On the stability of Approximate Taylor methods for ODE and their relationship with Runge-Kutta schemes
For numerical analysts, it provides theoretical validation of a practical ODE solver's stability and connections to established methods.
The paper proves two conjectures about approximate Taylor methods for ODEs: their absolute stability region matches that of exact Taylor methods, and they are equivalent to certain Runge-Kutta schemes.
In [Baeza et al., Computers and Fluids, 159, 156--166 (2017)] a new method for the numerical solution of ODEs is presented. This methods can be regarded as an approximate formulation of the Taylor methods and it follows an approach that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their high order derivatives, are needed. In this reference, the absolute stability region of the new methods is conjectured to be coincident with that of their exact counterparts. There is also a conjecture about their relationship with Runge-Kutta methods. In this work we answer positively both conjectures.