Project A15 • Localized methods for the wave equation with strong heterogeneities in space and time

Principal investigators

Project summary

In this project, we consider multiscale approaches for the efficient simulation of solutions to the wave equation in possibly strongly heterogeneous media, encoded by highly oscillatory (rough) coefficients in the equation. We consider the case where oscillations are space- and time-dependent, which is for instance of interest in view of recent progress in the understanding and simulation of wave processes in (spatiotemporally modulated) metamaterials. Due to the highly oscillatory nature of the problem, simulations with standard discretization methods quickly become unfeasible, since they need to globally resolve relevant features. Therefore, appropriate strategies such as multiscale methods or domain decomposition approaches are required.

Within this project, we pursue two different strategies to deal with the highly oscillatory behavior of the heterogeneous wave equation and its solution. First, we will develop and analyze a multiscale approach that provides reasonable approximations on coarse scales in space and time and which combines spatial and temporal multiscale approaches. More precisely, we will employ recent progress in reducing the bottleneck in the construction of spatial problem-adapted approximation spaces with deep learning strategies. This allows for a fast generation of reliable time-dependent coarse-scale spaces, which are then combined with a specifically designed coarse time-stepping procedure based on a variational formulation in time. The general idea is to conduct multiple local and parallelizable computations to capture microscopic effects. The gathered information is then used to construct a fast method that achieves good approximations of the effective behavior of solutions. Second, we consider iterative domain decomposition strategies to directly operate on fine scales that take into account relevant features of the problem, but we divide the computations into sub-problems to make them feasible. The construction of appropriate boundary conditions is essential to reduce the number of iterations, in particular in the multiscale setting involving highly oscillatory coefficients. We will make use of our recent ideas regarding the localization of implicit time discretization schemes for the wave equation. More precisely, we exploit the finite speed of propagation of solutions to the wave equation within a certain time frame as well as superposition properties. We will further refine these ideas and use them to reliably define boundary information for the approximation of exact transmission conditions in the context of non-overlapping iterative domain decomposition strategies.