Efficient Explicit Taylor ODE Integrators with Symbolic-Numeric Computing

TL;DR

This paper introduces a Julia-based Taylor series ODE solver that leverages symbolic-numeric code generation and adaptive algorithms to outperform traditional Runge-Kutta methods in efficiency and robustness.

Contribution

The paper presents a novel implementation of Taylor ODE integrators combining automatic differentiation, symbolic-numeric code generation, and adaptive strategies for improved performance.

Findings

Outperforms explicit Runge-Kutta methods in efficiency.

Achieves better run time performance with compiler-based tooling.

Provides a versatile adaptive time and order algorithm.

Abstract

Taylor series methods show a newfound promise for the solution of non-stiff ordinary differential equations (ODEs) given the rise of new compiler-enhanced techniques for calculating high order derivatives. In this paper we detail a new Julia-based implementation that has two important techniques: (1) a general purpose higher-order automatic differentiation engine for derivative evaluation with low overhead; (2) a combined symbolic-numeric approach to generate code for recursively computing the Taylor polynomial of the ODE solution. We demonstrate that the resulting software's compiler-based tooling is transparent to the user, requiring no changes from interfaces required to use standard explicit Runge-Kutta methods, while achieving better run time performance. In addition, we also developed a comprehensive adaptive time and order algorithm that uses different step size and polynomial degree across the integration period, which makes this implementation more efficient and versatile in a broad range of dynamics. We show that for codes compatible with compiler transformations, these integrators are more efficient and robust than the traditionally used explicit Runge-Kutta methods.