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Approximate Taylor methods for ODEs

Antonio Baeza, Sebastiano Boscarino, Pep Mulet, Giovanni Russo, David Zorío

TL;DR

An approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their derivatives, are needed, just as in classical Runge–Kutta schemes.

Abstract

A new method for the numerical solution of ODEs is presented. This approach is based on an approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their derivatives, are needed, just as in classical Runge-Kutta schemes. Compared to Runge-Kutta methods, the number of function evaluations to achieve a given order is higher, however with the present procedure it is much easier to produce arbitrary high-order schemes, which may be important in some applications. In many cases the new approach leads to an asymptotically lower computational cost when compared to the Taylor expansion based on exact derivatives. The numerical results that are obtained with our proposal are satisfactory and show that this approximate approach can attain results as good as the exact Taylor procedure with less implementation and computational effort.

Approximate Taylor methods for ODEs

TL;DR

An approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their derivatives, are needed, just as in classical Runge–Kutta schemes.

Abstract

A new method for the numerical solution of ODEs is presented. This approach is based on an approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their derivatives, are needed, just as in classical Runge-Kutta schemes. Compared to Runge-Kutta methods, the number of function evaluations to achieve a given order is higher, however with the present procedure it is much easier to produce arbitrary high-order schemes, which may be important in some applications. In many cases the new approach leads to an asymptotically lower computational cost when compared to the Taylor expansion based on exact derivatives. The numerical results that are obtained with our proposal are satisfactory and show that this approximate approach can attain results as good as the exact Taylor procedure with less implementation and computational effort.

Paper Structure

This paper contains 10 sections, 2 theorems, 54 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Taylor methods
  3. Computational cost
  4. Approximate Taylor methods
  5. Computational cost
  6. Low order approximate Taylor methods
  7. Numerical experiments
  8. Scalar equations
  9. Systems of ODE
  10. Conclusions

Key Result

Theorem 1

Let $f:\mathbb{R}^m\rightarrow\mathbb{R},$$u:\mathbb{R}\rightarrow\mathbb{R}^m$$r$ times continuously differentiable. Then where $\mathcal{P}_{r}=\{ s\in\mathbb N^{r} / \sum_{j=1}^{r} j s_j=r \}$, $|s|=\sum_{j=1}^{r} s_j$, $\left(\right)=\frac{r!}{s_1!\cdots s_r!}$, $D^s u(t)$ is an $m\times |s|$ matrix whose ($\sum\limits_{l<j}s_l+i$)-th column is given by and the action of the $k$-th derivativ

Figures (7)

  • Figure 1: Problem \ref{['eq:ex1']}. Left: Error vs CPU time, Right: Error vs. $h$. (a)-(b) $R=2$; (c)-(d) $R=4$; (e)-(f) $R=6$.
  • Figure 2: Problem \ref{['eq:ex2']}. Left: Error vs CPU time, Right: Error vs. $h$. (a)-(b) $R=2$; (c)-(d) $R=4$; (e)-(f) $R=6$.
  • Figure 3: Problem \ref{['eq:ex3']}. Left: Error vs CPU time, Right: Error vs. $h$. (a)-(b) $R=2$; (c)-(d) $R=4$; (e)-(f) $R=6$.
  • Figure 4: Problem \ref{['eq:pendol']}. Left: Error vs CPU time, Right: Error vs. $h$. (a)-(b) $R=2$; (c)-(d) $R=4$; (e)-(f) $R=6$; (g)-(h) $R=8$.
  • Figure 5: Problem \ref{['eq:toggle']}. Left: Error vs CPU time, Right: Error vs. $h$. (a)-(b) $R=2$; (c)-(d) $R=4$; (e)-(f) $R=6$; (g)-(h) $R=8$.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Theorem 1: Faà di Bruno's formula
  • Theorem 2
  • proof