On Energy Stable dG Approximation of the PML for the Wave ...users.monash.edu/~jdroniou/MWNDEA/slides/slides-duru.pdf · -2 -1 0 1 2 3-1-0.5 0 0.5 1 U PML x PML! Kenneth Duru: On

On Energy Stable dG Approximation of the PML forthe Wave Equation

Monash Workshop on Numerical Differential Equations and Applications

February 2020

Kenneth Duru

Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications1 / 34

Waves are everywhere

Simulations of seismic waves to quantify and assess earthquake risks and hazards.


Wave propagation:

Forward Modeling

Efficient Time Domain Wave Propagation Tool

Accurate and stable volume discretizations

Efficient and reliable absorbing boundaries

Accurate source generation

Efficient and scalable implementation on modern HPC platforms.

My research has penetrated all components.


Truncated Domain

Which boundary conditions ensure that numerical simulations converge to the infinite domainproblem? (old but relevant: Engquist and Majda (1977)).


Reflections from boundaries

A solution of the acoustic pressure with a point source.

0 2 4 6 8 10t[s]

-0.4

-0.2

0

0.2

0.4p[M

Pa]

∆x = 5/9∆x = 5/27

Analytical

Classical absorbing boundary condition


Absorbing Layer

Equations must be perfectly matched: J. P. Berenger (1994).


Anechoic chamber

”non-reflective”, ”non-echoing”, ”echo-free”.

A room designed to completely absorb reflections of either sound or electromagnetic waves.


Complex coordinate stretching: Chew and Weedon (1994)

∂/∂x → 1/Sx∂/∂x , Sx := dx/dx = 1 + dx (x)/s, dx ≥ 0.Simplifies PML construction for hyperbolic systems

−2 −1 0 1 2 3−1

−0.5

0

0.5

1U

PM

L

x

PML

!


Acoustic wave equation in first order form

1κ

∂p∂t

+∇ · v = 0, ρ∂v∂t

+∇p = 0.

(x , y , z) ∈ Ω = [−1, 1]3,

BCs:1− rη

2Zvη ∓

1 + rη2

p = 0, |rη | ≤ 1, at η = ±1.

dEdxdydz

=12

1κ|p|2 + ρ

∑η=x,y,z

|vη|2 > 0, E(t) =

∫Ω

dE > 0.

ddt

E(t) = −∮∂Ω

p (n · v) dS = −∑

η=x,y,z

∫ 1

−1

∫ 1

−1BT(η) dydzdx

dη≤ 0,

BT(η) =1− |rη |2

Z|χ(−η)|2

∣∣∣η=−1

+1− |rη |2

Z|χ(+η)|2

∣∣∣η=1≥ 0.



1κ

∂p∂t

+∇ · v = 0, ρ∂v∂t

+∇p = 0.

(x , y , z) ∈ Ω = [−1, 1]3,

BCs:1− rη

2Zvη ∓

1 + rη2

p = 0, |rη | ≤ 1, at η = ±1.

dEdxdydz

=12

1κ|p|2 + ρ

∑η=x,y,z

|vη|2 > 0, E(t) =

∫Ω

dE > 0.

ddt

E(t) = −∮∂Ω

p (n · v) dS = −∑

η=x,y,z

∫ 1

−1

∫ 1

−1BT(η) dydzdx

dη≤ 0,

BT(η) =1− |rη |2

Z|χ(−η)|2

∣∣∣η=−1

+1− |rη |2

Z|χ(+η)|2

∣∣∣η=1≥ 0.



1κ

∂p∂t

+∇ · v = 0, ρ∂v∂t

+∇p = 0.

