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. van den Berg, Jan Bouwe; Sheombarsing, Ray: Validated computations for connecting orbits in polynomial vector fields (2020)
  2. Gómez-Serrano, Javier: Computer-assisted proofs in PDE: a survey (2019)
  3. Perdomo, Oscar M.: The round Taylor method (2019)
  4. Schaumburg, Herman D.; Al Marzouk, Afnan; Erdelyi, Bela: Picard iteration-based variable-order integrator with dense output employing algorithmic differentiation (2019)
  5. Al Khawaja, U.; Al-Mdallal, Qasem M.: Convergent power series of (\operatornamesech(x)) and solutions to nonlinear differential equations (2018)
  6. Al Sakkaf, Laila Y.; Al-Mdallal, Qasem M.; Al Khawaja, U.: A numerical algorithm for solving higher-order nonlinear BVPs with an application on fluid flow over a shrinking permeable infinite long cylinder (2018)
  7. Baeza, A.; Boscarino, S.; Mulet, P.; Russo, G.; Zorío, D.: Reprint of: “Approximate Taylor methods for ODEs” (2018)
  8. Breden, Maxime; Lessard, Jean-Philippe: Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs (2018)
  9. Castelli, Roberto; Lessard, Jean-Philippe; James, Jason D. Mireles: Parameterization of invariant manifolds for periodic orbits. II: A posteriori analysis and computer assisted error bounds (2018)
  10. Karouma, Abdulrahman; Nguyen-Ba, Truong; Giordano, Thierry; Vaillancourt, Rémi: A new class of efficient one-step contractivity preserving high-order time discretization methods of order 5 to 14 (2018)
  11. Li, Xiaoming; Liao, Shijun: Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems (2018)
  12. Marzouk, Afnan Al; Erdelyi, Bela: Collisional (N)-body numerical integrator with applications to charged particle dynamics (2018)
  13. Schmitt, Jeremy; Shingel, Tatiana; Leok, Melvin: Lagrangian and Hamiltonian Taylor variational integrators (2018)
  14. Baeza, A.; Boscarino, S.; Mulet, P.; Russo, G.; Zorío, D.: Approximate Taylor methods for ODEs (2017)
  15. Castillo, Vanessa; Lázaro, J. Tomás; Sardanyés, Josep: Dynamics and bifurcations in a simple quasispecies model of tumorigenesis (2017)
  16. Churkina, Tatyana E.; Stepanov, Sergey Y.: On the stability of periodic Mercury-type rotations (2017)
  17. Gonzalez, J. L.; Mireles James, J. D.: High-order parameterization of stable/unstable manifolds for long periodic orbits of maps (2017)
  18. Groothedde, C. M.; Mireles James, J. D.: Parameterization method for unstable manifolds of delay differential equations (2017)
  19. Kehlet, Benjamin; Logg, Anders: A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations (2017)
  20. Mireles James, J. D.; Murray, Maxime: Chebyshev-Taylor parameterization of stable/unstable manifolds for periodic orbits: implementation and applications (2017)

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