  Multiscale problems are computationally costly to solve by direct simulation
because the smallest scales must be represented over a domain determined by the
largest scales of the problem. We have developed and analyzed new numerical
methods for multiscale wave propagation following the framework of the
heterogeneous multiscale method. The numerical methods couple simulations on
macro- and microscales for problems with rapidly fluctuating material
coefficients. The computational complexity of the new method is significantly
lower than that of traditional techniques. We focus on HMM approximation
applied to long time integration of one-dimensional wave propagation problems
in both periodic and non-periodic medium and show that the dispersive effect
that appear after long time is fully captured.
