A Priori Adaptive Numerical Methods for Estimating Blow-up Times of Autonomous ODEs

TL;DR

This paper introduces a priori adaptive numerical methods for estimating blow-up times of autonomous ODEs, leveraging sensitivity analysis to improve efficiency and accuracy, supported by theoretical and numerical validation.

Contribution

It proposes a novel adaptive approach based on auxiliary hitting time sensitivity, enhancing estimation accuracy and computational efficiency over existing methods.

Findings

The methods achieve theoretical error rates.

Adaptive algorithms reduce computational effort.

Numerical experiments confirm practical effectiveness.

Abstract

We present effective a priori adaptive numerical methods for estimating the blow-up time for solutions of autonomous ODEs. The novelty of our approach is to base our adaptive steps on the sensitivity of an auxiliary hitting time. We provide results on the theoretical error rates, and show that there is a benefit in terms of computational effort in choosing our adaptive algorithm over alternative approaches. Numerical experiments support our theoretical results and show how the methods perform in practice.