Spectral analysis and computation for the Kuramoto-Sakaguchi integroparabolic equation

A spectral method is developed to numerically solve the so-calledKuramoto–Sakaguchi equation, which is a nonlinear integro-differentialequation of the parabolic type, governing the dynamical statisticalbehaviour of certain populations of nonlinearly coupled randomoscillators. The approach rests on explicit bounds for the spacederivatives of solutions, obtained via energy-like estimates.Bounds for the numerical approximations of solutions are given,and improved (sometimes appreciably) by means of an ‘aposteriori error analysis’. Plots are shown to illustratethe performance of the method, and comparison with a finitedifference approach is also made.