[Paper Review] The Adams-Bashforth-Moulton Integration Methods Generalized to an Adaptive Grid

Allan Hayes|arXiv (Cornell University)|Apr 16, 2011

Geophysics and Gravity Measurements3 citations

TL;DR

This paper generalizes the Adams-Bashforth-Moulton predictor-corrector methods to adaptive grids, enabling dynamic step size adjustment to maintain desired error tolerance. The method achieves high accuracy with minimal derivative evaluations—only two per step—demonstrating convergence and reproducing the Tolman-Oppenheimer-Volkoff neutron star mass limit of 0.71 solar masses in just 23 steps with 1% precision.

ABSTRACT

We present a generalization of the Adams-Bashforth-Moulton predictor-corrector numerical integration methods to an adaptive grid. The step size may be chosen dynamically in order to maintain a desired relative magnitude of error in each step. We demonstrate that the methods remain convergent to the expected degree, and apply various methods to the famous problem of determining the maximum possible mass of a neutron star supported by pure fermionic exclusion pressure. We reproduce the Tolman-Oppenheimer-Volkoff result of 0.71 solar masses using only 23 integration steps, and reproducing both mass and radius within 1% requires 27. We also present various optimizations and features of our implementation.

Motivation & Objective

• To generalize classical Adams-Bashforth-Moulton multistep methods to adaptive grids for improved efficiency in problems with expensive derivative evaluations.
• To enable dynamic step size control using error estimates from predictor-corrector pairs, maintaining a target relative error per step.
• To reduce computational cost compared to Runge-Kutta methods by requiring only two derivative evaluations per step, regardless of convergence order.
• To validate the method on a physically significant problem: determining the maximum neutron star mass via the Tolman-Oppenheimer-Volkoff equation.
• To demonstrate that high-order convergence and stability can be preserved even with adaptive, non-uniform grids, avoiding issues like Runge’s phenomenon through order tuning.

Proposed method

• The method combines explicit Adams-Bashforth (AB) predictor of order N with implicit Adams-Moulton (AM) corrector of order N+1 to form a predictor-corrector pair.
• At each integration step, the AB predictor provides an initial estimate using previous derivative values, which is then corrected by the AM method to achieve higher-order accuracy.
• The difference between the AB and AM results provides an error estimate used to dynamically adjust the step size to maintain a desired relative error tolerance.
• Weighting factors for the AB and AM methods are recalculated at each step to accommodate variable step sizes, though the computational cost remains low compared to derivative evaluation.
• The method avoids the Courant-Friedrichs-Lewy (CFL) condition by being implicit, enabling larger stable steps, especially in regions of smooth behavior.
• To prevent Runge’s phenomenon, the order of convergence is tuned based on the problem’s behavior, particularly near regions of rapid change such as the neutron star surface.

Experimental results

Research questions

• RQ1 Can the Adams-Bashforth-Moulton predictor-corrector method be generalized to an adaptive grid while preserving its convergence order and stability?
• RQ2 Does the dynamic step size adjustment based on error estimation maintain accuracy and efficiency in problems with expensive derivative evaluations?
• RQ3 How does the adaptive ABM method compare to traditional fixed-grid or Runge-Kutta methods in terms of integration steps and derivative evaluations for a stiff astrophysical problem?
• RQ4 What order of convergence and error tolerance yield optimal efficiency for solving the Tolman-Oppenheimer-Volkoff equation?
• RQ5 How does the step size adapt to capture sharp gradients, such as the pressure drop near the surface of a neutron star?

Key findings

• The adaptive ABM method achieves the expected order of convergence across fixed-grid, fixed-step, and adaptive-grid configurations.
• The method reproduces the Tolman-Oppenheimer-Volkoff neutron star mass limit of 0.71017188 solar masses with 1% precision in only 23 integration steps using a 4-5 order method and 10⁻² tolerance.
• Achieving 1% precision in both mass and radius requires 27 steps with a 4-5 order method and 10⁻² tolerance.
• For higher precision—10⁻⁸ in mass and 10⁻⁵ in radius—the method requires 131 steps with a 9-10 order method and 10⁻⁵ tolerance.
• Step size increases exponentially at first, then plateaus in the core, and sharply decreases near the neutron star surface due to rapid pressure decline, as confirmed by pressure profile visualization.
• The AM correction term remains stable throughout most of the integration, with significant increases only near the surface, indicating higher local error sensitivity.