Quadratized Taylor series methods for ODE numerical integration.

We focus on Taylor Series Methods (TSM) and Automatic Differentiation (AD) for the numerical solution of Ordinary Differential Equations (ODE) characterized by a vector field given by a finite composition of elementary and standard functions. We show that computational advantages are achieved if a kind of pre-processing said Exact Quadratization (EQ) is applied to the ODE before applying the TSM and the AD. In particular, when the ODE function is given by a formal polynomial (i.e. with real powers) of n variables and m monomials, the computational complexity required by our EQ based method for the calculation of the k -th order Taylor coefficient is O (k) whereas by using the existing AD methods it amounts to O (k 2). • The paper describes the QTSM method, and proves that it outperforms in terms of execution time the classic TSM method. • The capability of QTSM for more general nonlinear ODEs than formally polynomal ones, is discussed. • The paper illustrates, in a simulation example, that TSM correctly calculates the solution whereas lower order methods fails. • The paper illustrates the performance of QTSM vs TSM for the Kepler, and the Van der Pol ODEs. [ABSTRACT FROM AUTHOR]