Are Absorbing Boundary Conditions History Example and Perfectly - - PowerPoint PPT Presentation

SLIDE 1

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Are Absorbing Boundary Conditions and Perfectly Matched Layers Really so Different ?

Martin J. Gander martin.gander@unige.ch

University of Geneva

RICAM, November 2011 Joint work with Achim Sch¨ adle

SLIDE 2

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

History of Absorbing Boundary Conditions

Engquist and Majda (1977): Wave Propagation: “In

practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation. Unfortunately [transparent boundary conditions] necessarily have to be non-local in both space and time and thus are not useful for practical calculations”

Bayliss Turkel (1980): Wave-Like Equations: “In the

numerical computation of hyperbolic equations it is not practical to use infinite domains. Instead, one truncates the domain with an artificial boundary“

Halpern (Wave Propagation 1982, Diffusion 1987):

“. . . one often introduces artificial boundaries with boundary conditions chosen so that the problem one gets is well-posed and the solution is ’as close as possible’ to that of the original problem”

SLIDE 3

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Truncation of an Infinite Computational Domain

Airbus A340 in approach over a city Truncation of the unbounded domain using

◮ A transparent (or exact) boundary condition (TBC) or

an absorbing (or inexact) boundary condition (ABC)

◮ A perfectly matched layer (PML)

SLIDE 4

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Interest for Domain Decomposition Methods

Computation performed with an optimized Schwarz method

• n 16 subdomains

An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation (G, Halpern and Magoules, 2006)

SLIDE 5

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

An Advice from the Past

´ Emile Picard (1893): Sur l’application des m´ ethodes d’approximations successives ` a l’´ etude de certaines ´ equations diff´ erentielles ordinaires Les m´ ethodes d’approximation dont nous faisons usage sont th´ eoriquement susceptibles de s’appliquer ` a toute ´ equation, mais elles ne deviennent vrai- ment int´ eressantes pour l’´ etude des propri´ et´ es des fonctions d´ efinies par les ´ equations diff´ erentielles que si l’on ne reste pas dans les g´ en´ eralit´ es et si l’on envisage certaines classes d’´ equations.

SLIDE 6

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Construction of Transparent Boundary Conditions

We consider the model problem (η − ∆)u = f in Ω = R × (0, π) u(x, 0) = u(x, π) = with f compactly supported in Ωint = (0, 1) × (0, π), and u bounded at infinity. x y 1 π Γ0 Γ1 support

• f f

In order to solve this problem on a computer, the computational domain needs to be truncated in x, and an artificial boundary condition needs to be imposed at x = 0 and x = 1.

SLIDE 7

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Construction of Transparent Boundary Conditions

Based on the decomposition of Ω = Ω

− ∪ Ωint ∪ Ω +

x y 1 π Γ0 Γ1 Ωint Ω− Ω+ and the equivalent coupled problems (η − ∆)v − = in Ω− v − = v

• n Γ0

∂nv = ∂nv −

• n Γ0

(η − ∆)v = f in Ωint ∂nv = ∂nv +

• n Γ1

v + = v

• n Γ1

(η − ∆)v + = in Ω+ with homogeneous conditions at y = 0 and y = π.

SLIDE 8

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Solution of Exterior Problems

The exterior problems are independent of the data f and can be readily solved using Fourier series in y: for example on Ω+ we have (η + k2 − ∂xx)ˆ v + = 0 an ordinary differential equation, whose solution is ˆ v +(x, k) = A(k)e √

η+k2x + B(k)e−√ η+k2x

Since v + needs to stay bounded for x → +∞, we have A(k) = 0 and using the Dirichlet data v + = v at x = 1 we

• btain

ˆ v +(x, k) = ˆ v(1, k)e−√

η+k2(x−1)

This implies ∂x ˆ v +(x, k)|x=1 = −

• η + k2ˆ

v(x, k)|x=1 and similarly on the left boundary Γ0, we obtain −∂xˆ v −(x, k)|x=0 = −

• η + k2ˆ

v(x, k)|x=0

SLIDE 9

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Closing the Interior Problem

Using these solutions for the interior problem, we get (η − ∆)v = f in Ωint ∂nv = F−1(−

• η + k2ˆ

v)

• n Γ0

∂nv = F−1(−

• η + k2ˆ

v)

• n Γ1

where the inverse tranform in general is a convolution: F−1(−

• η + k2ˆ

v) = ∞

−∞

f (y − ξ)v(·, ξ)dξ with F(f (y)) = −

• η + k2.

