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

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 Söderlind et al. [32], 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.