High-order Numerical Homogenization for Dissipative Ordinary Differential Equations
Zeyu Jin, Ruo Li Ruo Li · Multiscale Modeling and Simulation · 2022
Abstract. We propose a high-order numerical homogenization method for dissipative ordinary differential equations (ODEs) containing two time scales. Essentially, only the first order homogenized model globally in time can be derived. To achieve a high-order method, we have to adopt a numerical approach in the framework of the heterogeneous multiscale method (HMM). By a successively refined microscopic solver, the accuracy improvement up to arbitrary order is attained providing input data smooth enough. Based on the formulation of the high-order microscopic solver we derived, an iterative formula to calculate the microscopic solver is then proposed. Using the iterative formula, we develop an implementation of the method in an efficient way for practical applications. Several numerical examples are presented to validate the new models and numerical methods.