On Optimal Finite-Difference Approximation of PML

A technique derived from two related methods suggested earlier by some of the authors for optimization of finite-difference grids and absorbing boundary conditions is applied to discretization of perfectly matched layer (PML) absorbing boundary conditions for wave equations in Cartesian coordinates. We formulate simple sufficient conditions for optimality and implement them. It is found that the minimal error can be achieved using pure imaginary coordinate stretching. As such, the PML discretization is algebraically equivalent to the rational approximation of the square root on [0,1] conventionally used for approximate absorbing boundary conditions. We present optimal solutions for two cost functions, with exponential (and exponential of the square root) rates of convergence with respect to the number of the discrete PML layers using a second order finite-difference scheme with optimal grids. Results of numerical calculations are presented.

[1]  Vladimir Druskin,et al.  Application of the Difference Gaussian Rules to Solution of Hyperbolic Problems , 2000 .

[2]  F. Harris.  On the use of windows for harmonic analysis with the discrete Fourier transform , 1978, Proceedings of the IEEE.

[3]  Vladimir Druskin,et al.  Application of the difference Gaussian rules to solution of hyperbolic problems: global expansion , 2002 .

[4]  Weng Cho Chew,et al.  A 3D perfectly matched medium from modified maxwell's equations with stretched coordinates , 1994 .

[5]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[6]  L. Trefethen,et al.  Well-Posedness of one-way wave equations and absorbing boundary conditions , 1986 .

[7]  John L. Tassoulas,et al.  CONTINUED-FRACTION ABSORBING BOUNDARY CONDITIONS FOR THE WAVE EQUATION , 2000 .

[8]  Arden Ruttan,et al.  NUMERICAL RESULTS ON BEST UNIFORM RATIONAL APPROXIMATION OF $ \vert x\vert$ ON $ \lbrack-1,\,+1\rbrack$ , 1993 .

[9]  A. Gončar,et al.  ON MARKOV'S THEOREM FOR MULTIPOINT PADÉ APPROXIMANTS , 1978 .

[10]  David H. Bailey,et al.  A Fortran 90-based multiprecision system , 1995, TOMS.

[11]  Vladimir Druskin,et al.  Gaussian Spectral Rules for the Three-Point Second Differences: I. A Two-Point Positive Definite Problem in a Semi-Infinite Domain , 1999, SIAM J. Numer. Anal..

[12]  Vladimir Druskin,et al.  Three-point finite-difference schemes, Padé and the spectral Galerkin method. I. One-sided impedance approximation , 2001, Math. Comput..

[13]  Vladimir Druskin,et al.  Optimal finite difference grids and rational approximations of the square root I. Elliptic problems , 2000 .

[14]  R. Varga.  Scientific Computations on Mathematical Problems and Conjectures , 1987 .

[15]  P. Monk,et al.  Optimizing the Perfectly Matched Layer , 1998 .

[16]  Herbert Stahl,et al.  Best uniform rational approximation ofxα on [0, 1] , 2003 .

[17]  Vladimir Druskin,et al.  Gaussian spectral rules for second order finite-difference schemes , 2000, Numerical Algorithms.

[18]  P. Petrushev,et al.  Rational Approximation of Real Functions , 1988 .

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

[20]  Peter G. Petropoulos,et al.  Reflectionless Sponge Layers as Absorbing Boundary Conditions for the Numerical Solution of Maxwell Equations in Rectangular, Cylindrical, and Spherical Coordinates , 2000, SIAM J. Appl. Math..

[21]  J. Cooper,et al.  Theory of Approximation , 1960, Mathematical Gazette.

[22]  Jian-Ming Jin,et al.  Complex coordinate stretching as a generalized absorbing boundary condition , 1997 .

[23]  A. Majda,et al.  Radiation boundary conditions for acoustic and elastic wave calculations , 1979 .

[24]  G. A. Baker,et al.  Padé Approximants Second Edition: Extensions of Padé approximants , 1996 .