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

The article of record as published by be found at http://dx.doi.org/10.1016/j.jcp.2015.09.048In 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

An Improved High Order Finite Difference Method for Non-conforming Grid Interfaces for the Wave Equation

This paper presents an extension of a recently developed high order finite difference method for the wave equation on a grid with non-conforming interfaces. The stability proof of the existing methods relies on the interpolation operators being norm-contracting, which is satisfied by the second and fourth order operators, but not by the sixth order operator. We construct new penalty terms to impose interface conditions such that the stability proof does not require the norm-contracting condition. As a consequence, the sixth order accurate scheme is also provably stable. Numerical experiments demonstrate the improved stability and accuracy property

Simulation of earthquake rupture dynamics in complex geometries using coupled finite difference and finite volume methods

We couple a node-centered finite volume method to a high order finite dif- ference method to simulate dynamic earthquake ruptures along nonplanar faults in two dimensions. The finite volume method is implemented on an unstructured mesh, providing the ability to handle complex geometries. The geometric complexities are limited to a small portion of the overall domain and elsewhere the high order finite dif- ference method is used, enhancing efficiency. Both the finite volume and finite differ- ence methods are in summation-by-parts form. Interface conditions coupling the nu- merical solution across physical interfaces like faults, and computational ones between structured and unstructured meshes, are enforced weakly using the simultaneous- approximation-term technique. The fault interface condition, or friction law, provides a nonlinear relation between fields on the two sides of the fault, and allows for the par- ticle velocity field to be discontinuous across it. Stability is proved by deriving energy estimates; stability, accuracy, and efficiency of the hybrid method are confirmed with several computational experiments. The capabilities of the method are demonstrated by simulating an earthquake rupture propagating along the margins of a volcanic plug