| Popis: |
Wave propagation is ubiquitous in science and engineering applications, but solving the second-order wave equation in a parallel way is still computationally challenging. Specifically, as efficiency gained from spatial domain decomposition is saturated, time-domain becomes the next candidate for parallelization. However, most parallel-in-time methods are not effective in solving hyperbolic problems, including the wave equation. Motivated by the simple parareal algorithm developed by Lion, Maday, and Turinici, we propose a new parallel scheme called [theta]-parareal that generalizes the original parareal. The convergence and stability analysis of the [theta]-parareal scheme reveal the deficiency of the parareal method when applying to highly oscillatory problems. We then develop a new parallel-in-time iterative method for solving the homogeneous second-order wave equation. The new approach is a data-driven strategy in which we use pre-computed data to stabilize the iteration by minimizing the wave energy residual. We propose two techniques, a linear algebra-based method, and a deep neural network method. Numerical examples, including a wave speed with discontinuities, are provided to demonstrate the effectiveness of the proposed methods on the wave equation. |
