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On approximate implicit Taylor methods for ordinary differential equations

Antonio Baeza, Raimund Bürger, María del Carmen Martí, Pep Mulet, David Zorío

TL;DR

The paper addresses the challenge of efficiently implementing high-order implicit Taylor methods for autonomous ODEs by introducing approximate implicit Taylor (AIT) schemes that replace symbolic derivative computations with finite-difference based derivative approximations. This yields Newton-based solvers that require only ${oldsymbol{f}}$ and ${oldsymbol{f}'}$, resulting in a simpler, more scalable approach for systems of any size while preserving $R$-th order accuracy. The authors provide a detailed derivation of the AIT formulation, analyze its linear stability, and compare its performance against exact implicit Taylor methods and approximate explicit Taylor methods through extensive numerical experiments, showing improved efficiency at higher orders and better stiffness handling. The work demonstrates that AIT can deliver high-order accuracy with reduced computational cost, making it attractive for applications demanding very high-order implicit integration, and it suggests potential extensions to PDEs where implicit methods are essential.

Abstract

An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a method that requires less evaluations of the function that defines the ODE and its derivatives than the usual version. On the other hand, an efficient numerical solution of the equation that arises from the discretization by means of Newton's method is introduced for an implicit scheme of any order. Numerical experiments illustrate that the resulting algorithm is simpler to implement and has better performance than its exact counterpart.

On approximate implicit Taylor methods for ordinary differential equations

TL;DR

The paper addresses the challenge of efficiently implementing high-order implicit Taylor methods for autonomous ODEs by introducing approximate implicit Taylor (AIT) schemes that replace symbolic derivative computations with finite-difference based derivative approximations. This yields Newton-based solvers that require only and , resulting in a simpler, more scalable approach for systems of any size while preserving -th order accuracy. The authors provide a detailed derivation of the AIT formulation, analyze its linear stability, and compare its performance against exact implicit Taylor methods and approximate explicit Taylor methods through extensive numerical experiments, showing improved efficiency at higher orders and better stiffness handling. The work demonstrates that AIT can deliver high-order accuracy with reduced computational cost, making it attractive for applications demanding very high-order implicit integration, and it suggests potential extensions to PDEs where implicit methods are essential.

Abstract

An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a method that requires less evaluations of the function that defines the ODE and its derivatives than the usual version. On the other hand, an efficient numerical solution of the equation that arises from the discretization by means of Newton's method is introduced for an implicit scheme of any order. Numerical experiments illustrate that the resulting algorithm is simpler to implement and has better performance than its exact counterpart.
Paper Structure (22 sections, 1 theorem, 79 equations, 3 figures, 4 tables)

This paper contains 22 sections, 1 theorem, 79 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Scope
  3. Related work
  4. Outline of the paper
  5. Taylor methods
  6. Preliminaries
  7. Faà di Bruno's formula
  8. Explicit Taylor methods
  9. Implicit Taylor methods
  10. Exact implicit Taylor methods
  11. Approximate implicit Taylor methods
  12. Linear stability
  13. Newton iteration
  14. Exact implicit Taylor method
  15. Approximate implicit Taylor method
  16. ...and 7 more sections

Key Result

proposition 1

Assume that the functions $f:\mathbb{R}^M \to \mathbb{R}$ and $\boldsymbol{u}: \mathbb{R} \to \mathbb{R}^M$ are $r$ times continuously differentiable. Then

Figures (3)

  • Figure 1: Example 1 (nonlinear scalar problem \ref{['eq:experiment_log_escalar']}): performance of the IT and the AIT methods.
  • Figure 2: Example 2 (nonlinear scalar problem \ref{['eq:experiment_log_escalar']}): performance of the IT and the AIT methods.
  • Figure 3: Example 4 (stiff linear problem \ref{['eq:experiment_system_2']}): performance of the AET and the AIT methods.

Theorems & Definitions (1)

  • proposition 1: Faà di Bruno's formula (Faà di Bruno, 1855)