Localized orthogonal decomposition method for the wave equation with a continuum of scales

This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with an L 2 L^2 -projection. We derive explicit convergence rates of the method in the L ∞ ( L 2 ) L∞(L^2) -, W 1 , ∞ ( L 2 ) W1,∞(L^2) - and L ∞ ( H 1 ) L∞(H^1) -norms without any assumptions on higher order space regularity or scale-separation. The order of the convergence rates depends on further graded assumptions on the initial data. We also prove the convergence of the method in the framework of G-convergence without any structural assumptions on the initial data, i.e. without assuming that it is well-prepared. This rigorously justifies the method. Finally, the performance of the method is demonstrated in numerical experiments.