A high-order super-grid-scale absorbing layer and its application to linear hyperbolic systems

“…This AL is a special case of the method presented by Appelö and Colonius in [3]. Appelö and Colonius also slowed down waves inside the layer by stretching the grid at the boundaries and included higher-order dissipation operators for better performance.…”

Section: The Continuous Problemmentioning

confidence: 99%

Atmospheric Sound Propagation Over Large-Scale Irregular Terrain

2014

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“…In [1] reflections from buffer zones due to under-resolution of outgoing waves, and the influence of different orders of artificial viscosity terms on the damping of reflections are studied. It is shown by numerical experiments for linear hyperbolic systems that artificial viscosity terms based on high-order undivided difference substantially reduce the reflections from the buffer zone.…”

Section: Introductionmentioning

confidence: 99%

Analysis of stretched grids as buffer zones in simulations of wave propagation

Kreiss

Krank

Efraimsson³

2016

Applied Numerical Mathematics

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A zone of increasingly stretched grid is a robust and easy-to-use way to avoid unwanted reflections at artificial boundaries in wave propagating simulations. In such a buffer zone there are two main damping mechanisms, dissipation and under-resolution that turns a traveling wave into an evanescent wave. We present analysis in one and two space dimensions showing that evanescent decay through under-resolution is a very efficient way to damp waves. The analysis is supported by numerical computations.

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“…Efficient and reliable domain truncation becomes essential, since it enables more accurate numerical simulations. More than thirty years of extensive research in this area has resulted in two standard, competing approaches for artificial boundary closures: high order local non-reflecting boundary condition (NRBC) [19,20], and damping layers such as the perfectly matched layer (PML) [1][2][3][4][5][6][7][8][9][10][11][12], grid stretching techniques [14] and others [13,15]. An NRBC is a boundary condition defined on an artificial boundary such that little or no spurious reflections occur as a wave passes the boundary.…”

Section: Introductionmentioning

confidence: 99%

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

Duru

Kozdon

Kreiss

2015

Journal of Computational Physics

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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.

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A high-order super-grid-scale absorbing layer and its application to linear hyperbolic systems

Cited by 71 publications

References 27 publications

Atmospheric Sound Propagation Over Large-Scale Irregular Terrain

Atmospheric Sound Propagation Over Large-Scale Irregular Terrain

Analysis of stretched grids as buffer zones in simulations of wave propagation

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

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