By construction, v coincides with u in Ωint, and the conditions obtained are called transparent boundary conditions (TBCs). They require a non-local convolution boundary condition.

SLIDE 10

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Using Transparent Boundary Conditions

Because of their non-local nature, one often approximates TBCs and obtains Absorbing Boundary Conditions (ABCs):

◮ by polynomial or rational approximations of the symbol

(Enquist, Majda, Halpern, Bruneau-Di Menza, Nataf, Japhet, Szeftel, Shibata . . . )

◮ by approximation of the convolution kernel (Hairer,

Lubich, Schlichte, Greengard, Strain, Sch¨ adle . . . )

◮ using quadrature rules (Mayfield, Baskakov and

Popov,. . . ) Remark: To couple an incoming field uin from Ω+: ∂x(v − uin) = F−1(−

• η + k2(ˆ

v − ˆ uin))

• n Γ1

which by linearity is equivalent to ∂xv −F−1(−

• η + k2ˆ

v) = ∂xuin−F−1(−

• η + k2ˆ

uin) on Γ1

SLIDE 11

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Perfectly Matched Layers

• J. Berenger (1994): A perfectly matched layer for the

absorption of electromagnetic waves Split-Field PML: Original idea, split the electromagnetic fields into two unphysical fields in the PML region x y 1 π Γ0 Γ1 support

• f f

S.D. Gedney (1996): An anisotropic perfectly matched layer absorbing media for the truncation of FDTD latices Uniaxial PML (UPML): The PML is described as an artificial anisotropic absorbing material

SLIDE 12

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Stretched-Coordinate PML

W.C. Chew and W. H. Weedon (1994): A 3d perfectly matched medium from modified Maxwell’s equations with stretched coordinates F.L. Teixeira and W.C. Chew (1998): General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media Idea: Analytic continuation of the solution and equation to a complex x contour which changes oscillating waves into exponentially decaying waves outside the region of interest: eikx − → eik(ℜ(x)+iℑ(x)) = eikℜ(x)e−kℑ(x) Then perform a coordinate transform to express the complex x as a function of a real coordinate, which leads to complex materials: ∂ ∂x − → 1 1 + iσ(x)

k

∂ ∂x

SLIDE 13

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Example of a Coordinate Stretching

1 2 3 4 5 6 7 8 9 10 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

x

1 2 3 4 5 6 7 8 9 10 −1 −0.5 0.5 1

x

1 2 3 4 5 6 7 8 9 10 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

x

1 2 3 4 5 6 7 8 9 10 −1 −0.5 0.5 1

x

SLIDE 14

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Alternative Formulation: the Pole Condition

Hohage, Schmidt and Zschiedrich (SIAM Math. Anal. 2003) “The pole condition is a general concept for the theoretical analysis and the numerical solution of a variety of wave propagation problems. It says that the Laplace transform of the physical solution in the radial direction has no poles in the lower complex halfplane.” Ωint Γ

SLIDE 15

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

The Pole Condition for our Model Problem

On Ω+ we had after a Fourier transform (η + k2 − ∂xx)ˆ v + = 0 Applying a Laplace transform in the radial, i.e. x direction, we obtain (η + k2 − s2)˜ v +(s, k) − ∂nˆ v +(1, k) − sˆ v +(1, k) = 0 Hence ˜ v +(s, k) = ∂nˆ v +(1, k) + sˆ v +(1, k) η + k2 − s2 This solution has poles at s = ±

