A Simple, Fast, and Accurate Time-Integrator for StronglyNonlinear Dynamical Systems

Abstract

Abstract: In this study, we consider Initial Value Problems (IVPs) for strongly nonlinear dynamical systems, and study numerical methods to analyze short as well as long-term responses. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables of positions as well as velocities. For each discrete-time interval Radial Basis Functions (RBFs) are as-sumed as trial functions for the mixed variables in the time domain. A simple col-location method is developed in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source points as well as collocation points. Three numerical exam-ples are provided to compare the present algorithm with explicit as well implicit methods in terms of accuracy, required size of time-interval (or step) and compu-tational cost. The present algorithm is compared against, the second order central difference method, the classical Runge-Kutta method, the adaptive Runge-Kutta-Fehlberg method, the Newmark-b and the Hilber-Hughes-Taylor methods. First the highly nonlinear Duffing oscillator is analyzed and the solutions obtained from