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Nonreflecting Boundary Conditions
for Time-Dependent Wave Propagation
Inauguraldissertation
zur
Erlangung der Würde eines Doktors der Philosophie
vorgelegt der
Philosophisch-Naturwissenschaftlichen Fakultät
der Universität Basel
von
Imbo Sim
aus Suwon (Südkorea)
Lausanne, 2010
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Genehmigt von der Philosophisch-Naturwissenschaftlichen Fakultät
auf Antrag von
Prof. Dr. Marcus J. Grote
Prof. Dr. Thomas Hagstrom (Southern Methodist University)
Basel, den 26. Mai 2009
Prof. Dr. Eberhard Parlow
Dekan
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Abstract
Many problems in computational science arise in unbounded domains and
thus require an artificial boundary B, which truncates the unbounded ex-
terior domain and restricts the region of interest to a finite computational
domain, Ω. It then becomes necessary to impose a boundary condition at
B, which ensures that the solution in Ω coincides with the restriction to Ω
of the solution in the unbounded region. If we exhibit a boundary condition,
such that the fictitious boundary appears perfectly transparent, we shall call
it exact. Otherwise it will correspond to an approximate boundary condi-
tion and generate some spurious reflection, which travels back and spoils the
solution everywhere in the computational domain. In addition to the trans-
parency property, we require the computational effort involved with such a
boundary condition to be comparable to that of the numerical method used
in the interior. Otherwise the boundary condition will quickly be dismissed
as prohibitively expensive and impractical. The constant demand for increas-
ingly accurate, efficient, and robust numerical methods, which can handle a
wide variety of physical phenomena, spurs the search for improvements in
artificial boundary conditions.
In the last decade, the perfectly matched layer (PML) approach [16] has
proved a flexible and accurate method for the simulation of waves in un-
bounded media. Standard PML formulations, however, usually require wave
equations stated in their standard second-order form to be reformulated as
first-order systems, thereby introducing many additional unknowns. To cir-
cumvent this cumbersome and somewhat expensive step we propose instead
a simple PML formulation directly for the wave equation in its second-order
form. Our formulation requires fewer auxiliary unknowns than previous for-
mulations [23, 94].
Starting from a high-order local nonreflecting boundary condition (NRBC)
for single scattering [55], we derive a local NRBC for time-dependent multi-
ple scattering problems, which is completely local both in space and time. To
do so, we first develop a high order exterior evaluation formula for a purely
outgoing wave field, given its values and those of certain auxiliary functions
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needed for the local NRBC on the artificial boundary. By combining that
evaluation formula with the decomposition of the total scattered field into
purely outgoing contributions, we obtain the first exact, completely local,
NRBC for time-dependent multiple scattering. Remarkably, the informa-
tion transfer (of time retarded values) between sub-domains will only occur
across those parts of the artificial boundary, where outgoing rays intersect
neighboring sub-domains, i.e. typically only across a fraction of the artificial
boundary. The accuracy, stability and efficiency of this new local NRBC is
evaluated by coupling it to standard finite element or finite difference meth-
ods.
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Acknowledgements
This work was supported by the Swiss National Science Foundation under
projects, Advanced Methods for Computational Electromagnetics (NF–Nr. :
200020 − 105135, 200020 − 113702).
My sincere thanks go to Prof. Marcus J. Grote for his indispensable support
during my doctoral studies and all his patience. His excellent knowledge
and experience of computational wave propagation have always been a great
source of motivation for me.
I’m very grateful to Prof. Thomas Hagstrom for his interest in my work,
and for his willingness to act as a co-referee for my thesis.
I wish to express my warmest and deepest gratitude to my family as well
for their support as for their patience during my study abroad.
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Contents
1 Introduction to Wave Propagation
in Unbounded Domains
1.1 Nonreflecting boundary conditions
on planar boundaries . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.1Engquist - Majda . . . . . . . . . . . . . . . . . . . . . 12
1.1.2Higdon . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.3 Givoli - Neta . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.4 Hagstrom - Warburton . . . . . . . . . . . . . . . . . . 13
1.2 Nonreflecting boundary conditions
on spherical boundaries . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1Bayliss - Turkel . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Grote - Keller . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3Hagstrom - Hariharan
1.3 Perfectly matched layers (PML) . . . . . . . . . . . . . . . . . 17
1.3.1Split formulation . . . . . . . . . . . . . . . . . . . . . 17
1.3.2 Unsplit formulation . . . . . . . . . . . . . . . . . . . . 18
11
. . . . . . . . . . . . . . . . . . 15
2 On Local Nonreflecting Boundary Conditions for Time De-
pendent Wave Propagation
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2Absorbing boundary conditions . . . . . . . . . . . . . . . . . 23
2.2.1 The One-dimensional Wave Equation . . . . . . . . . . 23
2.2.2 Absorbing Boundary Conditions in Higher Dimensions
2.2.3 High-order local nonreflecting boundary conditions
2.3 Multiple scattering problems . . . . . . . . . . . . . . . . . . . 32
2.3.1The one-dimensional case
2.3.2The three-dimensional case . . . . . . . . . . . . . . . . 35
2.4Numerical experiment . . . . . . . . . . . . . . . . . . . . . . 36
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
20
25
. . 31
. . . . . . . . . . . . . . . . 33
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3 Perfectly Matched Layers for Time-Dependent Wave Equa-
tions in Second-Order Form
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 PML formulation . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Extension to complex frequency shifted PML . . . . . . . . . . 50
3.5 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.1 Finite difference discretization . . . . . . . . . . . . . . 52
3.5.2 Discontinuous Galerkin Discretization . . . . . . . . . . 54
3.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 55
3.6.1 Point source in 2D . . . . . . . . . . . . . . . . . . . . 56
3.6.2 Heterogeneous medium in 2D . . . . . . . . . . . . . . 56
3.6.3Point source in 3D . . . . . . . . . . . . . . . . . . . . 58
3.7PML for elastodynamic equations in second-order form . . . . 59
3.7.1 Model problem . . . . . . . . . . . . . . . . . . . . . . 59
3.7.2PML formulation . . . . . . . . . . . . . . . . . . . . . 61
3.7.3Extension to complex frequency shifted PML . . . . . . 63
3.7.4 Discretization . . . . . . . . . . . . . . . . . . . . . . . 65
3.7.5 Numerical experiments . . . . . . . . . . . . . . . . . . 67
3.8PML for poroelastic wave equations in second-order form . . . 75
3.8.1 Model problem . . . . . . . . . . . . . . . . . . . . . . 75
3.8.2 PML formulation . . . . . . . . . . . . . . . . . . . . . 76
3.8.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . 78
3.8.4 Numerical experiments . . . . . . . . . . . . . . . . . . 81
3.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 81
41
4Local Nonreflecting Boundary Conditions for Time-Dependent
Multiple Scattering
4.1 Local boundary condition for single scattering . . . . . . . . . 86
4.2 Exterior evaluation formula. . . . . . . . . . . . . . . . . . . 87
4.3 Local boundary condition for multiple scattering . . . . . . . . 94
4.3.1Multiple scattering in spherical coordinate . . . . . . . 94
4.4 Finite difference formulation . . . . . . . . . . . . . . . . . . . 97
4.5 Interpolation of the evaluated solution
4.5.1 Akima spline interpolation . . . . . . . . . . . . . . . . 98
4.6Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 101
4.6.1Accuracy of the evaluation formula . . . . . . . . . . . 101
4.6.2Multiple scattering of an incident plane wave . . . . . . 103
86
. . . . . . . . . . . . . 98
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5Time-Dependent Multiple Scattering for Maxwell’s Equa-
tions
5.1 Local NRBC for single scattering . . . . . . . . . . . . . . . . 105
5.2 Local boundary condition for multipole fields . . . . . . . . . . 106
5.3 Exterior evaluation formula for multipole fields . . . . . . . . . 110
5.4 Time-dependent multiple scattering
for Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . 111
105
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List of Figures
1.1Sphere with a ball-shaped obstacle as a computational domain
(the mesh was generated by DistMesh [89]). Nonreflecting
boundary condtions are imposed on the outer surface of a sphere. 14
Top: a photonic crystal ([76]) with periodic dielectric holes
that affect the propagation of electromagnetic waves. Bottom:
numerical solution of the z-component of the time-dependent
electric field, Ez, which was implemented with the PML method.
