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In this talk, we conduct numerical simulations of (non)linear wave propagation in periodic medium.We provide an unconditionally stable numerical method to achieve the spectral convergence in space even when the periodic coefficients vary fast or/and are discontinuous.In the contrast, the traditional pseudo-spectral methods and finite difference/volume schemes have poor convergent rates especially when the coefficients are discontinuous and highly oscillatory.We also consider the wave propagation in disordered medium.We get the well-known Anderson localization in our numerical tests.