Enriched higher-order multiscale approaches with applications to wave propagation
We consider the numerical solution of partial differential equations with coefficients that are strongly heterogeneous in space. We provide an overview of higher-order localized orthogonal decomposition (LOD) methods for the elliptic setting, including recent advancements, and then present a generalization of the strategy to linear hyperbolic multiscale problems. We address the limitations of earlier constructions for the wave equation, which only achieve second-order convergence in space, independent of the chosen polynomial degree. Building on the methodology of enriched corrections recently developed for parabolic multiscale problems, we motivate and propose an enriched higher-order LOD method for the wave equation. The enriched corrections exhibit exponential decay and can be computed on patches. Under minimal assumptions on the coefficient and standard well-preparedness conditions on the data, we derive a priori error estimates that achieve optimal high-order convergence rates, thereby overcoming the previously observed saturation of the convergence rate. With the fifth-order Rosenbrock-Wanner (ROW) time integrator, we conduct a series of numerical examples to verify our theoretical results. We provide examples showing the optimal spatial convergence of the method including the localization errors for different polynomial orders. We also present examples showing the optimal convergence rates of the time discretization.