A class of robust numerical methods for solving dynamical systems with multiple time scales

Abstract

In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters.

Additional Information

The research of T. Hou is partially supported by the NSF Grants DMS-1613861, DMS-1907977, and DMS-1912654. The research Z. Wang is partially supported by the Hong Kong PhD Fellowship Scheme. The research of Z. Zhang is supported by Hong Kong RGC grants (Projects 27300616, 17300817, and 17300318), National Natural Science Foundation of China via grant 11601457, Seed Funding Programme for Basic Research (HKU), and Basic Research Programme (JCYJ20180307151603959) of The Science, Technology and Innovation Commission of Shenzhen Municipality. The computations were performed using the HK ITS research computing facilities that are supported in part by the Hong Kong UGC Special Equipment Grant (SEG HKU09).

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