Numerical integration of relativistic equations of motion for Earth satellites
type: Item
instance of: scholarly article
title: Numerical integration of relativistic equations of motion for Earth satellites
zbMATH Open document ID: 1173.70002
author: J. Martínez
published in: Celestial Mechanics and Dynamical Astronomy
publication date: 2009-04-14T00:00:00Z
review text: This paper is concerned with the numerical investigation of the motion of an artificial satellite around the Earth taking into account the relativistic effects due to the Earth oblateness and the gravitational action of the Sun according to the model proposed by \textit{V. A. Brumberg} [Celest. Mech. Dyn. Astron. 88, No.~2, 209--225 (2004; Zbl 1131.70311)]. Denoting by \( \mathbf{q} = \mathbf{q} (t) \) the position of the satellite in a geocentric frame, the differential equations have the form \( \mathbf{q}'' (t) = \nabla_{\mathbf{q}} V(t,\mathbf{q}) + \mathbf{f},\) where the first term in the right hand side includes the forces derivable from a potential force, and \( \mathbf{f} \) the remaining relativistic forces. Here the author writes the above equation in the Lagrangian form \( ( L_{\mathbf{q}'} )' - L_{\mathbf{q}} + \mathbf{f} = 0 \) with \( L = (1/2) \| \mathbf{q}' \|^2 + V \), and an auxiliary energy equation \( E' = - \partial_t L + \mathbf{f}\mathbf{q}'\). For the long-time numerical integration of these equations, the author proposes a partitioned Runge-Kutta symplectic formula derived by \textit{L. Jay} [SIAM J. Numer. Anal. 33, No.~1, 368--387 (1996; Zbl 0903.65066)], which combines the four-stage Lobatto IIIA and IIIB formulas for positions and velocities. The main advantage of this method, as shown in several numerical experiments with energy equations, is that due to its geometric properties it provides an accurate numerical integration in contrast with standard non-symplectic integrators of high order. Finally, since the Lobatto IIIA and IIIB methods are fully implicit, a starting algorithm proposed by \textit{I. Higueras} and \textit{T. Roldán} [Appl. Numer. Math. 56, No.~1, 1--18 (2006; Zbl 1083.65067)] is used to improve the computational efficiency of the new method.
reviewed by: Manuel Calvo
Mathematics Subject Classification ID: 70-08
Mathematics Subject Classification ID: 70M20
Mathematics Subject Classification ID: 70H40
Mathematics Subject Classification ID: 65P10
zbMATH DE Number: 5542930
zbMATH Keywords: relativistic effects
zbMATH Keywords: partitioned Runge-Kutta symplectic formula
zbMATH Keywords: Lobatto formulas
describes a project that uses: RODAS
describes a project that uses: MPFRCPP
MaRDI profile type: Publication
full work available at URL: s10569-008-9162-5
OpenAlex ID: W2141510470
cites work: Q3997876
cites work: On relativistic equations of motion of an earth satellite
cites work: Generating numerical algorithms using a computer algebra system
cites work: Q4714145
cites work: Q4401601
cites work: Stage value predictors for additive and partitioned Runge-Kutta methods
cites work: Analytical solutions to the four post-Newtonian effects in a near-Earth satellite orbit
cites work: Q5796405
cites work: Discrete mechanics and variational integrators
cites work: Q4117285
DOI: 10.1007/S10569-008-9162-5
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description: scientific article; zbMATH DE number 5542930