1 Spring 2003 Prof. Tim Warburton MA557/MA578/CS557 Lecture 30

2 Special Edition of ANUM on Absorbing Boundary Conditions Applied Numerical Mathematics Volume 27, Issue 4, Pages (August 1998) Special Issue on Absorbing Boundary Conditions Edited by Eli Turkel

3 Structure of PML Regions PEC

4 How Thick Does the PML Region Need To Be Suppose we consider the region in blue [a,a+delta] And we set: PEC x=a

5 Good Papers on PML Stability Eliane Becache, Peter G. Petropoulosy and Stephen D. Gedney, “On the long-time behavior of unsplit Perfectly Matched Layers”. E.Turkel, A. Yefet, “Absorbing PML Boundary Layers for Wave-Like Equations”, Applied Numerical Mathematics, Volume 27, pp , 1998.

6 Today Last class we examined Berenger’s split field, PML, TE Maxwell’s equations. Berenger introduced anisotropic dissipative terms which allow plane waves to pass into an absorbing region without reflection. However – Abarbanel, Gottlieb, and Hesthaven later showed that the split PML may suffer explosive instability due to the fact that it is only weakly well posed: S. Abarbanel, D. Gottlieb and J. S. Hesthaven, “Long Time Behavior of the Perfectly Matched Layer Equations in Computational Electromagnetics”, Journal of Scientific Computing,vol. 17, no. 1-4, pp , Yet – even later, Becache and Joly showed that the split PML has at worst a linearly growing solution in the late-time. E. Becache and P. Joly, “On the analysis of Berenger’s Perfectly Matched Layers for Maxwell’s equations”, Mathematical Modelling and Numerical Analysis, vol. 36, no. 1, pp , ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4164.pdf

7 Catalogue of Some PMLs There are three well known PML formulations. 1)Berenger’s split PML 2)Ziolkowski’s PML based on a Lorentz material. 3)Abarbanel & Gottlieb’s mathematically derived PML.

8 Recall Berenger’s Split PML

9 Recall: Wave Speeds In the previous notation we looked at eigenvalues of linear combination of the flux matrices: The eigenvalues computed by Matlab: i.e. 0,0,1,-1 under constraint on (alpha,beta) So for Lax-Friedrichs we take

10 Eigenvectors of C Using Matlab we can determine the eigenvectors of C So C does not have a full space of eigenvectors, which in turn means that C can not be diagonalized. So the split PML equations are hyperbolic but only weakly well posed. i.e.

11 Ziolkowski’s PML Ziolkowski proposed a method based on a physical polarized absorbing Lorenz material. R. W. Ziolkowski, “Time-derivative Lorentz material model-based absorbing boundary condition” IEEE Trans. Antennas Propagat., vol. 45, pp , Oct

12 Lorentz Material Model Based PML Introduce 3 new auxiliary variables K,Jx,Jy:

13 Ziolkowski’s Lorentz Material Model Based PML Modification proposed by Abarbanel and Gottlieb: Set:

14 Reconfigured Lorentz Material Model Based PML Note that now the corrections are all lower order terms:

15 Abarbanel and Gottlieb’s PML Abarbanel and Gottlieb proposed a mathematically derived PML. Like the Ziolkowski’s PML it is constructed by adding lower order terms and auxiliary variables.

16 Abarbanel & Gottlieb’s PML

17 Converting Berenger To An Unsplit PML It is possible to start from the Berenger split PML and return to the Maxwell’s TE equations with additional, lower order terms. See: E.Turkel, A. Yefet, “Absorbing PML Boundary Layers for Wave-Like Equations”, Applied Numerical Mathematics, Volume 27, pp , 1998.

18 Berenger To Robust PML We will start with the Berenger PML equations:

19 Berenger To Robust PML Next we Fourier transform in time:

20 Berenger To Robust PML Gather like terms:

21 Berenger To Robust PML Multiply Hzx, Hzy terms with new factors:

22 Berenger To Robust PML Eliminate split variables:

23 Berenger To Robust PML Expand out Hz terms in 3 rd equation:

24 Berenger To Robust PML Expand out Hz terms in 3 rd equation: Also divide 3 rd by i*w:

25 Berenger To Robust PML Create Auxiliary variables and substitute into PML +

26 Berenger To Robust PML Inverse Fourier transform:

27 Berenger To Robust PML Inverse Fourier transform:

28 Berenger To Robust PML Change of variables: Manipulate equations:

29 Comments Notice that the additional auxiliary variables Px,Py,Qz are defined as solutions of ODEs. Corrections to TE Maxwell’s are linear corrections in Ex,Ey,Hz,Px,Py,Qz  strongly hyperbolic equations  well posed. Technically, one should verify that this is still a PML.

30 Surce Term Stability We can verify that the source matrix is a non-positive matrix: Eigenvalues are:

31 Eigenvectors Full set of vectors (at least in the corners):

32 Message on Derivation of a General PML After reviewing the literature it appears that there is a certain art to constructing a PML for a given set of PDEs. For a possible generic approach see: Hagstrom et al: