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Efficient Explicit Taylor ODE Integrators with Symbolic-Numeric Computing

Songchen Tan, Oscar Smith, Christopher Rackauckas

TL;DR

The paper addresses efficient, high-order explicit Taylor ODE integrators for non-stiff problems by delivering a Julia-based implementation that couples a Taylor-mode automatic differentiation engine with a symbolic-numeric code-generation pipeline for Taylor polynomials, enabling seamless, high-order accuracy with minimal user intervention. It introduces adaptive strategies for both the time step $h$ and the polynomial degree $p$, aiming to minimize work per unit time and to cope with region-dependent convergence radii. The approach achieves $O(p^2)$ cost for derivative evaluation and benefits from compile-time symbolic simplification, yielding faster Taylor polynomial evaluation and robust performance across varied dynamics. Benchmarks against explicit Runge–Kutta methods show competitive or superior accuracy- and time-efficiency at high precision for non-stiff problems, with extensions to implicit Taylor methods suggested for stiff dynamics.

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.

Efficient Explicit Taylor ODE Integrators with Symbolic-Numeric Computing

TL;DR

The paper addresses efficient, high-order explicit Taylor ODE integrators for non-stiff problems by delivering a Julia-based implementation that couples a Taylor-mode automatic differentiation engine with a symbolic-numeric code-generation pipeline for Taylor polynomials, enabling seamless, high-order accuracy with minimal user intervention. It introduces adaptive strategies for both the time step and the polynomial degree , aiming to minimize work per unit time and to cope with region-dependent convergence radii. The approach achieves cost for derivative evaluation and benefits from compile-time symbolic simplification, yielding faster Taylor polynomial evaluation and robust performance across varied dynamics. Benchmarks against explicit Runge–Kutta methods show competitive or superior accuracy- and time-efficiency at high precision for non-stiff problems, with extensions to implicit Taylor methods suggested for stiff dynamics.

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.
Paper Structure (14 sections, 19 equations, 4 figures, 2 algorithms)

This paper contains 14 sections, 19 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theory
  3. Taylor-mode automatic differentiation for higher-order directional derivatives
  4. Computing Taylor polynomial of ODE solutions
  5. Adaptive polynomial degree and adaptive step size mechanisms
  6. Experiments
  7. Solving various non-stiff ODE problems
  8. Lotka-Volterra problem
  9. Fitzhugh-Nagumo problem
  10. Rigid body problem
  11. Random linear problem
  12. Adaptive polynomial degree tests
  13. Conclusions
  14. Comparison with TaylorIntegration.jl

Figures (4)

  • Figure 1: Effect of symbolic compilation on computing Taylor polynomial. For various degrees, performing symbolic simplification in advance (yellow bars) can achieve about one order of magnitude acceleration compared to naively computing (blue bars).
  • Figure 1: Work-precision diagrams for Taylor methods and Runge-Kutta methods in various non-stiff ODE problems
  • Figure 1: Effect of symbolic compilation on computing Taylor polynomial, with both this work and TaylorIntegration.jl. In addition to this work (where naive and simplified are shown in blue and yellow bars), computing with TaylorIntegration.jl is shown in green bars for naive and pink for simplified.
  • Figure 2: Comparison of adaptive-degree Taylor methods (degree can vary between 6 and 12) and fixed-degree Taylor method (6, 8, 10 and 12). At lower tolerances, the adaptive-degree method has a chance to use different degree at different region to achieve better performance.