On energy stable discontinuous Galerkin spectral element approximations of the perfectly matched layer for the wave equation

In this paper, we develop a provably energy stable discontinuous Galerkin spectral element method (DGSEM), approximatingthe perfectly matched layer (PML) for the three and two space dimensional (3D and 2D) linear acoustic wave equations, infirst order form, subject to well-posed linear boundary conditions. First, using the well-known complex coordinate stretching,we derive an efficient un-split modal PML for the 3D acoustic wave equation, truncating a cuboidal computational domain.Second, we prove asymptotic stability of the continuous PML by deriving energy estimates in the Laplace space, for the3D PML in a heterogeneous acoustic medium, assuming piece-wise constant PML damping. In the time-domain, the energyestimate translates to a bound for the solutions in terms of the initial data. Third, we develop a DGSEM for the wave equationusing physically motivated numerical flux, with penalty weights, which are compatible with all well-posed, internal and external,boundary conditions. When the PML damping vanishes, by construction, our choice of penalty parameters yields an upwindscheme and a discrete energy estimate analogous to the continuous energy estimate. Fourth, to ensure numerical stability ofthe discretization when PML is present, it is necessary to systematically extend the numerical fluxes, and the inter-elementand boundary procedures, to the PML auxiliary differential equations. This is critical for deriving discrete energy estimatesanalogous to the continuous energy estimates. Finally, we propose a procedure to compute PML damping coefficients such thatthe PML error converges to zero, at the optimal convergence rate of the underlying numerical method. Numerical solutionsare evolved in time using the high order Taylor-type time stepping scheme of the same order of accuracy of the spatialdiscretization. By combining the DGSEM spatial approximation with the high order Taylor-type time stepping scheme and theaccuracy of the PML we obtain an arbitrarily accurate wave propagation solver in the time domain. Numerical experimentsare presented in 2D and 3D corroborating the theoretical results.