AN ENERGY PRESERVING DISCRETIZATION METHOD FOR THE THERMODYNAMIC KURAMOTO MODEL AND COLLECTIVE BEHAVIORS

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 transitionfrom the proposed discrete model to the corresponding continuous model. The thermodynamic Kuramotomodel describes the temporal evolution of the phase and temperature at each lattice pointin a domain. To integrate the continuous model numerically, one needs to discretize the continuousmodel in a suitable way so that the resulting discrete model exhibits the same emergent features asthe corresponding continuous model. The naive forward Euler discretization for phase-temperatureconfiguration does not conserve a total energy, which causes inconsistency with the continuous model.Thus,