Solving chaotic ODEs with time-parallel algorithms

Thesis

This thesis studies the numerical integration of chaotic dynamical systems over long time spans, a core problem in fields such as fusion energy research, climate modelling, and fluid dynamics. Standard time-parallel algorithms work well for linear systems or over short time intervals, but perform poorly on long periods dominated by chaos.

To address this problem, we develop a framework for finite-time convergence based on the theory of contraction mappings. This framework uses an outer–inner ball analysis to give explicit bounds on the number of iterations needed to reach a chosen tolerance, thereby going beyond traditional asymptotic results. We also define a proximity function to act as a specialized stopping rule that enforces short-term accuracy while accommodating the natural long-term divergence of trajectories.

Next, we introduce the moving-window (MoWi) algorithm for the robust long-term integration of chaotic systems. MoWi splits the overall time domain into a sequence of overlapping windows. It then runs a time-parallel subroutine within each window. Through our framework, a rigorous tracking analysis shows that the starting guess for each window lies inside the outer ball of convergence of the time-parallel subroutine. This guarantees that the subroutine converges within a fixed number of iterations.

Lastly, we propose three adaptive variants (AMoWi) to handle systems with time-varying dynamics. These variants dynamically adjust the window length, the shift between windows, and the proximity function weights. We test these methods against the Lorenz, Lorenz–96, and Kuramoto– Sivashinsky equations. The results show that MoWi and its variants can outperform standard time-parallel solvers, ensuring scalable speedups while preserving the true statistical behaviour of the dynamics.

Authors: Antonucci, GA

• DOI: 10.5287/ora-xoevp0dez
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: parallel-in-time integration; moving-window algorithm; parareal algorithm; long-time integration of chaotic dynamical chaotic dynamical systems; contraction mappings
• Subjects: Applied mathematics; Parallel computing; Chaotic behavior in systems; Scientific computing; Numerical analysis