TIDES is a free software to integrate numerically Ordinary Differential Equations by using a Taylor Series method. This software, consists on a C (Fortran) library, libTIDES, and a Mathematica package, MathTIDES. (MathTIDES requires Mathematica version >= 7.0)

The main features of TIDES are the following:

• TIDES permits to integrate numerically ODE problems with multiple precision, that means that you can solve ODE problems up to any precision level in a reasonable computer time.
• TIDES may solve directly sensitivity equations with respect to initial conditions or parameters up to any any order.
• TIDES integrates by using the Taylor Series method with an optimized variable-stepsize and variable-order formulation, and extended formulas for variational equations.
• The software has been done to be extremely easy to use: with MathTIDES we write, in a natural way, the ODE and their parameters, together with the parameters of the integration. Then, MathTIDES writes the C (Fortran) code, that, compiled and linked with libTIDES, integrates the ODE. The derivatives and partial derivatives are obtained by using Automatic Differentiation (AD) techniques.
• There are four different versions of the code generated by MathTIDES:
  • Two minimal versions (in C or Fortran language). These are faster than standard versions and easy to use as subprograms, but they do not have most of the possibilities of the standard versions.
  • A standard version (only in C language) for double precision computation.
  • A standard version (only in C language) for multiple precision computations. This version uses MPFR, and GMP libraries to integrate ODEs with any arbitrary precision.
• MathTIDES writes automatically the code to compute partial derivatives of the solution of the ODE with respect to any variable or parameter (using AD and avoiding the use of any variational equation or sensitivity with respect to the parameters). TIDES may detect events of ODEs, i. e. points where a function of the solution of the ODE satisfies an event function, like it becomes zero or reaches an extremum.

More information at

http://gme.unizar.es/software/tides