• η + k2, and a partial

fraction expansion gives ˜ v +(s, k) = ˆ v +(1, k) − ∂nˆ

v +(1,k)

√

η+k2

2(s +

• η + k2)

+ ˆ v +(1, k) + ∂nˆ

v +(1,k)

√

η+k2

2(s −

• η + k2)

SLIDE 16

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

The Meaning of the Pole Condition

Now the pole condition states that this solution ˜ v +(s, k) = ˆ v +(1, k) − ∂nˆ

v +(1,k)

√

η+k2

2(s +

• η + k2)

+ ˆ v +(1, k) + ∂nˆ

v +(1,k)

√

η+k2

2(s −

• η + k2)

can not have any poles in the right half of the complex plane, ℜ(s) ≥ 0, and since

• η + k2 > 0, the second term

can not be present, which means that ˆ v +(1, k) + ∂nˆ v +(1, k)

• η + k2

= 0 which is equivalent to ∂nˆ v +(1, k) = −

• η + k2ˆ

v +(1, k) as we found before chosing the bounded solution in the right

• uter domain.

SLIDE 17

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Approximations from the Pole Condition

In order to satisfy the pole condition approximately, it is natural to expand the desired analytic function in the right half plane into a power series. To do so, we first map the right half plane into the unit disk using the M¨

• bius transform

˜ s = s − s0 s + s0 ⇐ ⇒ s = −s0 ˜ s + 1 ˜ s − 1 ℜs ℑs ℜ˜ s ℑ˜ s s0 and expand ˜ v +(˜ s, k) into a power series about ˜ s = 0.

SLIDE 18

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

An Educated Guess

We use a particular Ansatz for the power series (Sch¨ adle 2008): ˜ v +(˜ s, k) = (˜ s − 1)

• (˜

s − 1)

∞

• n=0

an˜ sn − ˆ v +(1, k) 2s0

• because then ˜

v +(˜ s, k) satisfies already the limit property of Laplace transforms lim

s→∞ s˜

v +(s, k) = ˆ v +(1, k) as one can se from (recall that s = −s0 ˜

s+1 ˜ s−1)

lim

˜ s→1 −s0

˜ s + 1 ˜ s − 1˜ v +(˜ s, k) = ˆ v +(1, k).

SLIDE 19

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Determination of the Coefficients

Inserting this power series Ansatz into the equation (η + k2 − s2)˜ v +(s, k) − ∂nˆ v +(1, k) − sˆ v +(1, k) = 0 satisfied by ˜ v +(˜ s, k) and comparing coefficients of powers of ˜ s, we find

˜ s0 : (η + k2)a0 + (η + k2) ˆ

v+(1,k) 2s0

− s2

0a0 + s0 2 ˆ

v +(1, k)=−∂nˆ v +(1,k) ˜ s1 : ( η+k2)( a1−2a0)−( η+k2)ˆ

v+(1,k) 2s0

−s2

0(

a1+2a0)+s0

2 ˆ

v +(1 ,k)=0 . . . . . . ˜ si : (η + k2)(ai − 2ai−1 + ai−2) − s2

0(ai + 2ai−1 + ai−2)=0

for i = 2, . . .. Truncating this recurrence relation for ai at some i = I, and setting ai = 0 for i ≥ I we obtain a linear system for the unknowns ai, and an approximate relation between ˆ v +(1, k) and ∂nˆ v +(1, k).

SLIDE 20

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

A Simple Example

For I = 0, all coefficients ai = 0, and only the relation from the s0 term remains: ∂nˆ v +(1, k) + ( η 2s0 + s0 2 + k2 2s0 )ˆ v +(1, k) = 0 It remains to choose the expansion point s0: ℜs ℑs

• η + k2

max

• η + k2

min

s0 The spectrum we try to approximate is at s =

• η + k2,

k ∈ {kmin, . . . , kmax}. If we choose s0 = √η, we get ∂nˆ v +(1, k) + (√η + k2 2√η)ˆ v +(1, k) = 0 a second order Taylor condition.

