Collisional \(N\)-body numerical integrator with applications to charged particle dynamics. (English) Zbl 1451.65249


65Z05 Applications to the sciences
37M99 Approximation methods and numerical treatment of dynamical systems
70F10 \(n\)-body problems


Taylor; Cosy; NBODY4; ATOMFT
Full Text: DOI


[1] S. Aarseth and F. Hoyle, Dynamical evolution of clusters of galaxies, I, Mon. Not. R. Astron. Soc., 126 (1963), pp. 223–255.
[2] S. J. Aarseth, Gravitational N-Body Simulations: Tools and Algorithms, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, UK, 2003.
[3] P. D. S. Abeyratne, New Computational Approaches to the N-Body Problem with Applications to Electron Cooling of Heavy Ion Beams, Ph.D. thesis, Northern Illinois University, DeKalb, 2016.
[4] C. Affane-Aji, S. Biaz, and N. Govil, On annuli containing all the zeros of a polynomial, Math. Comput. Model., 52 (2010), pp. 1532–1537. · Zbl 1205.65167
[5] A. Ahmad and L. Cohen, A numerical integration scheme for the N-body gravitational problem, J. Comput. Phys., 12 (1973), pp. 389–402. · Zbl 0259.70009
[6] R. Barrio, Performance of the Taylor series method for ODEs/DAEs, Appl. Math. Comput., 163 (2005), pp. 525–545. · Zbl 1067.65063
[7] M. Berz, Modern Map Methods in Particle Beam Physics, Academic Press, New York, 1999.
[8] M. Bidkham and E. Shashahani, An annulus for the zeros of polynomials, Appl. Math. Lett., 24 (2011), pp. 122–125. · Zbl 1203.30009
[9] S. Blanes and A. Iserles, Explicit adaptive symplectic integrators for solving Hamiltonian systems, Celestial Mech. Dynam. Astronom., 114 (2012), pp. 297–317. · Zbl 1266.37045
[10] H. Bücker, G. Corliss, P. Hovland, U. Naumann, and B. Norris, Automatic Differentiation: Applications, Theory, and Implementations, Lect. Notes Comput. Sci. Eng. 50, Springer, Berlin, 2006. · Zbl 1084.65002
[11] M. Bücker and P. Hovland, Tools for Automatic Differentiation, .
[12] Y. Chang and G. Corliss, ATOMFT: Solving ODEs and DAEs using Taylor series, Comput. Math. Appl., 28 (1994), pp. 209–233. · Zbl 0810.65072
[13] W. G. Choe and J. Guckenheimer, Computing periodic orbits with high accuracy, Comput. Methods Appl. Mech. Engrg., 170 (1999), pp. 331–341. · Zbl 0954.65092
[14] A. Dalal and N. Govil, On region containing all the zeros of a polynomial, Appl. Math. Comput., 219 (2013), pp. 9609–9614. · Zbl 1296.26048
[15] F. De Terán, F. M. Dopico, and J. Pérez, New bounds for roots of polynomials based on Fiedler companion matrices, Linear Algebra Appl., 451 (2014), pp. 197–230. · Zbl 1288.15010
[16] M. J. Duncan, H. F. Levison, and M. H. Lee, A multiple time step symplectic algorithm for integrating close encounters, Astronom. J., 116 (1998), p. 2067.
[17] B. Erdelyi, E. Nissen, and S. Manikonda, A differential algebraic method for the solution of the Poisson equation for charged particle beams, Commun. Comput. Phys., 17 (2015), pp. 47–78. · Zbl 1373.78044
[18] M. Fujiwara, Über die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung, Tohoku Math. J., 10 (1916), pp. 167–171. · JFM 46.0123.01
[19] L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), pp. 325–348. · Zbl 0629.65005
[20] E. Hairer, Variable time step integration with symplectic methods, Appl. Numer. Math., 25 (1997), pp. 219–227. · Zbl 0884.65073
[21] E. Hairer and G. Söderlind, Explicit, time reversible, adaptive step size control, SIAM J. Sci. Comput., 26 (2005), pp. 1838–1851. · Zbl 1081.65117
[22] J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, New York, 1999.
[23] À. Jorba and M. Zou, A software package for the numerical integration of ODEs by means of high-order Taylor methods, Experiment. Math., 14 (2005), p. 99. · Zbl 1108.65072
[24] S.-H. Kim, On the moduli of the zeros of a polynomial, Amer. Math. Monthly, 112 (2005), pp. 924–925. · Zbl 1142.30002
