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.

翻译：本文针对高振荡介质中的声波方程，提出了一种多尺度方法。我们采用局部正交分解方法的高阶推广形式，结合高阶时间步进格式，并在能量诱导范数下给出严格的先验误差估计。研究发现：在系数仅满足有界性而无其他额外假设的非常一般性设定中，无法预期获得任意阶收敛性，但提高多项式阶数仍可显著降低误差量级；在附加正则性假设条件下，亦可实现高阶收敛。文中数值算例验证了理论结果。