Algorithm 924: TIDES, a Taylor Series Integrator for Differential EquationS

This article introduces the software package TIDES and revisits the use of the Taylor series method for the numerical integration of ODEs. The package TIDES provides an easy-to-use interface for standard double precision integrations, but also for quadruple precision and multiple precision integrations. The motivation for the development of this package is that more and more scientific disciplines need very high precision solution of ODEs, and a standard ODE method is not able to reach these precision levels. The TIDES package combines a preprocessor step in MATHEMATICA that generates Fortran or C programs with a library in C. Another capability of TIDES is the direct solution of sensitivities of the solution of ODE systems, which means that we can compute the solution of variational equations up to any order without formulating them explicitly. Different options of the software are discussed, and finally it is compared with other well-known available methods, as well as with different options of TIDES. From the numerical tests, TIDES is competitive, both in speed and accuracy, with standard methods, but it also provides new capabilities.