[25] P. Kustaanheimo and E. Stiefel, Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math., 218 (1965), pp. 204–219. · Zbl 0151.34901
[26] C. Le Guyader, Solution of the N-body problem expanded into Taylor series of high orders. Applications to the solar system over large time range, Astronom. Astrophys., 272 (1993), pp. 687–694.
[27] J. Makino, Optimal order and time-step criterion for Aarseth-type N-body integrators, Astrophys. J., 369 (1991), pp. 200–212.
[28] K. Makino and M. Berz, COSY INFINITY version 9, Nucl. Instr. Meth. Phys. Res. A, 558 (2005), pp. 346–350.
[29] M. Marden, Geometry of Polynomials, Math. Surveys Monogr. 3, AMS, Providence, RI, 1949. · Zbl 0038.15303
[30] A. A. Marzouk and B. Erdelyi, An accurate and efficient numerical integrator for pair-wise interaction, in Proceedings of NAPAC’16, Chicago, IL, 2017, pp. 514–517.
[31] M. Mignotte and D. Stefanescu, On an Estimation of Polynomial Roots by Lagrange, hal - 00129675, 2002.
[32] A. Morbidelli, Modern integrations of solar system dynamics, Annu. Rev. Earth Planet. Sci., 30 (2002), pp. 89–112.
[33] Z. E. Musielak and B. Quarles, The three-body problem, Rep. Progr. Phys., 77 (2014), p. 065901. · Zbl 1369.70001
[34] K. Nanbu, Theory of cumulative small-angle collisions in plasmas, Phys. Rev. E, 55 (1997), p. 4642.
[35] K. Nitadori and J. Makino, Sixth-and eighth-order Hermite integrator for N-body simulations, New Astron., 13 (2008), pp. 498–507.
[36] G. Parker and J. Sochacki, Implementing the Picard iteration, Neural Parallel Sci. Comput., 4 (1996), pp. 97–112. · Zbl 1060.34501
[37] C. D. Pruett, J. W. Rudmin, and J. M. Lacy, An adaptive N-body algorithm of optimal order, J. Comput. Phys., 187 (2003), pp. 298–317. · Zbl 1047.70002
[38] H. Rein and D. S. Spiegel, IAS15: A fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits, Mon. Not. R. Astron. Soc., 446 (2014), pp. 1424–1437.
[39] M. Rosin, L. Ricketson, A. M. Dimits, R. E. Caflisch, and B. I. Cohen, Multilevel Monte Carlo simulation of Coulomb collisions, J. Comput. Phys., 274 (2014), pp. 140–157. · Zbl 1351.82085
[40] P. Saha and S. Tremaine, Long-term planetary integration with individual time steps, Astronom. J., 108 (1994), pp. 1962–1969.
[41] J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: An overview, Acta Numer., 1 (1992), pp. 243–286. · Zbl 0762.65043
[42] K. Särkimäki, E. Hirvijoki, and J. Terävä, Adaptive time-stepping Monte Carlo integration of Coulomb collisions, Comput. Phys. Commun., 222 (2018), pp. 374–383.
[43] H. D. Schaumburg, A. Al Marzouk, and B. Erdelyi, Picard iteration-based variable-order integrator with dense output employing algorithmic differentiation, Numer. Algorithms, (2018).
[44] C. Simó, Global dynamics and fast indicators, in Global Analysis of Dynamical Systems, B. K. H. W. Broer and G. Vegter, eds., IOP, Bristol, UK, 2001, pp. 373–390.
[45] R. D. Skeel and J. J. Biesiadecki, Symplectic integration with variable stepsize, Ann. Numer. Math., 1 (1994), pp. 191–198. · Zbl 0826.65070
[46] G. Teschl, Ordinary Differential Equations and Dynamical Systems, Grad. Stud. Math. 140, AMS, Providence, RI, 2012.
[47] S. von Hoerner, Die numerische Integration des n-körper-Problemes für Sternhaufen. I, Z. Astrophys., 50 (1960).
[48] Q. Wang, The global solution of the n-body problem, Celestial Mech. Dynam. Astronom., 50 (1991), pp. 73–88.
[49] W. Wang, M. Okamoto, N. Nakajima, and S. Murakami, Vector implementation of nonlinear Monte Carlo Coulomb collisions, J. Comput. Phys., 128 (1996), pp. 209–222. · Zbl 0862.65091
[50] Y. Zhao, A binary collision Monte Carlo model for temperature relaxation in multicomponent plasmas, AIP Adv., 8 (2018), 075016.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.