(x , y , z) ∈ Ω = [−1, 1]3,

BCs:1− rη

2Zvη ∓

1 + rη2

p = 0, |rη | ≤ 1, at η = ±1.

dEdxdydz

=12

1κ|p|2 + ρ

∑η=x,y,z

|vη|2 > 0, E(t) =

∫Ω

dE > 0.

ddt

E(t) = −∮∂Ω

p (n · v) dS = −∑

η=x,y,z

∫ 1

−1

∫ 1

−1BT(η) dydzdx

dη≤ 0,

BT(η) =1− |rη |2

Z|χ(−η)|2

∣∣∣η=−1

+1− |rη |2

Z|χ(+η)|2

∣∣∣η=1≥ 0.


Derive PML in the Laplace domain

u(x , y , z, s) =

∫ ∞0

e−st u (x , y , z, t) dt, s = a + ib, Res = a > 0,

1κ

sp +∇ · v = 0, ρsv +∇p = 0.

PML : ∂/∂η → 1/Sη∂/∂η, Sη = 1 +dη(η)

s, dη(η) ≥ 0, η = x , y , z.

1κ

sp +∇d · v = 0, ρsv +∇d p = 0,

∇d = (1/Sx∂/∂x , 1/Sy∂/∂y , 1/Sz∂/∂z)T .



u(x , y , z, s) =

∫ ∞0


1κ

sp +∇ · v = 0, ρsv +∇p = 0.

PML : ∂/∂η → 1/Sη∂/∂η, Sη = 1 +dη(η)

s, dη(η) ≥ 0, η = x , y , z.

1κ

sp +∇d · v = 0, ρsv +∇d p = 0,

∇d = (1/Sx∂/∂x , 1/Sy∂/∂y , 1/Sz∂/∂z)T .



u(x , y , z, s) =

∫ ∞0


1κ

sp +∇ · v = 0, ρsv +∇p = 0.

PML : ∂/∂η → 1/Sη∂/∂η, Sη = 1 +dη(η)

s, dη(η) ≥ 0, η = x , y , z.

1κ

sp +∇d · v = 0, ρsv +∇d p = 0,

∇d = (1/Sx∂/∂x , 1/Sy∂/∂y , 1/Sz∂/∂z)T .



u(x , y , z, s) =

∫ ∞0


1κ

sp +∇ · v = 0, ρsv +∇p = 0.

PML : ∂/∂η → 1/Sη∂/∂η, Sη = 1 +dη(η)

s, dη(η) ≥ 0, η = x , y , z.

1κ

sp +∇d · v = 0, ρsv +∇d p = 0,

∇d = (1/Sx∂/∂x , 1/Sy∂/∂y , 1/Sz∂/∂z)T .


Time-domain PML

Auxiliary variables: sσ =

(dx − dy

Sx

Sy

)∂v∂y, sψ =

(dx − dz

Sx

Sz

)∂w∂z

.

Modified PDE:

1κ

(∂p∂t

+dx p)

+∇ · v−σ − ψ = 0, ρ

(∂v∂t

+dv)

+∇p = 0, d =

dx 0 00 dy 00 0 dz

.

Auxiliary differential equation: ODE(∂σ

∂t+ dyσ

)+ (dy − dx )

∂v∂y

= 0,(∂ψ

∂t+ dzψ

)+ (dz − dx )

∂w∂z

= 0,

BCs:1− rη

2Zvη ∓

1 + rη2

p = 0, at η = ±1.


Time-domain PML

Auxiliary variables: sσ =

(dx − dy

Sx

Sy

)∂v∂y, sψ =

(dx − dz

Sx

Sz

)∂w∂z

.

Modified PDE:

1κ

(∂p∂t

+dx p)

+∇ · v−σ − ψ = 0, ρ

(∂v∂t

+dv)

+∇p = 0, d =

dx 0 00 dy 00 0 dz

.

Auxiliary differential equation: ODE(∂σ

∂t+ dyσ

)+ (dy − dx )

∂v∂y

= 0,(∂ψ

∂t+ dzψ

)+ (dz − dx )

∂w∂z

= 0,

BCs:1− rη

2Zvη ∓

1 + rη2

p = 0, at η = ±1.


