JuliaCon 2017

In this talk we shall present TaylorIntegration.jl, an ODE integration package using Taylor's method in Julia. The main idea of Taylor's method is to approximate locally the solution by means by a high-order Taylor expansion, whose coefficients are computed recursively using automatic differentiation techniques. One of the principal advantages of Taylor's method is that, whenever high accuracy is required, the order of the method can be increased, which is more efficient computationally than taking smaller time steps. The accuracy of Taylor's method permits to have round-off errors per integration step. Traditionally, it has been difficult to make a generic Taylor integration package, but Julia permits this beautifully. We shall present some examples of the application of this method to ODE integration, including the whole computation of the Lyapunov spectrum, use of jet transport techniques, and parameter sensitivity. Open issues related to improving performance will be described.