Output error behavior for discretizations of ergodic, chaotic ODE systems

TL;DR

This paper develops a model to predict the error behavior in numerical simulations of ergodic, chaotic ODE systems, accounting for discretization, sampling, and computational costs, to better understand simulation accuracy versus resource expenditure.

Contribution

It introduces a comprehensive framework linking simulation parameters, error, and computational costs for chaotic ODE systems, including parallel computing considerations.

Findings

Error models accurately predict simulation inaccuracies.

Sampling cost impacts overall simulation error.

Parallel computation affects total simulation time and accuracy.

Abstract

The use of numerical simulation for prediction of characteristics of chaotic dynamical systems inherently involves unpredictable processes. In this work, we develop a model for the expected error in the simulation of ergodic, chaotic ODE systems, which allows for discretization and statistical effects due to unpredictability. Using this model, we then generate a framework for understanding the relationship between the sampling cost of a simulation and the expected error in the result, and explore the implications of the various parameters of simulations. Finally, we generalize the framework to consider the total cost -- including unsampled spin-up timesteps -- of simulations and consider the implications of parallel computational environments, to give a realistic model of the relationship between wall-clock time and the expected error in simulation of a chaotic ODE system.