Boundary conditions and stability of a perfectly matched layer for the elastic wave equation in first order form

Abstract

In computations, it is now common to surround artificial boundaries of a computational domain with a perfectly matched layer (PML) of finite thickness in order to prevent artificially reflected waves from contaminating a numerical simulation. Unfortunately, the PML does not give us an indication about appropriate boundary conditions needed to close the edges of the PML, or how those boundary conditions should be enforced in a numerical setting. Terminating the PML with an inappropriate boundary condition or an unstable numerical boundary procedure can lead to exponential growth in the PML which will eventually destroy the accuracy of a numerical simulation everywhere. In this paper, we analyze the stability and the well-posedness of boundary conditions terminating the PML for the elastic wave equation in first order form. First, we consider a vertical modal PML truncating a two space dimensional computational domain in the horizontal direction. We freeze all coefficients and consider a left half-plane problem with linear boundary conditions terminating the PML. The normal mode analysis is used to study the stability and well-posedness of the resulting initial boundary value problem (IBVP). The result is that any linear well-posed boundary condition yielding an energy estimate for the elastic wave equation, without the PML, will also lead to a well-posed IBVP for the PML. Second, we extend the analysis to the PML corner region where both a horizontal and vertical PML are simultaneously active. The challenge lies in constructing accurate and stable numerical approximations for the PML and the boundary conditions. Third, we develop a high order accurate finite difference approximation of the PML subject to the boundary conditions. To enable accurate and stable numerical boundary treatments for the PML we construct continuous energy estimates in the Laplace space for a one space dimensional problem and two space dimensional PML corner problem. We use summation-by-parts finite difference operators to approximate the spatial derivatives and impose boundary conditions weakly using penalties. In order to ensure numerical stability of the discrete PML, it is necessary to extend the numerical boundary procedure to the auxiliary differential equations. This is crucial for deriving discrete energy estimates analogous to the continuous energy estimates. Numerical experiments are presented corroborating the theoretical results. Moreover, in order to ensure longtime numerical stability, the boundary condition closing the PML, or its corresponding discrete implementation, must be dissipative. Furthermore, the numerical experiments demonstrate the stable and robust treatment of PML corners.

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

“…Even when the geometric stability condition is satisfied, however, numerical experiments have also shown that the PML can be unstable [32,33]. For models that satisfy the geometric stability condition, like the acoustic wave equation, recent results [1,2] have revealed the impact of numerical boundary procedures on the stability of discrete PMLs, using high order summation-by-parts (SBP) finite difference method.By the results in [11,16,17] the PML for the acoustic wave equation can be proven well-posed and asymptotically stable. However, the PML and NRBC, involve auxiliary variables and equations that are often not covered by standard DGSEMs.…”

“…Even when the geometric stability condition is satisfied, however, numerical experiments have also shown that the PML can be unstable [32,33]. For models that satisfy the geometric stability condition, like the acoustic wave equation, recent results [1,2] have revealed the impact of numerical boundary procedures on the stability of discrete PMLs, using high order summation-by-parts (SBP) finite difference method.…”

A reflectionless discrete perfectly matched layer

“…It is known that simulations of PML can exhibit temporal instability depending on the choice of damping coefficients [11], how the PML is discretized [12], and how it is truncated into a finite domain [13]. It is therefore necessary to check the numerical stability of Eq.…”

“…Yet, by reformulation and carefully choosing the damping coefficients, researchers of [11,12,60] showed that it is possible to regain stability for anisotropic PMLs. More recent studies showed instabilities due to the far-end boundary of the PML, which are then stabilized by backing the PML with a dissipating boundary treatment [13,61,62].…”

“…Now, similar to the wave equation (13) being the restriction of its complex extension (14) on the real lines α , a PML equation is the restriction of (14) on a set of more general curves…”