The pictures show that the photonic crystal can be used to in-
duce a 90 degree bend in the direction of propagation. . . . . . 19
1.2
2.1A typical scattering problem consists of an obstacle, a source
term f, and incoming wave ui, and a scattered wave us. The
artificial boundary B defines the outer boundary of the com-
putational domain Ω. . . . . . . . . . . . . . . . . . . . . . . . 22
The one-dimensional wave equation: inside the computational
domain, Ω = [0,L] the problem can be arbitrarily complicated,
but in the exterior region, x ≥ L, we assume that f(x,t) = 0
for t > 0 and that u and ∂tu vanish at t = 0. . . . . . . . . . . 24
A traveling plane wave with an angle of incidence θ. . . . . . . 29
Amount of spurious reflection (in percent) caused by the use
of the boundary conditions (2.14) for a plane wave with angle
of incidence θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Multiple scattering in one space dimension. . . . . . . . . . . . 33
Left: the time dependence of the Gaussian point source. Right:
the velocity profile c(r). . . . . . . . . . . . . . . . . . . . . . 37
Scattering from a spherical wave guide: snapshots of the ref-
erence solution at different times. The three circles drawn are
located at r = 0.5,1,1.5. The Gaussian point source is located
outside the computational domain at r = 0.45, θ = 0. . . . . . 39
The numerical solutions computed using the boundary condi-
tions (4.2) with P = 0, P = 1, and P = 5, are compared with
the exact solution at r = 0.75, θ = 3π/4. . . . . . . . . . . . . 40
2.2
2.3
2.4
2.5
2.6
2.7
2.8
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3.1 The damping profile ζi(xi) given by (3.15) is shown for differ-
ent values of¯ζi, with c = 1 and Li= 0.1 . . . . . . . . . . . . . 47
Point source in 2D: snapshots of the numerical solutions at
different times in Ω = [−0.5, 0.5]2, surrounded by a PML of
width L = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Point source in 2D: time evolution of the L2–error for different
damping coefficients¯ζi. . . . . . . . . . . . . . . . . . . . . . . 57
Heterogeneous medium in 2D: varying wave speed c given by
(3.51). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Heterogeous medium in 2D: snapshots of the numerical solu-
tion are shown at different times in Ω = [−1, 1]2, surrounded
by a PML of width L = 0.2. . . . . . . . . . . . . . . . . . . . 59
Point source in 3D: snapshots of the numerical solution are
shown at different times in Ω = [−0.5, 0.5]3, surrounded by a
PML of width L = 0.1. . . . . . . . . . . . . . . . . . . . . . . 60
Left: the orientation of the slowness vector s is the same as
the group velocity vswith respect to the direction k1, Right:
the orientations of s and vsare different with respect to the
direction k1(see more details [15]). . . . . . . . . . . . . . . . 62
Slowness curves for different materials. . . . . . . . . . . . . . 68
The snapshots of ?u?2in material I . . . . . . . . . . . . . . . 70
3.10 The snapshots of ?u?2in material II . . . . . . . . . . . . . . 71
3.11 The snapshots of ?u?2in material III
3.12 The snapshots of ?u?2in material IV . . . . . . . . . . . . . . 73
3.13 The snapshots of ?u?2in material V
3.14 Numerical solution, uh
computational domain Ω = [−3, 3]2surrounded by PML of
width L = 0.6.It was implemented with finite difference
method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.15 Numerical solution, wh
3.16 Numerical solution, ph
. . . . . . . . . . . . . . . . . . . . . . 84
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
. . . . . . . . . . . . . 72
. . . . . . . . . . . . . . 74
s,1with the pressure source, fp in the
1. . . . . . . . . . . . . . . . . . . . . . 83
4.1Wave scattering from an obstacle Γ. The computational do-
main, Ω, is bounded by the artificial boundary, B, where the
local NRBC (4.2) is imposed. Subsequent evaluation of the so-
lution in other sub-domains, Q1and Q2, is possible via (4.16)
by using past values of u and wkat B. . . . . . . . . . . . . . 95
Local coordinates (r1,θ1) and (r2,θ2) . . . . . . . . . . . . . . 96
Evaluation of solution on the other computational domain: we
evaluate the exterior solution on P2, P3, and P4based on the
auxiliary functions on P0, and P1. . . . . . . . . . . . . . . . . 98
4.2
4.3
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4.4We calculate the auxiliary functions, wk, k = 0,...,p of Hagstrom-
Hariharan’s NRBC (4.2) at the green points. Then we obtain
the exterior solutions at the blue points, using representation
formula (4.39), and if needed, interpolate the exterior solution
at the red points using the local spline interpolation (4.50)
with (4.59). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Contour lines across B obtained either from the numerical
solution for 0.5 ≤ r ≤ 1 or the evaluation formula (4.39) for
r > 1; the source is located at (0.4,0). . . . . . . . . . . . . . . 102
Evaluation of the solution at θ =π
The total L2-error is shown vs. the mesh size h for varying p. . 103
Plane wave scattering from two sound-soft spheres. The com-
putation is restricted to the two disjoint regions. . . . . . . . . 104
4.5
4.6
4.7
4.8
2and t = 1 for varying p. . . 102
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Dedicated to Hyunmyung & Gyuseong
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Chapter 1
Introduction to Wave
Propagation
in Unbounded Domains
Abstract
domains and thus require an artificial boundary B, which truncates the
unbounded exterior domain and restricts the region of interest to a finite
computational domain, Ω. It then becomes necessary to impose a bound-
ary condition at B, which ensures that the solution in Ω coincides with
the restriction to Ω of the solution in the unbounded region. If we exhibit
a boundary condition, such that the fictitious boundary appears perfectly
transparent, we shall call it exact. Otherwise it will correspond to an ap-
proximate boundary condition and generate some spurious reflection, which
travels back and spoils the solution everywhere in the computational domain.
In addition to the transparency property, we require the computational ef-
fort involved with such a boundary condition to be comparable to that of the
numerical method used in the interior. Otherwise the boundary condition
will quickly be dismissed as prohibitively expensive and impractical. The
constant demand for increasingly accurate, efficient, and robust numerical
methods, which can handle a wide variety of physical phenomena, spurs the
search for improvements in artificial boundary conditions. In this section we
give a brief review of nonreflecting boundary conditions (NRBC).
Many problems in computational science arise in unbounded
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1.1Nonreflecting boundary conditions
on planar boundaries
1.1.1 Engquist - Majda
In the late 1970s, Engquist and Majda [32, 33] contributed to the construction
and analysis of a hierachy of local boundary conditions, whose second-order
version is still widely used. By using the Laplace-Fourier transform in time
and in the plane tangential to the artificial boundary, they derived the ex-
act boundary condition in terms of a pseudo-differential operator, which in
practice needs to be localized through a Padé approximation.
1.1.2 Higdon
Higdon [71, 72] derived a Nonreflecting Boundary Condition (NRBC) of the
form
p?
This boundary condition is exact for any linear combination of plane waves
whose angles of incidence are ±αjwith wave speed c, i.e., each term of the
product in (1.1) annihilates the two plane waves u = u(t−cosαjx−sinαjy)
and u = u(t−cosαjx+sinαjy). These plane waves leave the computational
domain without reflections, but all other waves produce some reflections. Its
reflection coefficient is
p?
for plane waves propagating at the angle of incidence θ. This implies that
the reflection coefficient becomes smaller as the order p is increased. The
Higdon NRBCs can be applied to a variety of wave problems including those
in dispersive or in layered media. We note that Engquist-Majda ABC’s are
equivalent to (1.1) for αj= 0, j = 1,...,p.
j=1
?
cosαj∂
∂t− c∂
∂x
?
u = 0.
(1.1)
j=1
?cosαj− cosθ
cosαj+ cosθ
?
(1.2)
1.1.3Givoli - Neta
Based on a reformulation of the Higdon NRBCs, Givoli and Neta [43] derive
a new boundary scheme, which does not involve any high derivatives beyond
second order. In contrast to the exponential computational effort in Higdon’s
NRBCs, the effort with the Givoli-Neta NRBCs increases just with the order.
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1.1.4 Hagstrom - Warburton
Hagstrom and Warburton [58] propose a new formulation of local high-order
NRBC with several attractive features in comparison to the Givoli-Neta re-
formulation of Higdon-NRBC. They introduce new local auxiliary variables,
which satisfy a symmetrizable system of second-order wave equations on the
absorbing boundary and allows the straightforward derivation of the corre-
sponding high-order corner compatibility conditions.
1.2 Nonreflecting boundary conditions
on spherical boundaries
1.2.1 Bayliss - Turkel
Bayliss and Turkel [12] derived an alternative sequence of local operators,
which annihilate increasingly many terms in the large distance expansion of
an outgoing solution to the wave equation. Their boundary condition was
extended by Peterson [90] to Maxwell’s equations. The sequence of local
operators, which was introduced by Bayliss and Turkel, is as follows:
Bp=
p?
?1
j=1
?1
∂
∂t+∂
c
∂
∂t+∂
∂r+2j − 1
r
?
=
c ∂r+2p − 1
r
?
Bp−1.
(1.3)
1.2.2 Grote - Keller
The exact NRBC, which is local in time on a spherical boundary, was con-
tributed by Grote-Keller [45, 46] in 1995. It has the following general form
?∂
∂t+∂
∂r
?
[ru](r,θ,φ,t) = −1
R
N
?
n=1
n
?
m=−n
dn· ψnm(t)Ynm(θ,φ),
(1.4)
d
dtψnm(t) =
ψnm(0) = 0,
1
RAnψnm(t) + (u(r,θ,φ,t)|r=R,Ynm(θ,φ))en,
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Figure 1.1: Sphere with a ball-shaped obstacle as a computational domain
(the mesh was generated by DistMesh [89]). Nonreflecting boundary cond-
tions are imposed on the outer surface of a sphere.
with
dn= {dj
en= {ej
n}n
n}n
j=1,dj
n
=
n(n + 1)j
2
= δ1j,
,
(1.5)
j=1,ej
n
(1.6)
An= {Aij
n}n
i,j=1,Aij
n
=
−n(n+1)
2
(n+i)(n+1−i)
2i
0,
,i = 1
, i = j + 1
otherwise,
(1.7)
and
(u(r,θ,φ,t)|r=R,Ynm(θ,φ)) =
2π
?
0
π
?
0
u(R,θ,φ,t)Ynm(θ,φ)sinθ dθdφ,
(1.8)
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for an N ≥ 1. Here Ynmare the spherical harmonics:
Ynm(θ,φ) :=
?
2n + 1
4π
(n − |m|)!
(n + |m|)!P|m|
n (cosθ)eimφ
(1.9)
for m = −n,...,n, n = 0,1,2,.... The spherical harmonics are eigenfunc-
tions of the Laplace-Beltrami operator:
∆SYnm= −n(n + 1)Ynm.
(1.10)
The boundary condition is exact for N → ∞ and local in time, but non-local
in space. It is based on the following Fourier representation of the solution
u:
u(r,θ,φ,t) =
∞
?
2π
?
0
n=0
n
?
π
?
0
m=−n
unm(r,t)Ynm(θ,φ),
unm(r,t) =u(r,θ,φ,t)Ynm(θ,φ)sinθ dθdφ.
This derivation was later extended to electromagnetic and elastic waves [47]
- [49].
