Periodic waveform relaxation solutions of nonlinear dynamic equations

We present a simple theorem to safeguard the convergence of waveform relaxation (WR) solutions of a dynamic system described by nonlinear ordinary differential equations (ODEs) with a periodic constraint. Namely, if a basic expression of certain constants issued from the system is less than one, the proposed WR algorithm is convergent to the exact solution. It is the first time that WR is used to treat periodic solutions of nonlinear dynamic systems. A numerical example is provided to confirm the theoretic work of the paper.