Application of the Parareal Algorithm for Acoustic Wave Propagation

We present an application of the parareal algorithm [1] to solve wave propagation problems in the time domain. The parareal algorithm is based on a decomposition of the integration time interval in time slices. It involves a serial prediction step based on a coarse approximation, and a correction step (computed in parallel) based on a fine approximation within each time slice. In our case, the spatial discretization is based on a spectral element approximation which allows flexible and accurate wave simulations in complex geological media. Fully explicit time advancing schemes are classically used for both coarse and fine solvers. In a first stage, we solve the 1D acoustic wave equation in an homogeneous medium in order to test stability and convergence properties of the parareal algorithm. We confirmed the stability problems outlined by Bal [2] and Farhat et al. [3] for hyperbolic problems. These stability issues are mitigated by a time-discontinuous Galerkin discretization of the coarse solver. It may also involve a coarser spatial discretization (hp-refinement) which helps to preserve stability and allows more significant computer savings. Besides, we explore the contribution of elastodynamic homogenization to build consistent coarse grid solvers. Extension to 2D/3D realistic geological media is an ongoing work.