A Stable, perfectly matched layer for linearized Euler equations in unslit physical variables

Abstract The instability of an earlier perfectly matched layer (PML) formulation for the linearized Euler equations is investigated. It is found that, in the presence of a mean flow, there exist acoustic waves that have a positive group velocity but a negative phase velocity in the direction of the mean flow and these waves become actually amplified in the previous formulation, thus giving rise to the instability. A new stable PML formulation that is perfectly matched to the Euler equations and does not entail exponentially growing solution is presented. Furthermore, the new formulation is given in unsplit physical variables which should facilitate its implementation in many practical schemes. In addition, the well-posedness of the new formulation is also considered. It is shown that the proposed equations are well-posed for horizontal y -layers but weakly well-posed for vertical x -layers and corner layers. However, it is further shown that they can be easily modified to be symmetrizable, thus strongly well-posed, by an addition of arbitrarily small terms. Numerical examples that verify the stability and effectiveness of the proposed PML equations, such as an absorbing boundary condition, are given.

[1]  D. Gottlieb,et al.  Regular Article: Well-posed Perfectly Matched Layers for Advective Acoustics , 1999 .

[2]  Peter Monk,et al.  The Perfectly Matched Layer in Curvilinear Coordinates , 1998, SIAM J. Sci. Comput..

[3]  E. Turkel,et al.  Absorbing PML boundary layers for wave-like equations , 1998 .

[4]  Christopher K. W. Tam,et al.  Perfectly Matched Layer as an Absorbing Boundary Condition for the Linearized Euler Equations in Open and Ducted Domains , 1998 .

[5]  J. Hesthaven.  On the Analysis and Construction of Perfectly Matched Layers for the Linearized Euler Equations , 1998 .

[6]  Richard W. Ziolkowski,et al.  Time-derivative Lorentz material model-based absorbing boundary condition , 1997 .

[7]  David Gottlieb,et al.  A Mathematical Analysis of the PML Method , 1997 .

[8]  Fang Q. Hu,et al.  Application of PML Absorbing Boundary Conditions to the Benchmark Problems of Computational Aeroacoustics , 1997 .

[9]  S. Gedney.  An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices , 1996 .

[10]  Andreas C. Cangellaris,et al.  GT-PML: generalized theory of perfectly matched layers and its application to the reflectionless truncation of finite-difference time-domain grids , 1996, 1996 IEEE MTT-S International Microwave Symposium Digest.

[11]  Fang Q. Hu,et al.  On perfectly matched layer as an absorbing boundary condition , 1996 .

[12]  H. Kreiss,et al.  Time-Dependent Problems and Difference Methods , 1996 .

[13]  S. I. Hariharan,et al.  A potential-theoretic method for far-field sound radiation calculations , 1995 .

[14]  F. Hu.  On Absorbing Boundary Conditions for Linearized Euler Equations by a Perfectly Matched Layer , 1995 .

[15]  Shlomo Ta'asan,et al.  An absorbing buffer zone technique for acoustic wave propagation , 1995 .

[16]  M.Y. Hussaini,et al.  Low-Dissipation and Low-Dispersion Runge-Kutta Schemes for Computational Acoustics , 1994 .

[17]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[18]  C. Tam,et al.  Dispersion-relation-preserving finite difference schemes for computational acoustics , 1993 .

[19]  P. Moin,et al.  Boundary conditions for direct computation of aerodynamic sound generation , 1993 .

[20]  T. Poinsot.  Boundary conditions for direct simulations of compressible viscous flows , 1992 .

[21]  Michael B. Giles,et al.  Nonreflecting boundary conditions for Euler equation calculations , 1990 .

[22]  K. Thompson.  Time-dependent boundary conditions for hyperbolic systems, II , 1990 .

[23]  Craig L. Streett,et al.  Spectral multi-domain for large-scale fluid dynamic simulations , 1989 .

[24]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[25]  Thomas Hagstrom,et al.  Accurate Boundary ConditiOns for Exterior Problems in Gas Dynamics , 1988 .

[26]  Robert L. Higdon,et al.  Numerical absorbing boundary conditions for the wave equation , 1987 .

[27]  S. Orszag,et al.  Approximation of radiation boundary conditions , 1981 .

[28]  A. Bayliss,et al.  Radiation boundary conditions for wave-like equations , 1980 .

[29]  J. Gary,et al.  On boundary conditions for hyperbolic difference schemes , 1978 .

[30]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[31]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .