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

Håkon Hoel, Johannes Vincent Meo

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.

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

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.
Paper Structure (15 sections, 8 theorems, 81 equations, 7 figures, 2 tables, 2 algorithms)

This paper contains 15 sections, 8 theorems, 81 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. An adaptive numerical method for estimation of blow-up times for 1D ODEs
  3. Comparing adaptive and uniform time steps
  4. Extension to higher dimensions
  5. Dynamics with slower growth
  6. Numerical experiments
  7. Note on methods used for comparison
  8. Comparing several methods for a 1D ODE
  9. Using the adaptive method on a 1D ODE without analytical solution
  10. Extending the adaptive method for 1D ODEs
  11. An uncoupled problem in multiple dimensions
  12. Estimating blow-up of a coupled multidimensional problem
  13. Estimating blow-up of slow processes in multiple dimensions
  14. Semi-discretized reaction-diffusion equation
  15. Conclusion and future work

Key Result

Theorem 1

Let Assumption asmp:b hold and let $\bar{\tau}$ be calculated using Algorithm alg:1d. Then, for sufficiently small $\epsilon > 0$, it holds that $\left\lvert\bar{\tau}-\tau\right\rvert = \mathcal{O}(\epsilon)$ and the computational cost of the method is $\mathcal{O}\left(\epsilon^{-1}\right)$.

Figures (7)

  • Figure 1: Estimation of blow-up time for $x'(t) = x(t)^2$. Left: Computational cost plotted against the tolerance. Right: Absolute error plotted against the tolerance.
  • Figure 2: Estimation of blow-up time for $x'(t) = e^{x(t)^2}$. Left: Computational cost plotted against the tolerance. Right: Absolute error with regards to a pseudo solution plotted against the tolerance.
  • Figure 3: Estimation of blow-up time for $x'(t) = x(t)\log(x(t))^{1+c}$. Left plots: Computational cost plotted against the tolerance. Right plots: Absolute error plotted against the tolerance.
  • Figure 4: Illustration of example studied in Section \ref{['sec:numExp_UncoupledProblem']}
  • Figure 5: Estimation of blow-up time for the uncoupled multidimensional problem studied in Section \ref{['sec:numExp_UncoupledProblem']}. Left: Computational cost plotted against the tolerance. Right: Absolute error plotted against the tolerance.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Remark 1: Alternative to Assumption \ref{['asmp:b.FandG']}
  • Theorem 1: Properties of Algorithm \ref{['alg:1d']}
  • proof
  • Remark 2: Alternative to Assumption \ref{['asmp:b.FandG']} revisited
  • Remark 3: Extension to separable ODEs
  • Remark 4: Higher-order methods
  • Remark 5: Monotonicity of $\left\lvert\bar{x}_n\right\rvert$
  • Lemma 1
  • proof
  • Theorem 2: Properties of Algorithm \ref{['alg:Rn']}
  • ...and 13 more