2D: SBP finite difference approximation

0 1000 2000 3000 4000 500010

−3

10−2

10−1

100

time

‖E

z‖h

2nd−order

4th−order

6th−order

(b) Discrete PML

0 1000 2000 3000 4000 500010

−3

10−2

10−1

100

time

‖E

z‖h

2nd−order

4th−order

6th−order

(c) No PML (dx = 0).


A nightmare for finite element and dG practitioners!

-60 -40 -20 0 20 40 60

x[km]

0

10

20

30

40

50

y[k

m]

t=150 s

0

0.02

0.04

0.06

0.08

0.1

GLL.

-60 -40 -20 0 20 40 60

x[km]

0

10

20

30

40

50y[k

m]

t=35 s

0

0.02

0.04

0.06

0.08

0.1

GL.

-60 -40 -20 0 20 40 60

x[km]

0

10

20

30

40

50

y[k

m]

t=10 s

0

0.02

0.04

0.06

0.08

0.1

GLR.


Some pioneering work: Cauchy problem

Abarbanel and Gottlieb (1997 & 98), Hesthaven et al. (1999),

Collino and Tsogka (2001),

Becache et al. (2003): Geometric stability condition,

Appelo, Hagstrom and Kreiss (2006),

Diaz and Joly (2006), Halpern et al. (2011),

Duru and Kreiss (2012), ..., Duru (2016)

Skelton, et al. (2007): Destabilizing effects of boundary conditions


Summary of literature: IBVP

Assume constant coefficients and consider:

1. IVP PML : (x , y) ∈ (−∞,∞)× (−∞,∞). Geometric stability condition Becache et al.2003. Classical Fourier analysis

2. IBVP PML : PML is asymmetric.2a. Lower half–plane PML problem: (x, y) ∈ (−∞,∞)× (−∞, 0), with Ly U = 0, y = 0.2b . Left half–plane PML problem: (x, y) ∈ (−∞, 0)× (−∞,∞), with Lx U = 0, x = 0.

Assumption: In the absence of the PML, dx = 0, the IBVPs are stable in the sense of Kreiss.All eigenvalues are in the stable half of the complex plane.

What will happen when the PML is present, dx > 0?

TheoremIf the geometric stability condition is satisfied then the PML IBVPs with dx ≥ 0 areasymptotically stable. The PML damping dx > 0 will move all eigenvalues further into thestable complex plane.

Duru & Kreiss SINUM (2014), Duru SISC (2016), Duru et al. JCP (2016).

PML is asymptotically stable ! Too technical to be extended to numerical approximations inmultiple dimensions
































Need to extend theory

1 Energy estimates + useful for designing stable numerical methods.

2 Energy estimate must account for BCs.

3 PML is asymmetric. Energy estimate in the time-domain technically difficult.

4 Energy estimate in the Laplace space.

5 Numerical boundary/inter-element procedures.























Energy estimate in the Laplace space

Forcing:(fp, f, fσ , fψ

)T with f = (fx , fy , fz ).

Laplace transform + Eliminate PML auxiliary variables

1κ

sp +∇d · v =1κ

Fp, ρsv +∇d p = ρf,

fη =1

Sηfη , Fp =

(1

Sxfp −

κ

sSy Sxfσ −

κ

sSzSxfψ

),

BCs:1− rη

2Z vη ∓

1 + rη2

p = 0, at η = ±1.

1κ

s2p −∇d ·(

1ρ∇d p

)=

sκ

Fp −∇d · f,

BCs:1 + rη

2Zsp ±

1− rη2

1Sη

∂p∂η

= 0, at η = ±1,

s∗

Sηρp∗∂p∂η≥ 0, η = −1;

s∗

Sηρp∗∂p∂η≤ 0, η = 1.






1κ

sp +∇d · v =1κ


fη =1

Sηfη , Fp =

(1

Sxfp −

κ

sSy Sxfσ −

κ

sSzSxfψ

),

BCs:1− rη

2Z vη ∓

1 + rη2

p = 0, at η = ±1.