1.2.3Hagstrom - Hariharan
According to the expansion theorem of Wilcox [107], the scattered solution of
the wave equation can be represented as a series, which converges absolutely
and uniformly in r > R:
u(r,θ,φ,t) =1
r
∞
?
k ≥ 1, are determined from the function,
k=0
fk(θ,φ,r − ct)
rk
.
(1.11)
The functions fk= fk(θ,φ,r−ct),
f0by the recursion formula
2k∂fk
∂t
= −(∆S+ k(k − 1))fk−1,k ≥ 1,
(1.12)
with the Laplace-Beltrami operator
∆S=
1
sinθ
∂
∂θ
?
sinθ∂u
∂θ
?
+
1
sin2θ
∂2u
∂φ2.
(1.13)
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The function fkcan be represented by a Fourier series as
fk(θ,φ,t) =
∞
?
n=0
n
?
m=−n
fk
nm(t)Ynm(θ,φ).
(1.14)
Hence, the solution u is expressed as Fourier series:
u(r,θ,φ,t) =
∞
?
n=0
n
?
m=−n
unmYnm(θ,φ), r > R,
(1.15)
where the Fourier coefficients
unm(r,t) =1
r
n
?
k=0
fk
nm(r − ct)
rk
.
(1.16)
Substituting (1.16) into (1.15) with (1.9),we obtain the recursion formula
fk
dt
nm
= −k(k − 1) + n(n + 1)
2k
fk−1
nm, k = 1,2,....
(1.17)
We consider the sequence of local operators, which was introduced by Bayliss
and Turkel
?1
=
c
Set Bjunm= wnm
Bp=
p?
?1
j=1
c
∂
∂t+∂
∂r+2j − 1
r
?
∂
∂t+∂
∂r+2p − 1
r
?
Bp−1.
(1.18)
j . We define the auxiliay functions as
wj(r,θ,φ,t) =
∞
?
n=j
n
?
m=−n
ξj
nm(r,t)Ynm(θ,φ)
(1.19)
with
ξj
nm(r,t) =
n
?
k=j
aj
kr−k−j−1fk
nm(ct − r),
(1.20)
where aj
Hagstrom and Hariharan [55] use the functions wj to formulate the local
boundary conditions:
?1
?1
k= (−1)j21−j
k!
(k−j)!.
c
∂
∂t+∂
∂r+1
∂
∂t+j
r
?
?
wj=
u = w1,
cr
1
4r2
?
j(j − 1) + ∆S
?
wj−1+ wj+1, j = 1,2,...,p,
(1.21)
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with wp+1= 0. The boundary condition (1.21) is local in space and time and
does not involve high-order derivatives. For this reason, this local boundary
condition is easily combined with standard numerical methods and enables
arbitrarily high order implementations. Recently, it was extended to the time
dependent Maxwell equations [53].
1.3Perfectly matched layers (PML)
1.3.1Split formulation
An alternative to nonreflecting boundary conditions are absorbing layers. We
consider the wave equation as a first-order system with the wave speed c = 1
and without source terms:
∂u
∂t
∂v
∂t
where v = (v1,v2)⊤. Then Bérenger’s PML formulation [16] is based on
splitting the solution, u = ux+ uyas follows:
= ∇ · v,
(1.22)
= ∇u,
(1.23)
∂ux
∂t
∂uy
∂t
∂v
∂t
=
∂v1
∂x,
∂v2
∂y,
(1.24)
=
(1.25)
= ∇u.
(1.26)
We now formulate Bérenger’s PML system with damping in the x-direction.
Adding the damping terms in the equations involving uxand v1, we obtain
the PML equations:
∂ux
∂t
∂uy
∂t
∂v1
∂t
∂v2
∂t
Inside the absorbing layer a damping term ζ is added to the wave equation,
which acts only in the direction orthogonal to the layer. The initial formula-
tions in [16] were based on splitting the electromagnetic fields into two parts,
+ ζux =
∂v1
∂x,
∂v2
∂y,
∂u
∂x,
∂u
∂y.
(1.27)
=
(1.28)
+ ζv1 =
(1.29)
=
(1.30)
17
Page 20
the first containing the tangential derivatives and the second containing the
normal derivatives. Damping is then enforced only upon the normal direc-
tion. Later Abarbanel and Gottlieb [2] showed that Bérenger’s approach was
only weakly well-posed due to the unphysical splitting of the field variables.
1.3.2 Unsplit formulation
The Zhao-Cangellaris formulation [109] avoids splitting the solution, u. In-
stead, we apply the operator ∂tto (1.27) and the operator ∂t+ ζ to (1.28),
and add up the two equations:
??∂
The damping parameter, ζ, does not depend on y, and the operators ∂t+ ζ
and ∂ycommute each other. We introduce a new variable, v∗
the equation
∂v∗
2
∂t∂t
Finally we get the Zhao-Cangellaris formulation
∂
∂t ∂t+ ζ
?
u −∂v1
∂x
?
−∂
∂y
?∂
∂t+ ζ
?
v2= 0.
(1.31)
2, which satisfies
=∂v2
+ ζv2.
(1.32)
∂u
∂t+ ζu =
∂v∗
∂t
∂v1
∂t
∂v2
∂t
∂v1
∂x+∂v∗
∂v2
∂t
∂u
∂x,
∂u
∂y.
2
∂y,
(1.33)
2
=+ ζv2,
(1.34)
+ ζv1 =
(1.35)
=
(1.36)
In [14] it is shown that the Zhao-Cangellaris formulation is equivalent to the
Bérenger’s formulation.
18
Page 21
Figure 1.2: Top: a photonic crystal ([76]) with periodic dielectric holes that
affect the propagation of electromagnetic waves. Bottom: numerical solution
of the z-component of the time-dependent electric field, Ez, which was imple-
mented with the PML method. The pictures show that the photonic crystal
can be used to induce a 90 degree bend in the direction of propagation.
19
Page 22
Chapter 2
On Local Nonreflecting Boundary
Conditions for Time Dependent
Wave Propagation
Abstract
erally requires an artificial boundary to truncate the unbounded exterior and
limit the computation to a finite region. At the artificial boundary a bound-
ary condition is then needed, which allows the propagating waves to exit the
computational domain without spurious reflection. In 1977, Engquist and
Majda proposed the first hierarchy of absorbing boundary conditions, which
allows a systematic reduction of spurious reflection without moving the artifi-
cial boundary farther away from the scatterer. Their pioneering work, which
initiated an entire research area, is reviewed here from a modern perspective.
Recent developments such as high-order local conditions and their extension
to multiple scattering are also presented. Finally, the accuracy of high-order
local conditions is demonstrated through numerical experiments.
The simulation of wave phenomena in unbounded domains gen-
2.1 Introduction
Unbounded domains are often encountered in scientific and engineering appli-
cations. Examples are radar and sonar technology, wireless communication,
and seismic imaging. Typically the phenomenon of interest is local but em-
bedded in a vast surrounding medium. Although the exterior region may not
be truly unbounded, the boundary effects are often negligible, so that one
further simplifies the problem by replacing the vast exterior by an infinite
medium.
Mathematical models of natural phenomena usually consist of partial dif-
20
Page 23
ferential equations, whose derivation is based on physical conservation laws.
Many standard numerical methods, such as finite differences and finite ele-
ments, can approximately solve partial differential equations. In fact, they
can even handle complicated geometries, inhomogeneous media, and nonlin-
earity. However, they typically require an artificial boundary, which trun-
cates the unbounded exterior domain, to fit the infinite region on a finite
computer. This immediately raises the question:
Which boundary condition guarantees that the solution to the
initial-boundary value problem inside the artificial boundary co-
incides with the solution of the original problem in the unbounded
region ?
If we exhibit a boundary condition, such that the fictitious boundary appears
perfectly transparent, we shall call it “exact”. Otherwise it will correspond to
an approximate boundary condition1and generate some spurious reflection,
which travels back and perturbs the solution everywhere in the computa-
tional domain. The resulting error in the computer simulation then consists
of two independent error components: the discretization error of the numer-
ical method used in the interior and the spurious reflection generated at the
fictitious boundary. Unless both error components are reduced systemati-
cally, the numerical solution will not converge to the solution of the original
problem in the unbounded region. In this article, we shall restrict ourselves to
time dependent scattering problems. Typically a scattering problem consists
of an obstacle, a source term f, and possibly an incident wave ui– see Figure
1. Scattering problems are common in acoustic, electromagnetic, and elastic
wave propagation. Our goal is to calculate numerically the time-dependent
wave field usscattered from the complex, possibly nonlinear, but bounded
scatterering region.
In 1974 Smith suggested perhaps the first exact method to restrict the com-
putation to a finite region [95]. Let the computational domain Ω be bounded
by a convex boundary of n line segments (or planar facets in IR3). Then the
restricition to Ω of the solution in unbounded space consists of a linear com-
bination of 2nsolutions which satisfy all possible combinations of Dirichlet
or Neumann boundary conditions. Unfortunately, this approach has but lit-
tle practical value, since a rectangular domain requires 26= 64 independent
numerical solutions. This example illustrates a key aspect in the design of
improved absorbing boundary conditions: it is not sufficient to construct a
new boundary condition; in addition, the computational effort involved must
1“...also called radiating, absorbing, silent, transmitting, transparent, open, free-space,
and one-way boundary conditions.”,Givoli, 1992 [37]
21
Page 24
?
?
?
?
?
?
?
Figure 2.1: A typical scattering problem consists of an obstacle, a source term
f, and incoming wave ui, and a scattered wave us. The artificial boundary
B defines the outer boundary of the computational domain Ω.
be comparable to that of the numerical method used in the interior. Other-
wise it will quickly be dismissed as prohibitively expensive and impractical.
