Structure preserving schemes for the continuum Kuramoto model: phase transitions

TL;DR

This paper develops structure-preserving numerical schemes for the continuum Kuramoto model that accurately capture phase transitions and stationary states, even with multiple frequencies and diffusion, enabling detailed numerical investigation of synchronization phenomena.

Contribution

The paper introduces new numerical schemes that preserve the structural properties of the Kuramoto model, addressing challenges posed by high dimensionality and multiple frequencies.

Findings

Successfully capture phase transitions in numerical simulations

Efficiently handle multiple frequencies in the Kuramoto model

Accurately reproduce stationary states and synchronization behavior

Abstract

The construction of numerical schemes for the Kuramoto model is challenging due to the structural properties of the system which are essential in order to capture the correct physical behavior, like the description of stationary states and phase transitions. Additional difficulties are represented by the high dimensionality of the problem in presence of multiple frequencies. In this paper, we develop numerical methods which are capable to preserve these structural properties of the Kuramoto equation in the presence of diffusion and to solve efficiently the multiple frequencies case. The novel schemes are then used to numerically investigate the phase transitions in the case of identical and non identical oscillators.