Discrete wave equation upscaling

We present homogenization technique for the uniformly discretized wave equation, based on the derivation of an effective equation for the low-wavenumber component of the solution. The method produces a down-sampled, effective medium, thus making the solution of the effective equation less computationally expensive. Advantages of the method include its conceptual simplicity and ease of implementation, the applicability to any uniformly discretized wave equation in 1-D, 2-D or 3-D, and the absence of any constraints on the medium properties.We illustrate our method with a numerical example of wave propagation through a 1-D multiscale medium and demonstrate the accurate reproduction of the original wavefield for sufficiently low frequencies.