In 1977, Engquist and Majda [32, 33] proposed the first hierarchy of absorb-
ing boundary conditions, which allows a systematic reduction of spurious
reflection while keeping the artificial boundary at a fixed distance from the
scatterer. Their pioneering work, still very much in use even today, initiated
an entire research area that led to a wide variety of different approaches, such
as perfectly matched layers (PML) [16], fast integral based formulations [84],
semi-local formulations [45, 46] and high-order local conditions [55, 58, 53]
– see [56, 104, 39] for review articles and additional references. All these
approaches lead to convergent numerical schemes while treating the open
boundary at a computational cost comparable to that in the interior.
In this article we shall focus on local absorbing (or nonreflecting) bound-
ary conditions, which are completely local both in space and time. First in
Section 2, we introduce the fundamental ideas underlying the derivation of
nonreflecting boundary conditions by considering the simple one-dimensional
case. Next, we review the original Engquist-Majda conditions [32] for wave
propagation in more than one space dimensions, where we exhibit the trade-
off between exactness and locality. We also present recent developments of
high-order local boundary conditions without high-order derivatives, both
for acoustic and electromagnetic waves. Next, in Section 3, we consider the
extension of local NBC to multiple scattering, first in one and then in three
22
Page 25
space dimensions. Finally, in Section 4, we demonstrate the accuracy high-
order local conditions via numerical experiments.
2.2 Absorbing boundary conditions
To illustrate the fundamental ideas underlying the derivation of absorbing
boundary conditions, we begin with a simple one-dimensional problem. In
this special situation many basic notions, in particular the exact boundary
condition, appear in a very simple form. Nonetheless, we hasten to point out
that its appealing simplicity is also misleading: the real challenges in deriving
effective absorbing boundary conditions appear only in higher dimensions.
Indeed a one-dimensional wave can only propagate in two directions, to the
left or to the right. In two or more dimensions, however, waves propagate in
infinitely many directions.
2.2.1 The One-dimensional Wave Equation
Consider the one-dimensional wave equation on the positive real axis,
∂2u
∂t2−∂2u
∂x2= f,x > 0, t > 0.
(2.1)
At the left boundary, x = 0, we require that the solution satisfies
u(0,t) = 0, t > 0.
(2.2)
Thus, u(x,t) describes the position of an infinitely (or just very) long vibrat-
ing string, attached at its left end; hence, u = 0 corresponds to the state
at rest. The one-dimensional wave equation (2.1) describes the propagation
of small perturbations induced by the applied forcing f(x,t). Here we have
normalized the propagation speed to one by rescaling time appropriately.
The initial conditions of the vibrating string are defined by its position and
velocity at t = 0:
u(x,0) = U0(x),
∂
∂tu(x,0) = V0(x), x > 0.
(2.3)
It can be shown that the initial-boundary value problem (2.1)–(2.3) is well-
posed: it has a unique solution, which depends continously on U0, V0, and f.
We now make the following assumption, which defines the local character of
the problem: let the forcing vanish outside a bounded region next to the left
boundary, that is let f(x,t) = 0 for x ≥ L and for all time t > 0. Then
23
Page 26
??????
?
?
?
Figure 2.2: The one-dimensional wave equation: inside the computational
domain, Ω = [0,L] the problem can be arbitrarily complicated, but in the
exterior region, x ≥ L, we assume that f(x,t) = 0 for t > 0 and that u and
∂tu vanish at t = 0.
the positive real line separates into two distinct regions: the bounded interval
Ω = [0,L] and the interval [L,∞), unbounded yet where the forcing vanishes
identically. Both regions meet at the artificial boundary {x = L}, which
consists only of a single point. Furthermore, we assume that the string is
at rest in the exterior at t = 0: U0(x) = 0 and V0(x) = 0 for x ≥ L.
We now wish to simulate numerically the time dependent behavior of the
vibrating string in the computational domain Ω. Unfortunately, we cannot
apply our favorite numerical scheme in Ω and simply ignore the new artificial
boundary point. On the contrary, we must pay close attention to the new
boundary point at x = L: without a boundary condition at x = L, the
initial value problem (2.1)–(2.3) restricted to Ω is not even well-posed. To
derive a boundary condition, we first need to better understand its role at
the artificial boundary. Suppose a wave propagates to the right inside Ω
and reaches the right boundary at x = L. It must not be reflected, for any
spurious reflection will travel back into the computational domain and spoil
the solution everywhere. This spurious reflection, caused by an inaccurate
treatment of the artificial boundary, is not due to finite precision, unlike
discretization errors present in any computation.
condition, which lets the waves hit the boundary without any reflection, the
solution inside Ω, with that boundary condition imposed at x = L, coincides
with the restriction to Ω of the solution in the unbounded region. Hence
such a boundary condition is exact.
Inside the computational domain Ω waves propagate both to the left and to
If we find a boundary
24
Page 27
the right. In the exterior region, however, the absence of any forcing and the
zero initial conditions preclude the appearance of any waves traveling to the
left: there all waves propagate eastward towards infinity – see Figure 2. To
derive the exact boundary condition at x = L we first need to separate the
incoming from the outgoing waves. To do so, we let v and w be defined by
v =∂u
∂t+∂u
∂x,
w =∂u
∂t−∂u
∂x.
(2.4)
Since u satisfies the wave equation (2.1) in x ≥ L, we conclude that
∂v
∂t−∂v
Thus we can rewrite (2.1) as the first-order hyperbolic system:
?
Its general solution is
∂x= 0,
∂w
∂t+∂w
∂x= 0.
∂
∂t
v
w
?
+
?−1 0
01
?
∂
∂x
?
v
w
?
= 0.
(2.5)
v(x,t) = φ(x + t),
and
w(x,t) = ψ(x − t),
where φ and ψ are arbitrary functions, which are determined by initial and
boundary conditions. Therefore, v is constant on the characteristics x + t =
c, whereas w remains constant on the characteristics x − t = c. Thus v
corresponds to incoming waves, whereas w corresponds to outgoing waves.
Since there are no incoming waves in x ≥ L, we have
v(L,t) = 0, t > 0.
(2.6)
By applying the definition (2.4) of v in (2.6) we thus obtain the exact non-
reflecting boundary condition for the displacement u(x,t),
?∂
Note that the problem inside Ω can be arbitrarily complicated, since the
derivation of the (exact) nonreflecting boundary condition (2.7) depends only
on properties in the exterior region.
∂t+
∂
∂x
?
u = 0,x = L, t > 0.
(2.7)
2.2.2Absorbing Boundary Conditions in Higher Dimen-
sions
We consider a highly complex but local scatterer in unbounded two space
dimensions.Although we shall restrict ourselves to the two-dimensional
25
Page 28
case, much of the present discussion carries over immediately to the three-
dimensional case. Thus we consider the wave equation on the two-dimensional
infinite plane,
∂2u
∂t2−∂2u
with the initial conditions
∂x2−∂2u
∂y2= f,t > 0,
(2.8)
u(x,y,0) = U0(x,y),
∂
∂tu(x,y,0) = U1(x,y),t = 0.
By scaling time appropriately we have normalized the speed of propagation
to one. Again the phenomenon of interest is very complicated, possibly non-
linear, but local. Next, we truncate the unbounded exterior by an artificial
boundary and restrict the computation to the square Ω = [−L,L] × [−L,L]
– see Figure 1. Outside Ω we assume that neither source terms nor initial
perturbations are present:
U0(x,y) = U1(x,y) = 0,
and
f(x,y,t) = 0, t > 0,(x,y) ∈ IR2\ Ω.
Again we seek a boundary condition at (x,y) ∈ B, which ensures that all
waves reach the exterior region unharmed and without generating any un-
physical reflection at the fictitious interface. Because of symmetry we only
need to consider a single edge of the rectangle, here the right edge at x = L.
Hence the exterior region lies to the right and the computational domain Ω
to the left of the artificial boundary {(x,y) ∈ IR2|x = L}. Since the initial
conditions and the forcing vanish identically in the exterior, all waves in the
region x ≥ L are purely outgoing and must propagate eastward. To avoid
any spurious reflection at x = L, the exact boundary condition must anni-
hilate all incoming waves. In the previous section we easily derived such an
exact nonreflecting boundary condition for the one-dimensional wave equa-
tion. Unfortunately, the same approach does not apply in two dimensions. In
contrast to the one-dimensional case, any fixed location (L,y) at the artificial
boundary receives incoming waves from not one but infinitely many angles
of incidence, which propagate in infinitely many directions. The distinction
between incoming and outgoing waves is now “infinitely more complicated”.
Let ˆ u(x,ξ,ω) denote the Fourier transform of the solution u(x,y,t) in time
and in the tangential plane, parallel to the artificial boundary,
ˆ u(x,ξ,ω) =
∞
?
−∞
∞
?
−∞
u(x,y,t)ei(ωt+ξy)dy dt.
(2.9)
26
Page 29
Here we have set the solution u(x,y,t) to zero for all previous time t < 0.
Then u is related to ˆ u via the inverse Fourier transform, which resembles
(2.9) after exchanging u and ˆ u. Since u satisfies the wave equation (2.8) with
f = 0 for x ≥ L, its Fourier transform satisfies
∂2
∂x2ˆ u = (ξ2− ω2) ˆ u,x ≥ L.
(2.10)
To derive an exact nonreflecting boundary condition at x = L we need to
relate the normal derivative – here ∂xu – to tangential and time derivatives
– here ∂yu and ∂tu. From (2.10) we conclude that ∂xˆ u is determined by
±?ξ2− ω2ˆ u. The sign in front of the square root discriminates precisely
exact boundary condition:
incoming from outgoing waves; here the correct choice leads to the following
∂
∂xˆ u = −iω
?
1 − (ξ/ω)2ˆ u,x = L.
