Operator Upscaling for the Acoustic Wave Equation

Modeling of wave propagation in a heterogeneous medium requires input data that varies on many different spatial and temporal scales. Operator-based upscaling allows us to capture the effect of the fine scales on a coarser domain without solving the full fine-scale problem. The method applied to the constant density, variable sound velocity acoustic wave equation consists of two stages. First, we solve small independent problems for approximate fine-scale information internal to each coarse block. Then we use these subgrid solutions to define an upscaled operator on the coarse grid. The fine-grid velocity field is used throughout the process (i.e., no averaging of input fields is required). An equivalence between the variational form of the problem and a staggered finite-difference scheme allows us to use finite differences to solve the subgrid wave propagation problems. Due to the homogeneous Neumann boundary conditions imposed on each coarse block, the subgrid problems decouple, which leads to the natural parallelization of the first stage of the method. The algorithm requires none of the ghost cell (edge) communication required by standard data parallelism. Timing studies indicate that the parallel algorithm has near optimal speedup. Three variable velocity numerical experiments illustrate that operator-based upscaling captures the essential fine-scale information (even details contained within a single coarse grid block) and models wave propagation quite accurately at considerably less expense than full finite differences. The resulting coarse-grid solution has the advantage of being less prohibitive to store and manipulate.