1κ

s2p −∇d ·(

1ρ∇d p

)=

sκ

Fp −∇d · f,

BCs:1 + rη

2Zsp ±

1− rη2

1Sη

∂p∂η

= 0, at η = ±1,

s∗

Sηρp∗∂p∂η≥ 0, η = −1;

s∗

Sηρp∗∂p∂η≤ 0, η = 1.






1κ

sp +∇d · v =1κ


fη =1

Sηfη , Fp =

(1

Sxfp −

κ

sSy Sxfσ −

κ

sSzSxfψ

),

BCs:1− rη

2Z vη ∓

1 + rη2

p = 0, at η = ±1.

1κ

s2p −∇d ·(

1ρ∇d p

)=

sκ

Fp −∇d · f,

BCs:1 + rη

2Zsp ±

1− rη2

1Sη

∂p∂η

= 0, at η = ±1,

s∗

Sηρp∗∂p∂η≥ 0, η = −1;

s∗

Sηρp∗∂p∂η≤ 0, η = 1.



Re(

(sSη)∗

Sη

)= a + εη(s, dη), εη(s, dη) :=

2dηb2

|sSη |2≥ 0.

Define the energy

E2p (s, dη) =

∥∥∥sp∥∥∥2

1/κ+

∑η=x,y,z

∥∥∥ 1Sη

∂p∂η

∥∥∥2

1/ρ> 0,

E2f (s, dη) =

∥∥∥sFp

∥∥∥2

1/κ+

∑η=x,y,z

∥∥∥ 1Sη

∂ fη∂η

∥∥∥2

κ> 0.

TheoremConsider the PML IBVP with s 6= 0, Res = a > 0 and piecewise constant dη ≥ 0.

aE2p (s, dη) +

∑η=x,y,z

∥∥∥ 1Sη

∂p∂η

∥∥∥2

εη/ρ+ BT (s, dη) ≤ 2Ep (s, dη) Ef (s, dη),

BT (s, dη) =

∫ 1

−1

∫ 1

−1

∑η=x,y,z

(s∗

Sηρp∗∂p∂η

∣∣∣η=−1

−s∗

Sηρp∗∂p∂η

∣∣∣η=1

)dxdydz

dη≥ 0.


Well-posedness & Stability

TheoremThe problem, with dη ≥ 0, is asymptotically stable in the sense that no exponentially growingsolutions are supported.

Proof:Make the ansatz Q(x , y , z, t) = est Q(x , y , z), with s = a + ib,

aE2p (s, dη) +

∑η=x,y,z

∥∥∥ 1Sη

∂p∂η

∥∥∥2

εη/ρ+ BT (s, dη) = 0.

With Res = a > 0 the expression in the left hand side is positive. The conclusion is thatQ(x , y , z) ≡ 0.

TheoremLet the energy norms E2

p (t , dη) > 0, E2f (t , dη) > 0. For any a > 0 and T > 0

∫ T

0e−2at E2

p (t , dη) dt ≤4a2

∫ T

0e−2at E2

f (t , dη) dt , a > 0.

Key: Construct a discrete approximation that, as far as possible, mimics the energy estimate.


Well-posedness & Stability

TheoremThe problem, with dη ≥ 0, is asymptotically stable in the sense that no exponentially growingsolutions are supported.

Proof:Make the ansatz Q(x , y , z, t) = est Q(x , y , z), with s = a + ib,

aE2p (s, dη) +

∑η=x,y,z

∥∥∥ 1Sη

∂p∂η

∥∥∥2

εη/ρ+ BT (s, dη) = 0.

With Res = a > 0 the expression in the left hand side is positive. The conclusion is thatQ(x , y , z) ≡ 0.

