Sub-ODEs Simplify Taylor Series Algorithms for Ordinary Differential Equations

2025-03-27

AI Summary

This paper addresses the complexity and poor scalability of manually deriving recurrence relations for Taylor coefficients of composite functions in high-order Taylor methods for solving ordinary differential equation (ODE) initial value problems. We propose the Sub-ODEs method, which transforms built-in functions (e.g., exp, sin) into inline differential relations, automatically generating coefficient recurrences and eliminating per-function manual derivation. We provide the first rigorous proof that Sub-ODE systems—regardless of nesting depth—guarantee mathematical completeness of the Taylor coefficient recurrence. This shifts the function library extension paradigm from “one recurrence per function” to “a few ODE rules per function.” Leveraging symbolic automatic differentiation and differential-algebraic modeling, we implement a custom solver in MATLAB. Experiments demonstrate superior accuracy and larger maximum allowable step sizes compared to mainstream MATLAB ODE solvers, achieving a new balance between generality and computational efficiency for high-order Taylor methods.

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Abstract

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 $dot v=vdot u$. 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.

Problem

• Simplify Taylor series algorithms for ODEs using sub-ODEs
• Transform ODEs into differential-algebraic systems via sub-ODEs
• Ensure sound recurrence systems when incorporating sub-ODEs

Innovation

• Uses Taylor series for ODE solutions
• Applies sub-ODEs to simplify operations
• Ensures sound recurrence system automatically