TITLE:
A note on the local behavior of the Taylor method for stiff ODEs

AUTHORS:
Philip P. Forrier^(1), Joan Gimeno^(1) and Angel Jorba^(1,2)

(1) Departament de Matematiques i Informatica, Universitat de Barcelona
(2) Centre de Recerca Matematica (CRM), Barcelona

ABSTRACT:
In this note we study the behavior of the coefficients of the Taylor
method when computing the numerical solution of stiff Ordinary
Differential Equations. First, we derive an asymptotic formula for the
growth of the stability region w.r.t. the order of the Taylor
method. Then, we analyze the behavior of the Taylor coefficients of
the solution when the equation is stiff. Using jet transport, we show
that the coefficients computed with a floating point arithmetic of
arbitrary precision have an absolute error that depends on the
variational equations of the solution, which can have a dominant
exponential growth in stiff problems. This is naturally related to the
characterization of stiffness presented by Soderlind et al. and allows
to discuss why explicit solvers need a stepsize reduction when dealing
with stiff systems. We explore how high-order methods can alleviate
this restriction when high precision computations are required. We
provide numerical experiments with classical stiff problems and
perform extended precision computations to demonstrate this behavior.