SLIDE 21

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Formulation for the More General Case

Discretization of v + in x using finite differences: (η − Dyy)v +

ij − 1

h2 (v +

i+1,j − 2v + ij + v + i−1,j) = fn,j

x y 1

v +

11

v +

13

v +

n1

v +

n3

a01 a03 a11 a13 a21 a23

Representation of the recurrence relation ai on the same grid: (η − Dyy − s2

0)a0j + (s0 2 + η−Dyy 2s0 )v + nj + Dnv + nj = 0

(η−Dyy)(a1j −2a0j)−s2

0(a1j +2a0j)+(s0 2 − η−Dyy 2s0 )v + nj = 0

. . . (η−Dyy)(ai+1,j −2ai,j +ai−1,j)−s2

0(ai+1,j +2ai,j +ai−1,j) = 0

SLIDE 22

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Equivalence With Stretched-Coordinate PML

A general finite difference discretization of the PML layer equation with the coordinate stretching γ ∂xγ(x)∂xu − (η − Dyy)u = 0 is given by γj uj+1 − uj hj − γj−1 uj − uj−1 hj−1 h∗

j

− (η − Dyy)uj = 0

Result (G, Sch¨ adle 2010)

The Pole condition gives a recurrence relation which can be identified with a geometrically stretched-coordinate PML.

SLIDE 23

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Operator in the General Case

Need to understand the general recurrence relation (η + k2)(ai − 2ai−1 + ai−2) − s2

0(ai + 2ai−1 + ai−2) = 0

which is equivalent to (η + k2 − s2

0)ai−1 − 2(η + k2 + s2 0)ai + (η + k2 − s2 0)ai+1 = 0

• r with

b := η + k2 + s2 η + k2 − s2 we have the simple recursion ai−1 − 2bai + ai+1 = 0 which will be truncated for some i = I, aI−1 − 2baI = 0.

SLIDE 24

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Equivalence with Absorbing Boundary Conditions

Result (G, Sch¨ adle 2010)

The recurrence relation is equivalent to the continued fraction expansion 1 4s0

• 2b − 2

1 2b −

1 2b−...

• ˆ

v +(1, k) = − ∂nˆ v +(1, k) η + k2 − s2 . In the limit, independantly of s0, we recover the DtN

• perator:
• η + k2ˆ

v +(1, k) = −∂nˆ v +(1, k).

Result (G, Sch¨ adle 2010)

The Pole condition truncated at step I gives an absorbing boundary condition which is the (2I+2,2I) Pad´ e approximation at s0 of the symbol of the DtN operator.

SLIDE 25

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Pole Condition as ABC

Ω = (0.5, 1) × (0, 1) with truncation at x = 0.5, uex = sin(πky)e √

π2k2+η(x−1), η = 0, s0 = 1.5, I = 0

0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 u

x y

0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 u

x y

0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 −5 5 10 15 20 x 10

−4

error

x y

SLIDE 26

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Error Level of the ABC

10 10

1

10

2

10

3

10

−5

10

−4

10

−3

10

−2

10

−1

error

• rder 2

N error

10 10

1

10

2

10

3

10

−5

10

−4

10

−3

10

−2

10

−1

error

• rder 2

N error

I = 0, s0 = 1.5 I = 2, s0 = 1.5

10 10

1

10

2

10

3

10

−5

10

−4

10

−3

10

−2

10

−1

error

• rder 2

N error

10 10

1

10

2

10

3

10

−5

10

−4

10

−3

10

−2

10

−1

error

• rder 2

N error

I = 0, s0 = 2 I = 2, s0 = 3

SLIDE 27

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Varying s0: s0 =

• η + k2

min

0.8 0.85 0.9 0.95 1 0.5 1 1 2 3 4 5 6 x y u 0.8 0.85 0.9 0.95 1 0.5 1 0.2 x y error

SLIDE 28

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Varying s0: s0 =

• η + k2

2

0.8 0.85 0.9 0.95 1 0.5 1 1 2 3 4 5 6 x y u 0.8 0.85 0.9 0.95 1 0.5 1 −0.02 0.02 0.04 0.06 0.08 x y error

