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 nonidentical oscillators.
Bibliographical noteFunding Information:
JAC was partially supported by the EPSRC grant number EP/P031587/1 . YPC was supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (Nos. 2017R1C1B2012918 and 2017R1A4A1014735 ) and POSCO Science Fellowship of POSCO TJ Park Foundation . LP acknowledges the support of Imperial College London thanks to the Nelder visiting fellowship. The authors warmly thank Professor Julien Barré and Guy Métivier for helpful discussions and valuable comments.
© 2018 The Authors
All Science Journal Classification (ASJC) codes
- Numerical Analysis
- Modelling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics