Sparse-grid solvers for space-time discretization of the wave equation
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In this talk, we consider space-time Galerkin discretizations of the acoustic wave equation. Unlike traditional time-stepping methods, the time variable is treated as an additional dimension. We focus on the unconditionally stable numerical schemes with exponential weights proposed and analyzed in [Ferrari, Perugia 2026] and [Ferrari, Perugia, Zampa 2026]. Our goal is to analyze sparse-grid solvers based on the combination technique. The idea is to properly combine Galerkin solutions computed on properly chosen small full tensor-product spaces. We prove that this technique achieves the same order of convergence as the Galerkin approximation on the finest tensor-product space, while keeping the computational cost drastically lower. This solver also allows for easy parallelization of these space-time schemes. We present recent theoretical and numerical results. This talk is based on joint work with A. Moiola, C. Perinati, and I. Perugia.