(2.11)
Although this boundary condition ensures the absoute transparency of the
artificial boundary, this formulation has but little value in practice. Indeed,
we do not seek a boundary condition for ˆ u but instead for u. In theory we
can always compute the inverse transform and thus determine ∂xu. However,
unlike a polynomial expression, whose inverse Fourier transform yields a local
differential operator, the inverse transform of the above expression does not
result in a simple differential operator because of the square root. Instead,
we obtain a so-called pseudo-differential operator, which cannot be evaluated
without forward and inverse Fourier transform. As a consequence, the normal
derivative ∂xu at any given point on the boundary (L,y) depends on past
values of u on the entire line x = L, and cannot be computed locally either
in space or time.
“...unfortunately, these [perfectly absorbing] boundary conditions
have to be nonlocal in both space and time”, Engquist & Majda, 1977
To overcome the difficulties associated with the nonlocal nature of the exact
boundary condition (2.11), we can replace the above pseudo-differential op-
erator by an approximate differential operator. In doing so we give up on the
absolute transparency of the artificial boundary and accept some spurious
reflection. Such absorbing boundary conditions were proposed by Engquist
and Majda [32] in 1977, and we now briefly recall the fundamental ideas
underlying this popular approach.
The Fourier transform of a differential operator always results in a polynomial
expression in the frequency domain. For instance the Fourier transform of the
27
Page 30
differential operator ∂yyyields the polynomial −ξ2. Conversely every polyno-
mial in Fourier space corresponds to a (local) differential operator in physical
space. Thus, the inverse Fourier transform of a polynomial in s = ξ/ω, which
approximates
as an (approximate) absorbing boundary condition in physical space.
For s sufficiently small, we approximate√1 − s2by the first few terms of its
Taylor expansion:
√1 − s2, will yield a differential operator, which can be used
√1 − s2= 1 −s2
2+ O(s4),|s| → 0.
We now replace the square root in (2.11) by the leading term in the Taylor
expansion, that is
to obtain
√1 − s2≃ 1, and perform the inverse Fourier transform
∂ˆ u
∂x
≃ −iω ˆ u
⇒
?∂
∂t+
∂
∂x
?
u = 0,x = L.
This is the first-order Engquist-Majda boundary condition, which contains
only first derivatives of the solution. It coincides with the exact boundary
condition (2.7) for the one-dimensional wave equation. Therefore, it remains
exact for solutions of the two-dimensional wave equation, which depend only
on x and t and thus impinge on the artificial boundary with a normal angle
of incidence. Next, we include one additional term of the Taylor expansion
in the approximation,
Engquist-Majda boundary condition,
√1 − s2≃ 1 − s2/2. This yields the second-order
∂ˆ u
∂x
≃ −iω(1 − (ξ/ω)2/2) ˆ u
?∂2
⇒
∂t2+
∂2
∂x∂t−1
2
∂2
∂y2
?
u = 0,x = L.
(2.12)
Equation (2.12) remains exact at normal incidence, since we can rewrite it
in the equivalent form as
(∂t+ ∂x)(∂t+ ∂x) u = 0,x = L,
(2.13)
by using (2.8). The inclusion of even higher order terms of the Taylor expan-
sion to improve the accuracy of the approximation ceases to yield well-posed
boundary conditions. Although this difficulty can be overcome by the use of
rational (Padé) approximations, the high-order derivatives involved in these
28
Page 31
?
x
y
Figure 2.3: A traveling plane wave with an angle of incidence θ.
boundary conditions greatly complicate their use in any numerical scheme.
As a result, first- and second-order boundary conditions are most commonly
used in practice. Various other (e.g. Chebychev) approximations of√1 − s2
were proposed to design improved local boundary conditions. Eventually,
Higdon [72] showed that all these boundary conditions are particular cases
of the following general class of boundary operators, where α1,...,αp are
arbitrary parameters:
?
For instance, the second-order Engquist-Majda boundary condition (2.13)
results from setting α1= 0◦and α2= 0◦in (2.14). This general formulation,
written as the product of first-order differential operators (cosαi∂t+∂x), pro-
vides a new, more intuitive, interpretation for the effectiveness of absorbing
boundary conditions. Since any such differential operator perfectly annihi-
lates plane waves with angle of incidence ±αi, the artificial boundary will
appear absolutely transparent at the discrete angles of incidence α1,...,αp.
The choice of α1,...,αpis arbitrary and can be adapted to any given situa-
tion.
Nevertheless, all absorbing boundary conditions remain only approximations
to the exact boundary condition (2.11); therefore, they generate some spuri-
ous reflection at x = L. How large is the amount of reflection for a specific
boundary condition ? Recall that any solution of the (homogeneous) wave
equation can be represented by the superposition of plane waves. In Figure 3
cosαp
∂
∂t+
∂
∂x
?
...
?
cosα1
∂
∂t+∂
∂x
?
u = 0,x = L.
(2.14)
29
Page 32
0 10 2030 4050 607080 90
0
10
20
30
40
50
60
70
80
90
100
???
?
?
??
Æ
?
???
?
??
?
??
Æ
??
?
??
Æ
??
?
???
Æ
??????? ??????????
????????????Æ????? ??℄
Figure 2.4: Amount of spurious reflection (in percent) caused by the use of
the boundary conditions (2.14) for a plane wave with angle of incidence θ.
we observe a plane wave, which impinges on the artificial boundary at x = L
with an angle of incidence θ. The linearity of both the wave equation (2.8)
and the boundary condition (2.14) imply that any reflected wave necessarily
propagates with the same frequency as the incident wave. Hence the solution
consists of an outgoing wave, whose amplitude we normalize to one, and an
incoming spurious wave with amplitude |r|:
u(x,y,t) = ei(kx+ℓy−ωt)+ rei(−kx+ℓy−ωt),k,ω ≥ 0,
(2.15)
Here r = r(θ;α1,...,αp) depends on the angle of incidence θ, defined by
tanθ = ℓ/k, and the fixed parameters α1,...,αp. In Figure 4 we compare
the effectiveness of three absorbing boundary conditions by displaying the
amount of reflection |r| versus the angle of incidence θ. The choice α1= 0◦
corresponds to the first, whereas α1= 0◦und α2= 0◦corresponds to the
second Engquist-Majda boundary condition. Alternatively, the popular pa-
rameter choice α1= 0◦and α2= 60◦annihilates incoming waves at normal
incidence and at 60◦angle of incidence. All other angles of incidence will gen-
erate some spurious reflection, which is very small close to normal incidence
but rapidly increases as grazing incidence is approached.
30
Page 33
2.2.3 High-order local nonreflecting boundary conditions
The local absorbing boundary conditions described in the previous section
can be made arbitrarily accurate, but in practice the resulting increasingly
higher order derivatives greatly complicate their use in any numerical scheme.
As a result, first- and second-order boundary conditions are most commonly
used in practice. If even higher accuracy is needed, the artificial boundary
then needs to moved farther away from the scatterer. Hence the absorbing
boundary conditions from Section 2 do not fully satisfy the demand for in-
creasingly accurate and efficient modern numerical methods to solve complex
time-dependent scattering problems in unbounded domains.
Starting from a convergent series representation of the scattered field in in-
verse powers of distance, Hagstrom and Hariharan [55] derived a nonreflecting
boundary condition of arbitrarily high order, in the special case when B is
a sphere. Thus, let B be the sphere of radius R and assume that u satisfies
the homogeneous wave equation,
∂2u
∂t2− c2∆u = 0
(2.16)
with zero initial condition outside B. Starting from the convergent expansion
u(r,θ,φ,t) =
∞
?
j=1
fj(t − r,θ,φ)
rj
, r > R,
(2.17)
where r, θ, φ are spherical coordinates, Hagstrom and Hariharan [55] derived
the following exact local NRBC:
?1
?1
for k = 1,2,..., and w0 = 2u. Here, ∆S denotes the Laplace-Beltrami
operator in spherical coordinates (r,θ,φ),
c
∂
∂t+∂
∂r+1
∂
∂t+k
r
?
?
wk=
u = w1,
(2.18)
cr
1
4R2
?
k(k − 1) + ∆S
?
wk−1+ wk+1
∆S=
1
sinθ
∂
∂θ
?
sinθ∂
∂θ
?
+
1
sin2θ
∂2
∂φ2.
(2.19)
In fact in 1980, Bayliss and Turkel [12] started from that same infinite series
representation and derived a hierarchy of local absorbing boundary conditions
in spherical coordinates. Similar to the boundary conditions of Engquist and
Majda [32, 33], it also requires increasingly higher order derivatives for im-
proved accuracy.
31
Page 34
The boundary condition (4.2) is local in space and time yet does not involve
high-order derivatives, but instead an infinite sequence of auxiliary variables
wkdefined on B. In practice, only a finite number of auxiliary functions wk,
k = 1,...,P is used setting wP+1= 0. Then, in general the boundary condi-
tion is no longer exact, but it remains exact for solutions which consist of a
finite combination of vector spherical harmonics up to order P. Imposition
of the boundary condition at any fixed radius R thus yields at least spectral
accuracy for smooth wave fields with increasing P. Therefore (4.2) is exact
in the same sense as the conditions proposed in [45, 46, 47], namely that P
can always be chosen sufficiently large so that the error introduced at B is
smaller than the discretization error inside Ω, without moving B farther away
from the scatterer. However, this new boundary condition does not require
any spherical harmonics or inner products with them; hence, it is somewhat
easier and cheaper to implement.
By combining ideas from [55] and [47], the above approach was recently ex-
tended to Maxwell’s equations in three space dimensions [53]. Again, outside
B, the medium is assumed to be linear, homogeneous, isotropic, of constant
electric permittivity ε, of constant magnetic permeability µ, and of zero con-
ductivity. In addition, we assume that at t = 0 the scattered field is confined
to the computational domain inside B. Then, the following exact nonreflect-
ing boundary condition holds [53]:
ˆ r × curlE −1
1
c ∂t
1
c∂t
Again, the boundary condition (2.20)–(2.22) is local both in space and time.
