Perfectly Matched Layers and High Order Difference Methods for Wave Equations

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2. Non-reflecting boundary conditions (NRBC)

1

2

tt

xx

yy

2

0

t

0

0

0

Exact boundary conditions for (2.1) were first derived in the pioneering work by Engquist and Majda  and have recently been reviewed [41, 46] from a modern perspective. The construction of the absorbing boundary conditions

1We note that all exact NRBCs are global, but all global conditions are not exact. Similarly all local NRBCs are approximate but all approximate NRBCs are not local.

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y

0

−i

q

ωc

2−k2yx

0

y

2

2

2y

y

y

y

2

y

y

y

−1

y

2

y

−1

y

−∞

−∞

i

ωt+kyy

y

y

2

y

2

2

2

xx

yy

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−1

y

2

y

1

1

−1

2

2

2

t

x

2

tt

tx

yy

m

1

1

m

1

2

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2

j

j

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3. Absorbing layers

1

The PML was first introduced for the Maxwell’s equations in the seminal paper  by J-P. Bérenger . In its original form , the PML is derived by

1A more general interpretation of perfect matching is that the restriction of the solution to the PML problem in the interior coincides with the solution to the original problem, see .

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tt

xx

yy

y

x

x

y

−∞

−iωt

2

xx

yy

y

x

x

y

0

−iω(k1x+k2y)

1

2

x

y

0

x

y

0

−k1Γ(x)

−iω(k1x+k2y)

0

−iω

 k1

x+Γ(x)  +k2y

1

1

1

1

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1

2

1

1

x

x

yy

1

y

x

1

x

y

1

−iω(k1(s1x)+k2y)

0

x

x

1

1

x

y

tt

1

t

xx

yy

y

x

x

y

1

x

1

y

t

x

1

t

y

1

t

x

y

1

2

3

T

1

t

2

x

y

3

y

x

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t

x

y

1

t

x

1

1

y

y

y

x

y

x

y

y

y

λ x

0

y

λ x

0

y

λ

x+1s Rx

0σ1(z)dz

0

y

λ

x+1s Rx

0σ1(z)dz

0

y

1

1

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t

0

2

0

2

κ (t−t0)

0

2Hs

0

s

0

0

t

1

x

2

y

1

3

1

2

3

x

y

2

2x

2y

x

y

x

1

y

2

1

3

t

1

x

2

y

1

3

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1

2

3

x

1

y

2

0

2

s

0

2Hs

0

s

0

0

j

x

y

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x

x

x

x

x

x

x

x

x

x

x

x

x

x

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x

x

y

x

2

1

x

x

. Firstly, we take Fourier transform in time and apply the complex change of variables (3.3) to the physical boundary con- dition (3.24). Secondly, we chose auxiliary variables and invert the Fourier transforms. Depending on the choice of our auxiliary variables we obtain two

2Note that when α 6= 0 the Cauchy PML can support growth independent of the boundary conditions. However, we have used |α| = 0.5 and 10 grid points in the PML such that the discrete Cauchy PML is stable, see Paper II for more details.

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y

x

1

y

x

1

1

−4

−2

0

2

2

ble discrete approximations. However, in the discrete setting, derivatives are

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time = 1.5

0.05 0.1 0.15 0.2

time = 3

0.05 0.1 0.15 0.2 0.25

time = 1.5

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

time = 3

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

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tt

xx

0

t

0

0

0

1

t

x

1

2

t

x

2

i

i

2

b

a

2

u

t

2

x

2

1

2

2

2

t

u

t

t

t

u

u

u

u

t

u

u

t

0

2

2

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2

2

1

0

2

2

2

2

1

0

2

2

2

2

2

j

j

j

1

2

3

N

T

H

T

1

2

1

2

2

2

1

2

1

−1

T

N

N

T

T

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2

−1

N

T

T

T

T

2

1

2

1

H

T

1

T

T

N

1

H

21

20

2

H

T

2

T

N

T

N

N

0

0

1

2

α

t

N

γ

α

1

2

N

γ

1

2

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2

tt

2

0

t

0

n

−1

α

t

N

γ

n

tt

t

0

t

0

−1

γ

−1

α

v

t

2H

T

1

21

2

2N

tT

v

t

H

t

H

t

H

H

v

v

v

v

t

H

v

v

t

0

H

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