The heterogeneous multiscale methods (HMM) is a general framework for
the numerical approximation of multiscale problem. It is here developed
for a class of ODEs containing different time scales. The theory of
obtaining higher order estimates of the effective force by kernels
satisfying certain moment conditions and regularity properties is
presented. Stability and convergence results for the proposed HMM
methods are also presented together with numerical tests. The new
methods have superior computational complexity compared to traditional
methods.