It only involves first time derivatives and second tangential derivatives of E
and of the (unknown) auxiliary functions wp, p ≥ 1, which satisfy (2.21)–
(2.22). Since at least two scalar potentials are necessary to represent the
general three-dimensional electro-magnetic field in free space, this bound-
ary condition is optimal in the sense that the number of auxiliary variables
required is minimal.
c
∂Etan
∂t
1
2r2
1
4r2(− →
= w1,
?− − →
∆S+p(p − 1))wp−1+ wp+1,
(2.20)
∂w1
+w1
r
= curlScurlSE +
?µ
εˆ r ×− − →curlScurlSH
?
+ w2,
(2.21)
∂wp
+p
rwp=p ≥ 2.
(2.22)
2.3Multiple scattering problems
When the scatterer consists of several obstacles, which are well separated
from each other, the use of a single artificial boundary to enclose the entire
32
Page 35
scattering region becomes too expensive. Instead it is preferable to enclose
every sub-scatterer by a separate artificial boundary Bi. Then we seek an
exact boundary condition on B =?Bi, where each Bisurrounds a single
outgoing waves leave Ωiwithout spurious reflection from Bi, but also propa-
gate the outgoing wave from Ωito all other sub-domains, which it may reen-
ter subsequently. To derive such an exact boundary condition, an analytic
representation of the solution everywhere in the exterior region is needed.
computational sub-domain Ωi. This boundary condition must not only let
2.3.1 The one-dimensional case
x
t
OB1
B2
Ω1
Ω2
u0
u0
u1
u2
u1
u2
u1= u
u2= u
u1= u − u2
u2= u − u1
L
utt− uxx= 0
Figure 2.5: Multiple scattering in one space dimension.
To illustrate the basic principle underlying the NRBC for multiple scatter-
ing problems, we first consider the following simple one-dimensional Cauchy
33
Page 36
problem:
∂2u
∂t2−∂2u
u(x,0) = u0(x),
ut(x,0) = v0(x).
∂x2= f(x,t), −∞ < x < ∞, t > 0,
(2.23)
We assume that the initial disturbance and the forcing are supported inside
the region Ω = Ω1∩ Ω2, with Ω1= [0, B1] and Ω2= [B2, L],
B2< L, that is supp{u0, v0, f(·,t)} ⊂ Ω – see Figure 5. We now wish to
restrict the computation to the sub-region Ω; therefore we need to impose
appropriate boundary conditions at x = 0, B1, B2, and L to ensure that the
solution in Ω coincides with the solution u of the original Cauchy problem
for all time. Because u is purely outgoing for x < 0 and x > L, the NBC at
x = 0 and x = L correspond to the standard artificial boundary conditions
for single scattering (see Section 2.1), that is
?∂
?∂
which require no further discussion. We now focus on the two remaining
artificial boundary points at x = B1 and x = B2, where u is not purely
outgoing. Because u satisfies the homogeneous wave equation in [B1, B2], it
is the superposition of a left and right moving wave there, that is
0 < B1<
∂x−∂
∂x+∂
∂t
?
?
u = 0,x = 0,
∂t
u = 0,x = L,
(2.24)
u(x,t) = u1(x,t) + u2(x,t),
(2.25)
with
u1(x,t) = f(x − t),u2(x,t) = g(x + t).
Moreover, if we require that supp{u1} ⊂ Ω1and supp{u2} ⊂ Ω2at t = 0,
u1and u2are uniquely defined for all time (see [51]). At x = B1, an exact
NRBC is
?∂
=
∂x+∂
∂t
?
u =
?∂
?∂
∂x+∂
∂x+∂
∂t
?
?
u1+
?∂
∂x+∂
∂t
?
u2
∂t
u2,
(2.26)
since u1is outgoing here.
Thus to impose the exact NRBC at x = B1, we must be able to evaluate u2
there. Here we need to distinguish initial times up to t = B2−B1from later
times t ≥ B2− B1:
34
Page 37
• 0 ≤ t < B2− B1: due to the finite propagation speed (here equal to
one) u2hat not reached Ω1yet; hence, it is still zero at x = B1and
(2.26) reduces to the standard NRBC for purely outgoing solutions.
• B2− B1 ≤ t: u2 no longer vanishes at x = B1, however it is fully
determined by its past values at x = B2through
u2(B1,t) = u2(B2,t − (B2− B1)).
(2.27)
How do we determine u2at x = B2? Recall that we are only computing u
(and not u1or u2) inside Ω. Again, during initial times t < B2−B1we have
u2= u at x = B2. To determine u2at later times we recall that
u(x,t) = u1(x,t) + u2(x,t),∀x ∈ [B1, B2], t > 0.
(2.28)
Therefore we obtain u2 at x = B2 by subtracting from u the value of u1
there, which again is determined by its past values on B1, that is
u2(B2,t) = u(B2,t) − u1(B2,t)
= u(B2,t) − u1(B1,t − (B2− B1)).
Hence in every time step of the numerical scheme, we concurrently update
the new values of u1and u2at x = B1, B2respectively. This requires the
additional storage of past values uiat x = Bi, i = 1,2, for the finite time
window [t − (B2− B1), t].
(2.29)
2.3.2 The three-dimensional case
For simplicity, we consider a scattering problem with two bounded disjoint
scatterers, each surrounded by a sphere Biof radius, Ri i = 1,2. Hence,
the entire artificial boundary B = B1∪ B2and the computational domain
Ω = Ω1∪ Ω2. In contrast to the situation of single scattering in Section 2,
we cannot simply expand u outside B as a superposition of purely outgoing
wave fields. In fact, since part of the scattered field leaving Ω1will reenter
Ω2 at later times, and vice versa, u is not outgoing ouside Ω. Thus, the
boundary condition we seek at B must not only let outgoing waves leave Ω1
without spurious reflection from B1, but also propagate those waves to Ω2,
and so forth, without introducing any spurious reflections.
Following [51], we first decompose the scattered field u in two wave fields,
u = u1+ u2, where uiis purely outgoing as seen from Ωi. The two wave
fields u1and u2both solve the homogeneous wave equation (4.1) outside Ω,
and their sum coincides with u. The outgoing field uout
1, as seen from Ω1, is
35
Page 38
fully determined by its boundary values on B1, while the incoming field uin
is fully determined by its boundary values on B2. Hence,
uout
1
+ uin
uout
2
+ uin
where uout
i
is the outgoing wave field from Ωiand uin
propagating from Ωjto Ωi.
Next, we apply c−1∂t+∂ri+r−1
i
in local spherical coordinates (ri,θi,φi) to u
on each artificial boundary component Bi, i = 1,2. This yields the following
exact local NBC for multiple scattering [54]:
?1
= B1uout
B2u|B2=
c
= B2uout
To evaluate B1uout
values for u2and the corresponding auxiliary functions on B1. The needed
past values of wkare stored on each Biat regular time and angular intervals
and calculated, as needed, via local spline interpolation [54]. Because those
values are time-retarded, they are already known, so that the entire scheme
remains explicit in time. Remarkably, the information transfer (of time re-
tarded values) between sub-domains occurs only across those parts of the
artificial boundary, where outgoing rays intersect neighboring sub-domains,
i.e. typically only across a fraction of the artificial boundary.
12
12= u|B1,
21= u|B2,
(2.30)
ijis the incoming wave
B1u|B1=
c
∂
∂t+
∂
∂r1
+
1
R1
?
?
u|B1
onB1,
1
∂
∂t+
+ B1uin
∂
∂r2
+ B2uin
12
?1
+
1
R2
u|B2
onB2.
2 21
(2.31)
1
we use (4.2) at B1, whereas to evaluate B1uin
12we use past
2.4Numerical experiment
We shall now illustrate the accuracy of local nonreflecting boundary con-
ditions via the following numerical experiment. Consider a spherical inclu-
sion of radius r0> 0 located inside an unbounded inhomogeneous acoustic
medium. At the sound-soft interface of the inclusion we impose a time-
dependent pressure field which corresponds to an outgoing spherical wave
field, initiated by an off-centered Gaussian point source. Located on the
z-axis at distance d < r0from the origin, its time dependence is shown in
Figure 6. Hence at the surface of the cavity, r = r0, the imposed time-
dependent acoustic field is determined by
u(r0,θ,t) =1
rde−(rd−cmint+0.2)2/σ2,
(2.32)
36
Page 39
where rd=
cmin= 0.5; here, rdcorresponds to the distance of any point (r0,θ) from the
point source at distance d fom the origin.
The sound speed in the surrounding medium varies from cminto cmaxas a
function of distance only; for r ≥ 1, the velocity profile shown in Figure 6
is constant and equal to cmaxnormalized to one. Hence the inhomogeneous
surrounding medium, initially at rest, is expected to act as a spherical wave
guide around the cavity. The unbounded exterior is now truncated at R =
1, where we apply the high-order local conditions (4.2) with varying P –
note that with P = 0 the boundary condition (4.2) corresponds to the first-
order Engquist-Majda condition in spherical coordinates. Although this test
?r2
0+ d2− 2r0d cos(θ),σ = 0.25/√−logα,α = 10−7, and
0 0.1 0.20.3 0.40.5
t
0.60.7 0.80.91
0
2
4
6
8
10
12
14
16
18
20
g(0.5, 0, t)
0.50.6 0.70.80.91
r
1.1 1.21.3 1.4 1.5
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
C
Figure 2.6: Left: the time dependence of the Gaussian point source. Right:
the velocity profile c(r).
problem is three-dimensional, it is axisymmetric about the z-axis, that is
the solution is independent of φ, so that we can restrict the computations
to the two-dimensional region Ω, determined by r0 ≤ r ≤ R, 0 ≤ θ ≤
π. Inside Ω we use standard second-order centered finite differences on a
80 × 480 polar equidistant mesh, combined with the explicit second-order
leap-frog scheme in time. At the artificial boundary B, located at r = R, the
boundary condition (4.2) is discretized in space using centered second-order
finite differences and in time as described in [55].
