NUMERICAL SOLUTION OF THE KURAMOTO–SAKAGUCHI EQUATION GOVERNING POPULATIONS OF COUPLED OSCILLATORS

A spectral method is developed to solve the Kuramoto–Sakaguchi nonlinear integro-differential equation numerically. This describes the dynamical behavior of populations of infinitely many nonlinearly coupled oscillators, and models a large number of phenomena in Biology, Medicine, and Physics. Some relevant bifurcation properties of solutions are investigated, and the numerical results are compared with those obtained from both the linearized equation and Monte–Carlo-type simulations of finitely many Langevin equations. In the numerical experiments, several frequency distributions, and several values of the bifurcation parameters are considered.

Download Full-text

The validity of the kinetic collection equation revisited – Part 2: Simulations for the hydrodynamic kernel

Atmospheric Chemistry and Physics Discussions ◽

10.5194/acpd-10-6219-2010 ◽

2010 ◽

Vol 10 (3) ◽

pp. 6219-6240

Author(s):

L. Alfonso ◽

G. B. Raga ◽

D. Baumgardner

Keyword(s):

Differential Equation ◽

Monte Carlo ◽

Monte Carlo Simulations ◽

Numerical Solution ◽

Mass Distribution ◽

Total Mass ◽

Analytic Solution ◽

Numerical Solutions ◽

Integro Differential Equation ◽

Simulation Results

Abstract. The kinetic collection equation (KCE) has been widely used to describe the evolution of the average droplet spectrum due to the collection process that leads to the development of precipitation in warm clouds. This deterministic, integro-differential equation only has analytic solution for very simple kernels. For more realistic kernels, the KCE needs to be integrated numerically. In this study, the validity time of the KCE for the hydrodynamic kernel is estimated by a direct comparison of Monte Carlo simulations with numerical solutions of the KCE. The simulation results show that when the largest droplet becomes separated from the smooth spectrum, the total mass calculated from the numerical solution of the KCE is not conserved and, thus, the KCE is no longer valid. This result confirms the fact that for realistic kernels appropriate for precipitation development within warm clouds, the KCE can only be applied to the continuous portion of the mass distribution.

Download Full-text

The validity of the kinetic collection equation revisited – Part 2: Simulations for the hydrodynamic kernel

Atmospheric Chemistry and Physics ◽

10.5194/acp-10-7189-2010 ◽

2010 ◽

Vol 10 (15) ◽

pp. 7189-7195 ◽

Cited By ~ 6

Author(s):

L. Alfonso ◽

G. B. Raga ◽

D. Baumgardner

Keyword(s):

Differential Equation ◽

Monte Carlo ◽

Monte Carlo Simulations ◽

Numerical Solution ◽

Mass Distribution ◽

Total Mass ◽

Analytic Solution ◽

Numerical Solutions ◽

Integro Differential Equation ◽

Simulation Results

Abstract. The kinetic collection equation (KCE) has been widely used to describe the evolution of the average droplet spectrum due to the collection process that leads to the development of precipitation in warm clouds. This deterministic, integro-differential equation only has analytic solution for very simple kernels. For more realistic kernels, the KCE needs to be integrated numerically. In this study, the validity time of the KCE for the hydrodynamic kernel is estimated by a direct comparison of Monte Carlo simulations with numerical solutions of the KCE. The simulation results show that when the largest droplet becomes separated from the smooth spectrum, the total mass calculated from the numerical solution of the KCE is not conserved and, thus, the KCE is no longer valid. This result confirms the fact that for kernels appropriate for precipitation development within warm clouds, the KCE can only be applied to the continuous portion of the mass distribution.

Download Full-text