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Multiscale problems are computationally costly to solve by direct simulationbecause the smallest scales must be represented over a domain determined by thelargest scales of the problem. We have developed and analyzed new numericalmethods for multiscale wave propagation following the framework of theheterogeneous multiscale method. The numerical methods couple simulations onmacro- and microscales for problems with rapidly fluctuating materialcoefficients. The computational complexity of the new method is significantlylower than that of traditional techniques. We focus on HMM approximationapplied to long time integration of one-dimensional wave propagation problemsin both periodic and non-periodic medium and show that the dispersive effectthat appear after long time is fully captured.

Analysis of HMM for One Dimensional Wave Propagation Problems Over Long Time