Since no simple analytical expression for the exact solution is available here,
we shall compute a reference solution in a much larger domain. Due to the
finite speed of propagation, any spurious reflection will then be postponed to
later times and thus not affect the reference solution inside Ω until T = 7.5.
In Figure 7 we observe how the spherical wave front penetrates the acoustic
medium to the right of the cavity and then progresses around it – note the
distorted wave front due to the varying velocity profile around t = 1.5. By
37
Page 40
t = 3 the main wave front has left the computational domain, yet part of the
wave energy remains trapped inside the wave guide which and continues to
travel around the cavity. We now compare the exact (numerical reference)
solution with that obtained by imposing the boundary condition (4.2) for
varying P at the fixed location r = 0.75, θ = 3π/4, located well inside Ω. In
Figure 8 the numerical solutions obtained with P = 0, P = 1, and P = 5
are shown. The numerical solution obtained with P = 0 strongly deviates
from the exact solution past t = 2. We recall that the error observed here
is solely due to the approximate nature of the boundary condition and thus
cannot be improved upon by refining the mesh. The solution obtained with
P = 1 clearly displays a significant improvement in accuracy. Nonetheless,
we find again deviations of 5-10% around t = 3.5, for instance. As we further
increase P, those spurious reflections essentially disappear and cannot be
observed anymore at this scale. Hence their amplitude now lies below the
discretization error. Further mesh refinement in the interior, however, would
generally require further increase in P, as both error components need to be
reduced systematically to achieve convergence.
2.5Conclusion
The constant demand for increasingly accurate, efficient, and robust numer-
ical methods, which can handle a wide variety of physical phenomena, spurs
the search for improvements in absorbing boundary conditions. The frustra-
tion is all too obvious, when the gains made in the interior by using sophis-
ticated numerical methods, such as high order and adaptive methods, are
annihilitated at the artificial boundary by the use of an inaccurate boundary
condition.
Among the many different approaches nowadays available to truncate the
unbounded exterior and achieve convergence at a reasonable computational
cost, local absorbing boundary conditions remain probably the simplest and
most flexible approach. Because they are completely local, they apply to all
(convex) artificial boundaries and require no special functions or damping
parameters in the exterior. Moreover, they are easily coupled with stan-
dard finite difference or finite element methods in the interior and have been
found very accurate in practical computations. In contrast to the popular
perfectly matched layer approach, high-order local nonreflecting boundary
conditions can also be extended to multiple scattering problems, as they
yield an efficient analytical representation of the solution everywhere outside
the computational domain.
38
Page 41
−1.5−1−0.50
x
0.511.5
−1.5
−1
−0.5
0
0.5
1
1.5
t= 0.6449
y
t= 1.5048
t= 2.5796 t= 3.117
t= 4.7293 t= 6.1266
Figure 2.7: Scattering from a spherical wave guide: snapshots of the ref-
erence solution at different times. The three circles drawn are located at
r = 0.5,1,1.5. The Gaussian point source is located outside the computa-
tional domain at r = 0.45, θ = 0.
39
Page 42
01234567
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
t
u
: EXACT
: P = 0
: P = 1
: P = 5
Figure 2.8: The numerical solutions computed using the boundary conditions
(4.2) with P = 0, P = 1, and P = 5, are compared with the exact solution
at r = 0.75, θ = 3π/4.
40
Page 43
Chapter 3
Perfectly Matched Layers for
Time-Dependent Wave Equations
in Second-Order Form
Abstract
has proved a flexible and accurate method for the simulation of waves in
unbounded media. Most PML formulations, however, usually require wave
equations stated in their standard second-order form to be reformulated as
first-order systems, thereby introducing many additional unknowns. To cir-
cumvent this cumbersome and somewhat expensive step, we instead propose
a simple PML formulation directly for the wave equation in its second-order
form. Inside the absorbing layer, our formulation requires only two auxiliary
variables in two space dimensions and four auxiliary variables in three space
dimensions; hence it is cheap to implement. Since our formulation requires
no higher derivatives, it is also easily coupled with standard finite difference
or finite element methods. Strong stability is proved while numerical exam-
ples in two and three space dimensions illustrate the accuracy and long time
stability of our PML formulation.
In the last decade, the perfectly matched layer (PML) approach
3.1Introduction
The accurate and reliable simulation of wave propagations in unbounded
media is of fundamental importance in a wide range of applications. The
perfectly matched layer (PML) approach [16] has proved a flexible and ac-
curate method for the simulation of waves. It consists in surrounding the
computational domain by an absorbing layer, which generates no reflections
at its interface with the computational domain; hence, it is perfectly matched.
41
Page 44
Inside the absorbing layer a damping term is added to the wave equation,
which acts only in the direction perpendicular to the layer. This approach is
analogous to the physical treatment of the walls of an anechoic chamber and
provides an alternative to absorbing or nonrelfecting boundary conditions
[45, 46, 56, 55, 60].
The initial PML formulation of Bérenger [16] was based on splitting the elec-
tromagnetic fields into two parts, the first containing the tangential deriva-
tives and the second containing the normal derivatives; damping was then
enforced only upon the normal component. Later Abarbanel and Gottlieb
[2] showed that Bérenger’s approach was only weakly well-posed due to the
unphysical splitting of the field variables. Several strongly well-posed ap-
proaches have been suggested since, some of which were shown to be linearly
equivalent [6, 109].
The PML approach has proved very successful in practice, because of its
simplicity, versatility, and robust treatment of corners. Once discretized and
truncated at a finite thickness, the layer is no longer perfectly absorbing and
the optimal damping parameters need to be determined via numerical ex-
periments. Stability properties of the PML approach has been analyzed in
several works, such as in [29, 2, 6, 15] among others.
The best implementation in the time domain is still under debate. Most
PML formulations require wave equations stated in their standard second-
order form to be reformulated as first-order hyperbolic systems, thereby in-
troducing many additional unknowns. Here we propose instead a simple
PML formulation directly for the second-order wave equation both in two
and in three space dimensions. Our formulation also requires fewer auxiliary
variables than previous formulations for the second-order wave equation –
see [8, 23, 94], for instance.
Our paper is organized as follows. In Section 2 we derive a PML formula-
tion for the wave equation in its standard second-order form. By judiciously
choosing the auxiliary variables in the Laplace transformed domain, the re-
sulting PML modified equations require only two auxiliary variables in two
dimensions and four auxiliary variables in three dimensions inside the ab-
sorbing layer. Next, in Section 3 we prove stability of our PML formulation
by using standard theory from [77]. In Section 4 we extend our method
to complex frequency shifted PML. The finite difference and discontinuous
Galerkin discretization of the PML modified wave equation is shown in Sec-
tion 5. In Section 6, our numerical results both in two and three space
dimensions demonstrate the accuracy and long time stability of the PML
formulation. The further applications of our method to elastodynamic and
poroelastic problems are shown in Section 7 and 8 with various numerical
experiments.
42
Page 45
3.2PML formulation
We consider a time dependent wave field u propagating through unbounded
three dimensional space and assume that all sources and initial disturbances
are confined to the rectangular domain Ω = [−a1,a1] × [−a2,a2] × [−a3,a3],
a1, a2, a3> 0. Outside Ω, we further assume the speed of propagation c > 0
to be constant; hence, all waves are purely outgoing in the unbounded exterior
R3\Ω. Inside Ω, the wave field u(x1,x2,x3,t) satisfies
utt− ∇ ·?c2∇u?= f
u = u0
ut= v0
t > 0,
(3.1)
t = 0,
(3.2)
t = 0.
(3.3)
We wish to truncate the unbounded exterior and thereby restrict the com-
putation to the finite computational domain Ω. In doing so, we need to
ensure that all waves propagating outward leave Ω without spurious reflec-
tion. Thus we shall surround Ω by a perfectly matched layer (PML) of
thickness Li, i = 1,2,3, in each coordinate which is designed to absorb the
waves exiting Ω. Inside the absorbing layer, u then satisfies a modified wave
equation whose solutions decay exponentially fast with distance from the
computational domain.
Following [2, 6], we let ˆ u denote the Laplace transform of u, defined as
?∞
Outside Ω, ˆ u then satisfies the Helmholtz equation,
?
Next, we introduce the coordinate transformation
?xi
where the damping profile ζiis positive inside the absorbing layer, |xi| > ai,
i = 1,2,3, but vanishes inside Ω. If we now require ˆ u to satisfy the modified
Helmholtz equation in those stretched coordinates,
?
ˆ u = ˆ u(x,s) =
0
estu(x,t)dt,s ∈ C.
(3.4)
s2ˆ u =
∂
∂x1
c2∂ˆ u
∂x1
?
+
∂
∂x2
?
c2∂ˆ u
∂x2
?
+
∂
∂x3
?
c2∂ˆ u
∂x3
?
.
(3.5)
xi?→ ˜ xi:= xi+1
s
0
ζi(x)dx,i = 1,2,3,
(3.6)
s2ˆ u =
∂
∂˜ x1
c2∂ˆ u
∂˜ x1
?
+
∂
∂˜ x2
?
c2∂ˆ u
∂˜ x2
?
+
∂
∂˜ x3
?
c2∂ˆ u
∂˜ x3
?
,
(3.7)
43
Page 46
it is well-known that u will remain unaltered inside Ω, but decay expo-
nentially fast inside the layer; hence the absorbing layer will be perfectly
matched. In fact, the (unsplit) PML modified Helmholtz equation (3.7) in
the Laplace transformed domain is standard [2, 6]. The difficulty lies in
transforming (3.7) back to the time domain, without introducing high order
derivatives or too many auxiliary variables.
From (3.6), we observe that partial differentiation with respect to ˜ xiis re-
lated to partial differentiation with respect to the physical coordinate, xi,
through
∂
∂˜ xi
s + ζi
We now let γi= γi(ζi; s), i = 1,2,3 denote
=
s
∂
∂xi.
