The Newmark scheme as a Velocity-Stress Time staggering: An efficient PML for Spectral Element simulations of elastodynamics

Perfectly matched layers (PMLs) provide an exponential decay, independent of the frequency, of any propagating field along an assigned direction without producing spurious reflections at the interface with the elastic volume. For this reason PMLs have been applied as absorbing boundary conditions (ABCs) and their efficiency in attenuating outgoing wave fields on the outskirts of numerical grids is more and more recognized. However, PMLs are designed for first-order differential equations and a natural extension to second-order Partial Differential Equations (PDEs) involves either additional variables in the time evolution scheme or convolutional operations. Both techniques are computationally expensive when implemented in a spectral element (SE) code and other ABCs (e.g. paraxial or standard sponge methods) still remain more attractive than PMLs. Here, an efficient second-order implementation of PMLs for SE is developed from interpreting the Newmark scheme as a time-staggered velocity–stress algorithm. The discrete equivalence with the standard scheme is based on an L2 approximation of the stress field, with the same polynomial order as the velocity. In this case, PMLs can be introduced as for first-order equations preserving the natural second-order time stepping. Subsequently, a non-classical frequency-dependent perfectly matched layer is introduced by moving the pole of the stretching along the imaginary axis. In this case, the absorption depends on the frequency and the layer switches from a transparent behaviour at low frequencies to a uniform absorption as the frequency goes to infinity. It turns out to be more efficient than classical PMLs in the absorption of the incident waves, as the grazing incidence approaches, at the cost of a memory variable for any split component. Finally, an extension of PMLs to general curvilinear coordinates is proposed.