TheoremLet the energy norms E2

p (t , dη) > 0, E2f (t , dη) > 0. For any a > 0 and T > 0

∫ T

0e−2at E2

p (t , dη) dt ≤4a2

∫ T

0e−2at E2

f (t , dη) dt , a > 0.

Key: Construct a discrete approximation that, as far as possible, mimics the energy estimate.


Dicretize domain

Discretize the domain (x , y , z) ∈ Ω = ∪Ωlmn with Ωlmn = [xl , xl+1]× [ym, ym+1]× [zn, zn+1],

(x , y , z)←→ (q, r , s) ∈ Ω = [−1, 1]3:

x = xl +∆xl

2(1 + q) , y = ym +

∆ym

2(1 + r) , z = zn +

∆zn

2(1 + s) ,

with

J =∆xl

2∆ym

2∆zn

2> 0, ∆xl = xl+1 − xl , ∆ym = ym+1 − ym, ∆zn = yn+1 − yn.

∫Ω

f (x , y , z)dxdydz =K∑

k=1

L∑l=1

M∑m=1

∫Ωklm


k=1

L∑l=1

M∑m=1

∫Ω

f (q, r , s)Jdqdrds.

Physics based flux fluctuations:

F x :=Z2

(vx − vx

)+

12

(p − p

)= 0, x = xl ,

Gx :=Z2

(vx − vx

)−

12

(p − p

)= 0, x = xl+1.

(1)

vx , p encode the solution of the IBVP at element boundaries.


Dicretize domain

Discretize the domain (x , y , z) ∈ Ω = ∪Ωlmn with Ωlmn = [xl , xl+1]× [ym, ym+1]× [zn, zn+1],

(x , y , z)←→ (q, r , s) ∈ Ω = [−1, 1]3:

x = xl +∆xl

2(1 + q) , y = ym +

∆ym

2(1 + r) , z = zn +

∆zn

2(1 + s) ,

with

J =∆xl

2∆ym

2∆zn

2> 0, ∆xl = xl+1 − xl , ∆ym = ym+1 − ym, ∆zn = yn+1 − yn.

∫Ω


k=1

L∑l=1

M∑m=1

∫Ωklm


k=1

L∑l=1

M∑m=1

∫Ω

f (q, r , s)Jdqdrds.

Physics based flux fluctuations:

F x :=Z2

(vx − vx

)+

12

(p − p

)= 0, x = xl ,

Gx :=Z2

(vx − vx

)−

12

(p − p

)= 0, x = xl+1.

(1)

vx , p encode the solution of the IBVP at element boundaries.


Integral form

∫Ω

φp

(1κ

(∂p∂t

+ dx p)

+∇ · v− σ − ψ)

Jdqdrds

= −∑η=q,r,s

∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φp

ZFη|η=−1 −

φp

ZGη|η=1

)dqdrds

dη,

∫Ω

θT(ρ

(∂v∂t

+ dv)

+∇p)

Jdqdrds

= −∑η=q,r,s

∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(θ

T nFη|η=−1 + θT nGη|η=1

) dqdrdsdη

,

∫Ω

φσ

(∂σ

∂t+ dyσ + (dy − dx )

2∆y

∂vy

∂r

)Jdqdrds

= −ωy (dy − dx )∑η=q,r,s

∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φσ

Zny Fη|η=−1 −

φσ

Zny Gη|η=1

)dqdrds

dη︸︷︷︸PML stabilizing flux fluctuation

,

∫Ω

φψ

(∂ψ

∂t+ dzψ + (dz − dx )

2∆z

∂vz

∂s

)Jdqdrds

= −ωz (dz − dx )∑η=q,r,s

∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φψ

Znz Fη|η=−1 −

φψ

Znz Gη|η=1

)dqdrds


.