(3.8)
γi:= 1 +ζi
s.
(3.9)
Then, by replacing partial derivatives according to (3.8) and multiplying the
resulting expression by γ1γ2γ3, we rewrite (3.7) in physical coordinates as
?
s2γ1γ2γ3ˆ u =
∂
∂x1
c2γ2γ3
γ1
∂ˆ u
∂x1
?
+∂
∂x2
?
c2γ3γ1
γ2
∂ˆ u
∂x2
?
+∂
∂x3
?
c2γ1γ2
γ3
∂ˆ u
∂x3
(3.10)
?
.
From (3.9) we derive after some algebra the following identities:
γ2γ3
γ1
γ3γ1
γ2
γ1γ2
γ3
= 1 +(ζ2+ ζ3− ζ1)s + ζ2ζ3
(s + ζ1)s
= 1 +(ζ3+ ζ1− ζ2)s + ζ3ζ1
(s + ζ2)s
= 1 +(ζ1+ ζ2− ζ3)s + ζ1ζ2
(s + ζ3)s
,
,
.
(3.11)
By using (3.11) in (3.10) we find
?
=
∂x1
∂x1
∂
∂x1
∂
∂x3
s2+ s(ζ1+ ζ2+ ζ3) + (ζ1ζ2+ ζ2ζ3+ ζ3ζ1) +ζ1ζ2ζ3
?
?
?
s
?
ˆ u
∂
c2∂ˆ u
?
+
∂
∂x2
?
c2∂ˆ u
∂x2
?
+
∂
∂x3
?
c2∂ˆ u
∂x3
?
?
+
c2?(ζ2+ ζ3− ζ1)s + ζ2ζ3
c2?(ζ1+ ζ2− ζ3)s + ζ1ζ2
(s + ζ1)s
?∂ˆ u
?∂ˆ u
∂x1
?
?
+
∂
∂x2
c2?(ζ3+ ζ1− ζ2)s + ζ3ζ1
(s + ζ2)s
?∂ˆ u
∂x2
?
+
(s + ζ3)s
∂x3
.
(3.12)
44
Page 47
Next, we introduce the auxiliary functions ψ and φ = (φ1,φ2,φ3)⊤,
?ψ =1
?φ1= c2
?φ2= c2
?φ3= c2
sˆ u,
?ζ2+ ζ3− ζ1
?ζ3+ ζ1− ζ2
?ζ1+ ζ2− ζ3
s + ζ1
+
ζ2ζ3
(s + ζ1)s
ζ3ζ1
(s + ζ2)s
ζ1ζ2
(s + ζ3)s
?∂ˆ u
?∂ˆ u
?∂ˆ u
∂x1,
s + ζ2
+
∂x2,
s + ζ3
+
∂x3,
or equivalently
s?ψ = ˆ u,
(s + ζ1)?φ1= c2
(s + ζ2)?φ2= c2
(s + ζ3)?φ3= c2
?
?
?
(ζ2+ ζ3− ζ1) +ζ2ζ3
(ζ3+ ζ1− ζ2) +ζ3ζ1
(ζ1+ ζ2− ζ3) +ζ1ζ2
s
?∂ˆ u
?∂ˆ u
?∂ˆ u
∂x1,
s
∂x2,
and
s
∂x3.
Finally, we use the above relations in (3.12) and transform the resulting
equations back to the time domain, which yields the PML modified wave
equation
utt+ (ζ1+ ζ2+ ζ3)ut+ (ζ1ζ2+ ζ2ζ3+ ζ3ζ1)u = ∇ ·?c2∇u?+ ∇ · φ − ζ1ζ2ζ3ψ,
φt= Γ1φ + c2Γ2∇u + c2Γ3∇ψ,
ψt= u,
(3.13)
where
Γ1=
−ζ1
0
0
00
0−ζ2
0−ζ3
,Γ2=
ζ2+ ζ3− ζ1
0
0
00
0ζ3+ ζ1− ζ2
0ζ1+ ζ2− ζ3
and
Γ3=
ζ2ζ3
0
0
00
0ζ3ζ1
0ζ1ζ2
.
In the interior of Ω, the damping profiles ζi, i = 1,2,3 and the auxiliary
variables φ, ψ vanish; hence, (3.13) reduces to (3.1) in Ω. Because our PML
45
Page 48
formulation (3.13) requires only four auxiliary scalar variables φ1, φ2, φ3, ψ
inside the layer and no high order derivatives, its implementation is not only
straightforward but also cheap to implement.
In two space dimensions, ζ3and φ3and ψ vanish and our PML formulation
reduces to
utt+ (ζ1+ ζ2)ut+ ζ1ζ2u = ∇ ·?c2∇u?+ ∇ · φ,
where
?−ζ1
Remarkably only two auxiliary functions are needed here.
The choice of the damping profiles ζi(x) ≥ 0,
can be constant, linear, or quadratic among others. In our computations, we
always use
Because ζi(x) is twice continuously differentiable throughout the interface at
|xi| = ai, no special transmission conditions are needed there. The constant
¯ζi depends on the discretization and the thickness of the layer, which in
practice is truncated by a homogeneous Dirichlet (or Neumann) boundary
condition. Then the relative reflection, R, is given by
?1
In Figure 1 we show damping profiles for different values of¯ζi.
φt= Γ1φ + c2Γ2∇u,
?ζ2− ζ1
(3.14)
Γ1=
0
0−ζ2
?
,Γ2=
0
0ζ1− ζ2
?
.
i = 1,2,3 is arbitrary; it
ζi(xi) =
0
for |xi| < ai,
for ai≤ |xi| ≤ ai+ Li,
i = 1,2,3
¯ζi
?
|xi−ai|
Li
−
sin
“2π |xi−ai|
Li
2π
”
?
i = 1,2,3.
(3.15)
¯ζi=
c
Li
log
R
?
,i = 1,2,3.
(3.16)
3.3Stability
We now establish the stability and well-posedness of our PML formulation,
first in two and then in three space dimensions, where we assume that the
absorbing layer extends to infinity. Here we follow standard stability theory
for hyperbolic systems [77], which we briefly recall below.
Consider a general Cauchy problem,
?∂
Ut= P
∂x
?
U,0 ≤ t ≤ T,U ∈ Rp,
(3.17)
46
Page 49
00.020.040.060.08 0.1
0
10
20
30
40
x
ζ(x)
ζ = 10
ζ = 25
ζ = 40
Figure 3.1: The damping profile ζi(xi) given by (3.15) is shown for different
values of¯ζi, with c = 1 and Li= 0.1 .
where P(∂x) denotes a linear differential operator, with initial conditions
U(x,0) = U0(x),x ∈ R3.
(3.18)
Following [77], the Cauchy Problem is weakly (resp. strongly) well-posed, if
the solution U(·, t) satisfies
?U(·, t)?L2≤ Keαt?U(·, 0)?Hs
with s > 0 (resp. s = 0). The Cauchy Problem is weakly (resp. strongly)
stable, if the solution U(·, t) satisfies
?U(·, t)?L2≤ K (1 + t)s?U(·, 0)?Hs
with s > 0 (resp. s = 0). A necessary and sufficient condition for weak well-
posedness (resp. stability) is that all eigenvalues λ of the operator P (ik))
satisfy
ℜ{λ(P(ik))} ≤ C,
with C > 0 (resp. C = 0) independent of k. For strong well-posedness (resp.
stability), the corresponding eigenvectors must also be complete.
By rewriting the PML-modified wave equations (3.13), (3.14) as a first-order
hyperbolic system and applying the stability theory from [77] delineated
above, we can prove the following two stability results.
(3.19)
(3.20)
k ∈ R,
(3.21)
Theorem 3.3.1 The Cauchy problem for the PML formulation (3.14) in
two space dimensions is strongly stable for ζ1, ζ2≥ 0.
proof)
For simplicity, we assume that ζ1, ζ2are constant; note, however, that the
47
Page 50
stability theory from [77] extends to smoothly varying coefficients. We intro-
duce the new variable v to rewrite the first equation in (3.14) equivalently
as
ut= −ζ2u + divv,
By using (3.22), we now rewrite (3.14) as a first order hyperbolic system:
vt= −ζ1v + c2∇u + φ.
(3.22)
Ut= AUx+ B Uy+ C,
(3.23)
where
Ut= (u, φ1, φ2, v1, v2)⊤,
000
000
ζ2
00
0ζ1
0
00ζ1
(3.24)
A =
00 0 1 0
c2(ζ2− ζ1) 0 0 0 0
0
c2
0
0 0 0 0
0 0 0 0
0 0 0 0
,B =
00 0 0 1
c2(ζ1− ζ2) 0 0 0 0
0
0
c2
0 0 0 0
0 0 0 0
0 0 0 0
,
and
C = −
ζ2
0
0
0
0
0
ζ1
0
0
0
.
(3.25)
By using a symbolic algebra program we find that the eigenvalues of the
principal part of P(ik) for (3.23) are
λ(P (ik)) = ±ic(k2
1+ k2
2)1/2.
(3.26)
Thus,
ℜ{λ(P (ik))} = 0,
(3.27)
while the corresponding eigenvectors are also complete for all ζ1, ζ2 ≥ 0.
Therefore, since C is a diagonal matrix with negative entries for ζ1, ζ2≥ 0,
we conclude that (3.14) is strongly stable.
Theorem 3.3.2 The Cauchy problem for the PML formulation (3.13) in
three space dimensions is strongly stable, if at least two ζj= 0, j = 1,2,3,
and weakly stable, otherwise.
proof)
We introduce the new variable v to rewrite the first equation in (3.13) as
ut= −ζ2u + divv − ζ3ψ,
vt= −ζ1v + c2∇u + φ.
(3.28)
48
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