Sub-ODEs Simplify Taylor Series Algorithms for Ordinary Differential Equations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909554436997120 |
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| author | Nedialkov, Nedialko S. Pryce, John D. |
| author_facet | Nedialkov, Nedialko S. Pryce, John D. |
| contents | A Taylor method for solving an ordinary differential equation initial-value problem $\dot x = f(t,x)$, $x(t_0) = x_0$, computes the Taylor series (TS) of the solution at the current point, truncated to some order, and then advances to the next point by summing the TS with a suitable step size.
A standard ODE method (e.g. Runge-Kutta) treats function $f$ as a black box, but a Taylor solver requires $f$ to be preprocessed into a code-list of elementary operations that it interprets as operations on (truncated) TS.
The trade-off for this extra work includes arbitrary order, typically enabling much larger step sizes.
For a standard function, such as $\exp$, this means evaluating $v(t)=\exp(u(t))$, where $u(t),v(t)$ are TS.
The sub-ODE method applies the ODE $d v/d u=v$, obeyed by $v=\exp(u)$, to in-line this operation as $\dot v=v\dot u$.
This gives economy of implementation: each function that satisfies a simple ODE goes into the "Taylor library" with a few lines of code--not needing a separate recurrence relation, which is the typical approach.
Mathematically, however, the use of sub-ODEs generally transforms the original ODE into a differential-algebraic system, making it nontrivial to ensure a sound system of recurrences for Taylor coefficients.
We prove that, regardless of how many sub-ODEs are incorporated into $f$, this approach guarantees a sound system.
We introduce our sub-ODE-based Matlab ODE solver and show that its performance compares favorably with solvers from the Matlab ODE suite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21078 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sub-ODEs Simplify Taylor Series Algorithms for Ordinary Differential Equations Nedialkov, Nedialko S. Pryce, John D. Numerical Analysis Mathematical Software 65L05, 65L80, 41A58 A Taylor method for solving an ordinary differential equation initial-value problem $\dot x = f(t,x)$, $x(t_0) = x_0$, computes the Taylor series (TS) of the solution at the current point, truncated to some order, and then advances to the next point by summing the TS with a suitable step size. A standard ODE method (e.g. Runge-Kutta) treats function $f$ as a black box, but a Taylor solver requires $f$ to be preprocessed into a code-list of elementary operations that it interprets as operations on (truncated) TS. The trade-off for this extra work includes arbitrary order, typically enabling much larger step sizes. For a standard function, such as $\exp$, this means evaluating $v(t)=\exp(u(t))$, where $u(t),v(t)$ are TS. The sub-ODE method applies the ODE $d v/d u=v$, obeyed by $v=\exp(u)$, to in-line this operation as $\dot v=v\dot u$. This gives economy of implementation: each function that satisfies a simple ODE goes into the "Taylor library" with a few lines of code--not needing a separate recurrence relation, which is the typical approach. Mathematically, however, the use of sub-ODEs generally transforms the original ODE into a differential-algebraic system, making it nontrivial to ensure a sound system of recurrences for Taylor coefficients. We prove that, regardless of how many sub-ODEs are incorporated into $f$, this approach guarantees a sound system. We introduce our sub-ODE-based Matlab ODE solver and show that its performance compares favorably with solvers from the Matlab ODE suite. |
| title | Sub-ODEs Simplify Taylor Series Algorithms for Ordinary Differential Equations |
| topic | Numerical Analysis Mathematical Software 65L05, 65L80, 41A58 |
| url | https://arxiv.org/abs/2503.21078 |