SLIDE 29

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Varying s0: s0 =

• η + k2

3

0.8 0.85 0.9 0.95 1 0.5 1 1 2 3 4 5 6 x y u 0.8 0.85 0.9 0.95 1 0.5 1 −0.01 −0.005 0.005 0.01 0.015 0.02 0.025 0.03 x y error

SLIDE 30

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Varying s0: s0 =

• η + k2

4

0.8 0.85 0.9 0.95 1 0.5 1 1 2 3 4 5 6 x y u 0.8 0.85 0.9 0.95 1 0.5 1 −4 −2 2 4 6 8 10 12 14 x 10

−3

x y error

SLIDE 31

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Varying s0: s0 =

• η + k2

5

0.8 0.85 0.9 0.95 1 0.5 1 1 2 3 4 5 6 x y u 0.8 0.85 0.9 0.95 1 0.5 1 −5 5 10 15 20 x 10

−3

x y error

SLIDE 32

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Varying s0: s0 =

• η + k2

6

0.8 0.85 0.9 0.95 1 0.5 1 1 2 3 4 5 6 x y u 0.8 0.85 0.9 0.95 1 0.5 1 −0.005 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 x y error

SLIDE 33

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Varying s0: s0 =

• η + k2

7

0.8 0.85 0.9 0.95 1 0.5 1 1 2 3 4 5 6 x y u 0.8 0.85 0.9 0.95 1 0.5 1 −0.01 0.01 0.02 0.03 0.04 0.05 0.06 x y error

SLIDE 34

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Varying s0: s0 =

• η + k2

8

0.8 0.85 0.9 0.95 1 0.5 1 1 2 3 4 5 6 x y u 0.8 0.85 0.9 0.95 1 0.5 1 −0.01 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 x y error

SLIDE 35

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Varying s0: s0 =

• η + k2

9

0.8 0.85 0.9 0.95 1 0.5 1 1 2 3 4 5 6 x y u 0.8 0.85 0.9 0.95 1 0.5 1 −0.02 0.02 0.04 0.06 0.08 0.1 0.12 x y error

SLIDE 36

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Varying s0: Error Levels Attained

2 4 6 8 10 12 14 16 18 10

−2

10

−1

10

error km

SLIDE 37

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Optimization of s0

Result (G, Sch¨ adle 2010)

Let Pj(k,s0)

Qj(k,s0) be the (2j + 2, 2j) Pad´

e approximation at s0 of the symbol of the DtN operator. Then the reflection coefficient of the jth pole condition approximation, ρ(k, s0) =

• η + k2 − Pj(k,s0)

Qj(k,s0)

• η + k2 + Pj(k,s0)

Qj(k,s0)

, is minimized for k ∈ [0, kmax], kmax large, if the expansion point is chosen to be s2

0 = η1/2

η + k2

max

1/2 .

SLIDE 38

ABS and PML Martin J. Gander Domain Truncation

History Example

TBS and ABS

Construction Application: ABCs

PML

Historical Stretched-Coordinate PML

Pole Condition

Meaning Approximations

Mathematical Equivalences

With PML With ABC Numerical Experiments Optimization of s0

Conclusions

Conclusions

◮ The Pole Condition leads to Pad´

e approximations of the symbol of the DtN operator, and hence gives an Absorbing Boundary Condition

◮ The Pole Condition also leads to a geometrically

stretched-coordinate Perfectly Matched Layer

◮ Complete error estimates for the Helmholtz case,

including optimal choice of s0 for any order.

Current efforts:

◮ An improved Pole Condition would use collocation in

• rder to obtain best approximation. For example:

s0=(η + k2

min)1/4(η + k2 max)1/4

s1= 1

2(η + k2 min)5/12(η + k2 max)1/12

s2=2(η + k2

min)1/12(η + k2 max)5/12

⇒ R = 1 − 4 (η + k2

min)1/12

(η + k2

max)1/12

◮ Use of the Pole Condition in Domain Decomposition