An efficient numerical method for solving dynamical systems with multiple time scales

Abstract

In this paper, we propose an efficient numerical method for solving dynamical systems with multiple time scales. Our approach involves representing the solution of a multiscale dynamical system as a transformation of a slowly varying solution, effectively capturing the fast oscillation of the solution. By assuming scale separation, we systematically construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide a convergence analysis for our method. To demonstrate its accuracy and efficiency, we present various numerical examples, including ODE systems with three-and four-separated time scales. The results demonstrate the robustness of our method in solving ODE systems with multiple time scales, as it allows for a fixed time step independent of the multiscale parameters.