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2025-07-01

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cc-by-nc-nd (c) Elsevier B.V., 2025
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/218951

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

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https://doi.org/10.1016/j.amc.2025.129344

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.

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Anàlisi numèrica, Equacions diferencials

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Numerical analysis, Differential equations

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Articles publicats en revistes (Matemàtiques i Informàtica)

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PITA FORRIER, Philip, GIMENO ALQUEZAR, Joan and JORBA I MONTE, Àngel. A note on the local behavior of the Taylor method for stiff ODEs. Applied Mathematics and Computation. 2025. Vol. 496. ISSN 0096-3003. [consulted: 9 of August of 2026]. Available at: https://hdl.handle.net/2445/218951

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