A parallelizable method for two-dimensional wave propagation using subdomains in time with Multigrid and Waveform Relaxation

In this paper we compare the implicit schemes for the solution of the two-dimensional wave equation using Singlegrid and Multigrid methods. The discretization is performed using the Finite Difference Method, weighted in time by an established parameter. The parallelization of the algorithms is ensured by employing the Waveform Relaxation method, where numerical stability is achieved by applying the method of subdomains in time. The primary innovation of this work lies in the development of a high-order method that harnesses the parallelizability and robustness of the Multigrid method, enabling efficient solutions to the 2D wave equation. These methods also effectively mitigate oscillations that would otherwise significantly increase the maximum residual, a concern arising from the application of the standard Waveform Relaxation method.