Heterogeneous multiscale methods for stiff ordinary differential equations

The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different time scales. Stability and convergence results for the proposed HMM methods are presented together with numerical tests. The analysis covers some existing methods and the new algorithms that are based on higher-order estimates of the effective force by kernels satisfying certain moment conditions and regularity properties. These new methods have superior computational complexity compared to traditional methods for stiff problems with oscillatory solutions.