Integral form

∫Ω

φp

(1κ

(∂p∂t

+ dx p)

+∇ · v− σ − ψ)

Jdqdrds

= −∑η=q,r,s

∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φp

ZFη|η=−1 −

φp

ZGη|η=1

)dqdrds

dη,

∫Ω

θT(ρ

(∂v∂t

+ dv)

+∇p)

Jdqdrds

= −∑η=q,r,s

∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(θ


) dqdrdsdη

,

∫Ω

φσ

(∂σ


2∆y

∂vy

∂r

)Jdqdrds


∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φσ

Zny Fη|η=−1 −

φσ

Zny Gη|η=1

)dqdrds


,

∫Ω

φψ

(∂ψ


2∆z

∂vz

∂s

)Jdqdrds


∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φψ

Znz Fη|η=−1 −

φψ

Znz Gη|η=1

)dqdrds


.


Integral form

∫Ω

φp

(1κ

(∂p∂t

+ dx p)

+∇ · v− σ − ψ)

Jdqdrds

= −∑η=q,r,s

∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φp

ZFη|η=−1 −

φp

ZGη|η=1

)dqdrds

dη,

∫Ω

θT(ρ

(∂v∂t

+ dv)

+∇p)

Jdqdrds

= −∑η=q,r,s

∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(θ


) dqdrdsdη

,

∫Ω

φσ

(∂σ


2∆y

∂vy

∂r

)Jdqdrds


∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φσ

Zny Fη|η=−1 −

φσ

Zny Gη|η=1

)dqdrds


,

∫Ω

φψ

(∂ψ


2∆z

∂vz

∂s

)Jdqdrds


∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φψ

Znz Fη|η=−1 −

φψ

Znz Gη|η=1

)dqdrds


.


Integral form

∫Ω

φp

(1κ

(∂p∂t

+ dx p)

+∇ · v− σ − ψ)

Jdqdrds

= −∑η=q,r,s

∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φp

ZFη|η=−1 −

φp

ZGη|η=1

)dqdrds

dη,

∫Ω

θT(ρ

(∂v∂t

+ dv)

+∇p)

Jdqdrds

= −∑η=q,r,s

∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(θ


) dqdrdsdη

,

∫Ω

φσ

(∂σ


2∆y

∂vy

∂r

)Jdqdrds


∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φσ

Zny Fη|η=−1 −

φσ

Zny Gη|η=1

)dqdrds


,

∫Ω

φψ

(∂ψ


2∆z

∂vz

∂s

)Jdqdrds


∫ 1

−1

∫ 1

−1J√η2

x + η2y + η2

z

(φψ

Znz Fη|η=−1 −

φψ

Znz Gη|η=1

)dqdrds


.


The dG Approximation

Galerkin approximation + Polynomial interpolants + GL, GLL, GLR, Collocation nodes:

κ−1(

dp(t)dt

+ dx p(t))

+∇D · v(t) + σ(t) + ψ(t) = −∑

η=x,y,z

H−1η

(eη(−1)

ZFη −

eη(1)

ZGη),

ρ

(dv(t)

dt+ dv(t)

)+∇Dp(t) = −

∑η=x,y,z

H−1η

(eη(−1)

ZnFη −

eη(1)

ZnGη

),

(dσ(t)

dt+ dyσ(t)

)+ (dy − dx )Dy vy (t) = −ωy (dy − dx )

∑η=x,y,z

H−1η

(eη(−1)

Zny Fη −

eη(1)

Zny Gη

)︸︷︷︸

PML stabilizing flux fluctuation

,

(dψ(t)

dt+ dyψ(t)

)+ (dz − dx )Dz vz (t) = −ωz (dz − dx )

∑η=x,y,z

H−1η

(eη(−1)

Znz Fη −

eη(1)

Znz Gη

)︸︷︷︸


.

∇D = (Dx ,Dy ,Dz )T, Dx =

2∆x

(D ⊗ I ⊗ I) , Hx =∆x2

(H ⊗ I ⊗ I) , ex (η) = (e(η)⊗ I ⊗ I) ,

D = H−1A ≈∂

∂q, H = diag[h1, h2, · · · , hP+1], Aij =

P+1∑m=1

hmLi (qm)L ′j (qm) =

∫ 1

−1Li (q)L ′j (q)dq,

e(η) = [Li (η),Li (η), · · · ,LP+1(η)]T .




