  We propose a high order numerical homogenization method for dissipative
ordinary differential equations (ODEs) containing two time scales. Essentially,
only 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 to the method in an efficient way for practical applications.
Several numerical examples are presented to validate the new models and
numerical methods.
