A higher order multiscale method for the wave equation

In this paper we propose a multiscale method for the acoustic wave equation in highly oscillatory media. We use a higher order extension of the localized orthogonal decomposition method combined with a higher order time stepping scheme and present rigorous a priori error estimates in the energy-induced norm. We find that in the very general setting without additional assumptions on the coefficient beyond boundedness arbitrary orders of convergence cannot be expected, but that increasing the polynomial degree may still considerably reduce the size of the error. Under additional regularity assumptions higher orders can be obtained as well. Numerical examples are presented that confirm the theoretical results.