy. In the first application, for different time periods and sampling rates (increments of time), it is shown that the orbit determined by the new method is—with respect to the Keplerian motion—at least 6 and 25,000,000 times more accurate than, respectively, Runge–Kutta and Adams–Bashforth–Moulton methods of the same degree and absolute tolerance. In the second application, a real orbit propagation problem is discussed for the GRACE satellites. The orbit is propagated by the new numerical solver, using perturbed satellite motion equations up to degree 280. The results are compared with another independent method, the Unscented Kalman Filter. It is shown that the orbit propagated by the numerical solver is approximately 50 times more accurate than the one propagated by the Unscented Kalman Filter approach.

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References

  • Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York

    Google Scholar 

  • Balmino G (1974) Coriolis forces in numerical integration of satellite orbits, and absolute reference system. Celest Mech 10:423–436

    Article  Google Scholar 

  • Berrut JP, Trefethen LN (2004) Barycentric Lagrange interpolation. SIAM Rev 46:501–517

    Article  Google Scholar 

  • Berry MM, Healy LM (2004) Implementation of Gauss-Jackson integration for orbit propagation. J Astronaut Sci 52:331–357

    Google Scholar 

  • Chiou JC, Wu SD (1999) On the generation of higher order numerical integration methods using lower order Adams–Bashforth and Adams–Moulton methods. J Comput Appl Math 108:19–29

    Article  Google Scholar 

  • Fukushima T (2005) Efficient orbit integration by orbital longitude methods. Proc Symp Celest Mech 36:123–129

    Google Scholar 

  • Gaposchkin EM (1986) Precision orbit computation in geodesy and geodynamics with KS variables. Adv Space Res 6:135–141

    Article  Google Scholar 

  • Julier SJ, Uhlmann JK (2004) Unscented filtering and nonlinear estimation. Proc IEEE 93:35–56

    Google Scholar 

  • Kahaner D, Moler C, Nash S (1989) Numerical methods and software. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Kaula WM (1966) Theory of satellite geodesy. Blaisdel Publishing Company, London

    Google Scholar 

  • Kincaid D, Cheney W (1990) Numerical analysis, mathematics of scientific computing. Cole Publishing Corporation, Pacific Grove

    Google Scholar 

  • Meyer O, Reichhoff B, Schneider M (2000) Analytical theories of satellite motion constructed with a new package formula manipulation and compared with numerical integration. Celest Mech Dyn Astron 76:121–129

    Article  Google Scholar 

  • Miguel AS (2009) Numerical integration of relativistic equations of motion for earth satellites. Celest Mech Dyn Astron 103:17–30

    Article  Google Scholar 

  • Montenbruck O, Gill E (2000) Satellite orbits: models, methods, and applications. Springer, Berlin

    Book  Google Scholar 

  • Mousavi SE, Xiao H, Sukumar N (2010) Generalized Gaussian quadrature rules on arbitrary polygons. J Numer Methods Eng 82:99–113

    Google Scholar 

  • Panou G, Korakitis R (2019) Geodesic equations and their numerical solution in Cartesian coordinates on a triaxial ellipsoid. J Geod Sci 9:1–12

    Article  Google Scholar 

  • Papanikolaou TD, Tsoulis D (2016) Assesment of numerical integration methods in the context of low Earth orbits and inter-satellite observation analysis. Acta Geod Geophys 51:619–641

    Article  Google Scholar 

  • Seeber G (2003) Satellite geodesy. Walter de Gruyter GmbH and Co. KG, Berlin

    Book  Google Scholar 

  • Shokri A, Walker JP, Van Dijk AIJM, Pauwels VRN (2019) On the use of adaptive ensemble Kalman filtering to mitigate error misspecifications in GRACE data assimilation. Water Resour Res 55:7622–7637

    Article  Google Scholar 

  • Somodi B, Foldvary L (2011) Application of numerical integration techniques for orbit determination of state-of-the-art LEO satellites. Period Polytech 55:99–106

    Article  Google Scholar 

  • Somodi B, Foldvary L (2012) Accuracy of numerical integration techniques for GOCE orbit determination. Proc Adv Res Sci Fields 1:1578–1583

    Google Scholar 

  • Strohmer G (2004) First order differential equations and the atmosphere. Coll Math J 35:93–96

    Article  Google Scholar 

  • Visser PNAM (2005) Low–low satellite to satellite tracking: a comparison between analytical linear orbit perturbation theory and numerical integration. J Geodesy 79:160–166

    Article  Google Scholar 

  • Xu P (2008) Position and velocity perturbations for the determination of geopotential from space geodetic measurements. Celest Mech Dyn Astron 100:231–249

    Article  Google Scholar 

  • Xu P (2018) Measurement-based perturbation theory and differential equation parameter estimation with applications to satellite gravimetry. Commun Nonlinear Sci Numer Simul 59:515–543

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Geodesy, School of Surveying and Geospatial Data Engineering, University of Tehran, Tehran, Iran

    Mostafa Kiani Shahvandi

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  1. Mostafa Kiani Shahvandi
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Correspondence to Mostafa Kiani Shahvandi.

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Kiani Shahvandi, M. Numerical solution of ordinary differential equations in geodetic science using adaptive Gauss numerical integration method. Acta Geod Geophys 55, 277–300 (2020). https://doi.org/10.1007/s40328-020-00293-6

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  • DOI: https://doi.org/10.1007/s40328-020-00293-6

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