Numerical Algorithms

, Volume 80, Issue 2, pp 377–396 | Cite as

Picard iteration-based variable-order integrator with dense output employing algorithmic differentiation

  • Herman D. SchaumburgEmail author
  • Afnan Al Marzouk
  • Bela Erdelyi
Original Paper


Motivated by the high accuracy requirements and the huge ratio of the largest to smallest time scales of Coulomb collision simulations of a considerable number of charges, we developed a novel numerical integration scheme, which uses algorithmic differentiation to produce variable, high-order integrators with dense output. We show that Picard iterations are not only a nice theoretical tool, but can also be successfully implemented to develop competitive integrators, especially when accuracies close to machine precision are required. The numerical integrators’ performance and applications to the electrostatic n-body problem are illustrated.


Picard iteration-based Algorithmic differentiation Variable order Dense output 

Mathematics Subject Classification (2010)

65L05 65P99 70F10 


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Funding Information

This work was supported by the United States Department of Energy, Office of Nuclear Physics, under contract no. DE-SC0005823.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsNorthern Illinois UniversityDeKalbUSA

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