Adaptive step size controllers for fractional product integration methods

Alexandru Fikl

Fourth International Nonlinear Dynamics Conference (NODYCON 2025)

url

Abstract

We propose an adaptive numerical algorithm for general nonlinear fractional initial value problems discretized using the popular Product Integration (PI) family of methods. The construction is based on rigorous error models for the rectangular and the trapezoidal PI methods that are available for both the implicit and the explicit variants on non-uniform grids. The adaptive algorithm mirrors similar classic developments for integer order initial value problems, while taking into account the history properties of the fractional operators. As in the classical case, adaptive time stepping algorithms are required in many physically relevant systems that permit fractional extensions. We showcase the numerical methods on different stiff systems originating from chemistry (Brusselator), fluid mechanics (Lorenz), and neuroscience (FitzHugh-Nagumo) with multiple components and different fractional orders. In all of these systems, we find that an adaptive algorithm gives more accurate results and more efficient solution methods.

BibTeX

@inproceedings{Fikl2025,
  title         = {Adaptive step size controllers for fractional product integration methods},
  author        = {Alexandru Fikl and Roberto Garrappa},
  year          = 2025,
  month         = jun,
  day           = 22,
  journal       = {Fourth International Nonlinear Dynamics Conference (NODYCON 2025)},
  url           = {https://nodycon.app.earendelplatform.com/},
  language      = {en},
  abstract      = {
    We propose an adaptive numerical algorithm for general nonlinear fractional
    initial value problems discretized using the popular Product Integration
    (PI) family of methods. The construction is based on rigorous error models
    for the rectangular and the trapezoidal PI methods that are available for
    both the implicit and the explicit variants on non-uniform grids. The
    adaptive algorithm mirrors similar classic developments for integer order
    initial value problems, while taking into account the history properties of
    the fractional operators. As in the classical case, adaptive time stepping
    algorithms are required in many physically relevant systems that permit
    fractional extensions. We showcase the numerical methods on different stiff
    systems originating from chemistry (Brusselator), fluid mechanics (Lorenz),
    and neuroscience (FitzHugh-Nagumo) with multiple components and different
    fractional orders. In all of these systems, we find that an adaptive
    algorithm gives more accurate results and more efficient solution methods.
  },
}