A software package for the numerical integration of ODEs by means of high-order Taylor methods. This paper revisits the Taylor method for the numerical integration of initial value problems of ordinary differential equations (ODEs). The main goal is to present a computer program that outputs a specific numerical integrator for a given set of ODEs. The generated code includes a function to compute the jet of derivatives of the solution up to a given order plus adaptive selection of order and step size at run time. The package provides support for several extended precision arithmetics, including user-defined types. par The authors discuss the performance of the resulting integrator in some examples, showing that it is very competitive in many situations. This is especially true for integrations that require extended precision arithmetic. The main drawback is that the Taylor method is an explicit method, so it has all the limitations of these kind of schemes. For instance, it is not suitable for stiff systems.

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  1. Breden, Maxime; Lessard, Jean-Philippe; Mireles James, Jason D.: Computation of maximal local (un)stable manifold patches by the parameterization method (2016)
  2. Pouly, Amaury; Graça, Daniel S.: Computational complexity of solving polynomial differential equations over unbounded domains (2016)
  3. Sánchez-Taltavull, Daniel; Vieiro, Arturo; Alarcón, Tomás: Stochastic modelling of the eradication of the HIV-1 infection by stimulation of latently infected cells in patients under highly active anti-retroviral therapy (2016)
  4. van den Berg, J.B.; Mireles James, J.D.: Parameterization of slow-stable manifolds and their invariant vector bundles: theory and numerical implementation (2016)
  5. Bartuccelli, Michele V.; Deane, Jonathan H.B.; Gentile, Guido: The high-order Euler method and the spin-orbit model (2015)
  6. Bartha, Ferenc A.; Munthe-Kaas, Hans Z.: Computing of B-series by automatic differentiation (2014)
  7. Cyranka, Jacek: Efficient and generic algorithm for rigorous integration forward in time of dPDEs. I (2014)
  8. Luque, Alejandro; Villanueva, Jordi: Quasi-periodic frequency analysis using averaging-extrapolation methods (2014)
  9. Pellegrini, Etienne; Russell, Ryan P.; Vittaldev, Vivek: $F$ and $G$ Taylor series solutions to the Stark and Kepler problems with Sundman transformations (2014)
  10. Huguet, Gemma; de la Llave, Rafael: Computation of limit cycles and their isochrons: fast algorithms and their convergence (2013)
  11. Abad, Alberto; Barrio, Roberto; Blesa, Fernando; Rodríguez, Marcos: Algorithm 924, TIDES, a Taylor series integrator for differential equations (2012)
  12. Bervillier, C.: Status of the differential transformation method (2012)
  13. Galeş, C.: A cartographic study of the phase space of the elliptic restricted three body problem. Application to the Sun-Jupiter-asteroid system (2012)
  14. Migaszewski, Cezary: The generalized non-conservative model of a 1-planet system revisited (2012)
  15. Rodríguez, Marcos; Barrio, Roberto: Reducing rounding errors and achieving Brouwer’s law with Taylor series method (2012)
  16. Thelwell, Roger J.; Warne, Paul G.; Warne, Debra A.: Cauchy-Kowalevski and polynomial ordinary differential equations (2012)
  17. Barrio, R.; Rodríguez, M.; Abad, A.; Blesa, F.: Breaking the limits: The Taylor series method (2011)
  18. Barrio, R.; Rodríguez, M.; Abad, A.; Serrano, S.: Uncertainty propagation or box propagation (2011)
  19. Broer, Henk; Dijkstra, Henk; Simó, Carles; Sterk, Alef; Vitolo, Renato: The dynamics of a low-order model for the Atlantic multidecadal oscillation (2011)
  20. Hayden, Kevin; Olson, Eric; Titi, Edriss S.: Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations (2011)

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