Locally implicit and stabilized explicit time schemes for transient visco-elastic wave propagation problems

In the context of numerical methods for time-domain wave propagation problems, combining high-order lumped finite elements with an explicit time scheme is a popular approach for either inviscid or visco-elastic models. This strategy has proven to be efficient in numerous cases. However, when dealing with non-uniform meshes or high-contrast materials, the stability condition on the time step becomes drastically stringent. One can encounter such configurations when meshing "unfortunate"CAD input file, e.g. when dealing with heterogeneous materials where neighboring heterogeneities produce very small elements in-between them, or when considering materials with high and localized wave velocities. To address efficiently these configurations, we propose to adapt the locally implicit and stabilized explicit methods to the Kelvin-Voigt, Maxwell and Zener visco-elastic models. We prove using energy arguments that the global stability condition of these schemes can be much more favorable compared to a leapfrog explicit scheme, decreasing the number of iterations for a fixed time window. We illustrate our approaches with 2D and 3D numerical test cases related to ultrasonic non-destructive testing experiments.