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Taylor series Integrator for Differential EquationS. This software is developed by Profs. A. Abad, R. Barrio, F. Blesa and M. Rodriguez, (GME, University of Zaragoza, Spain). It consists on a C (Fortran) library, libTIDES, and a Mathematica package, MathTIDES. (MathTIDES requires Mathematica version >= 7.0) .
Basic references:
* A. Abad, R. Barrio, F. Blesa, M. Rodriguez, 2012. Algorithm 924: TIDES, a Taylor series Integrator for Differential EquationS, ACM TOMS. 39, no. 1, art. 5. (Main reference)
* A. Abad, R. Barrio, F. Blesa, M. Rodriguez, 2011. TIDES tutorial: Integrating ODEs by using the Taylor Series Method., Monografías de la Academia de Ciencias de la Universidad de Zaragoza. 36, pp. 1-116.
* R. Barrio, 2005. Performance of the Taylor series method for ODEs/DAEs, Appl. Math. Comput. 163 (2), pp. 525--545
* R. Barrio, 2006. Sensitivity analysis of ODE's/DAE's using the Taylor series method, SIAM J. on Scientific Computing 27 (6), pp. 1929--1947
Features
- TIDES permits to integrate numerically ODE problems with double or multiple precision (using MPFR, and GMP libraries), 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.
- 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.