Communications in Mathematical Sciences

Volume 20 (2022)

Number 2

An energy preserving discretization method for the thermodynamic Kuramoto model and collective behaviors

Pages: 495 – 521

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n2.a9

Authors

Seung-Yeal Ha (Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, South Korea)

Woojoo Shim (Research Institute of Basic Sciences, Seoul National University, Seoul, South Korea)

Jaeyoung Yoon (Department of Mathematical Sciences, Seoul National University, Seoul, South Korea)

Abstract

We provide an energy preserving discretization method for the thermodynamic Kuramoto (TK) model on a lattice and investigate its emergent dynamics, and show a smooth transition from the proposed discrete model to the corresponding continuous model. The thermodynamic Kuramoto model describes the temporal evolution of the phase and temperature at each lattice point in a domain. To integrate the continuous model numerically, one needs to discretize the continuous model in a suitable way so that the resulting discrete model exhibits the same emergent features as the corresponding continuous model. The naive forward Euler discretization for phase-temperature configuration does not conserve a total energy, which causes inconsistency with the continuous model. Thus, we instead propose an implicit scheme which preserves energy and satisfies entropy principle, and provide several sufficient frameworks leading to the emergent collective behaviors and uniform-in-time smooth transition from the discrete model to the continuous model.

Keywords

emergence, entropy principle, Kuramoto model, thermodynamics

2010 Mathematics Subject Classification

34E10, 39A30, 65L05

Received 28 March 2021

Received revised 8 July 2021

Accepted 1 August 2021

Published 28 January 2022