κ−1(

dp(t)dt

+ dx p(t))

+∇D · v(t) + σ(t) + ψ(t) = −∑

η=x,y,z

H−1η

(eη(−1)

ZFη −

eη(1)

ZGη),

ρ

(dv(t)

dt+ dv(t)

)+∇Dp(t) = −

∑η=x,y,z

H−1η

(eη(−1)

ZnFη −

eη(1)

ZnGη

),

(dσ(t)

dt+ dyσ(t)


∑η=x,y,z

H−1η

(eη(−1)

Zny Fη −

eη(1)

Zny Gη

)︸︷︷︸


,

(dψ(t)

dt+ dyψ(t)


∑η=x,y,z

H−1η

(eη(−1)

Znz Fη −

eη(1)

Znz Gη

)︸︷︷︸


.


2∆x

(D ⊗ I ⊗ I) , Hx =∆x2

(H ⊗ I ⊗ I) , ex (η) = (e(η)⊗ I ⊗ I) ,

D = H−1A ≈∂

∂q, H = diag[h1, h2, · · · , hP+1], Aij =

P+1∑m=1


∫ 1


e(η) = [Li (η),Li (η), · · · ,LP+1(η)]T .




κ−1(

dp(t)dt

+ dx p(t))

+∇D · v(t) + σ(t) + ψ(t) = −∑

η=x,y,z

H−1η

(eη(−1)

ZFη −

eη(1)

ZGη),

ρ

(dv(t)

dt+ dv(t)

)+∇Dp(t) = −

∑η=x,y,z

H−1η

(eη(−1)

ZnFη −

eη(1)

ZnGη

),

(dσ(t)

dt+ dyσ(t)


∑η=x,y,z

H−1η

(eη(−1)

Zny Fη −

eη(1)

Zny Gη

)︸︷︷︸


,

(dψ(t)

dt+ dyψ(t)


∑η=x,y,z

H−1η

(eη(−1)

Znz Fη −

eη(1)

Znz Gη

)︸︷︷︸


.


2∆x

(D ⊗ I ⊗ I) , Hx =∆x2

(H ⊗ I ⊗ I) , ex (η) = (e(η)⊗ I ⊗ I) ,

D = H−1A ≈∂

∂q, H = diag[h1, h2, · · · , hP+1], Aij =

P+1∑m=1


∫ 1


e(η) = [Li (η),Li (η), · · · ,LP+1(η)]T .


Stability of the discrete undamped problem

Introduce the elemental energy density

dE (q, r , s, t) =12

1κ(q, r , s)

|p(q, r , s, t)|2 + ρ(q, r , s)∑

η=x,y,z

(|vη(q, r , s, t)|2

)and the corresponding semi-discrete energy

E (t) =∑l=1

∑m=1

∑n=1

P+1∑i=1

P+1∑j=1

P+1∑k=1

dE (qi , rj , sk , t)Jhi hj hk , J =∆xl

2∆ym

2∆zn

2.

TheoremConsider the semi-discrete approximation. When all the damping vanish, dη = 0, the solutionof the semi-discrete approximation satisfies the energy identity

d

dtE (t) = −

L∑l=1

M∑m=1

N∑n=1

∑η=q,r,s

N+1∑i=1

N+1∑j=1

(J√η2

x + η2y + η2

z

( 1

Z|Fη|2 +

1

Z|Gη|2 + BT(η)

))ij

hi hj

lmn

≤ 0.

The numerical approximation is asymptotically stable!

Result does not translate to the PML when dη > 0.


Stability of the discrete undamped problem

Introduce the elemental energy density

dE (q, r , s, t) =12

1κ(q, r , s)

|p(q, r , s, t)|2 + ρ(q, r , s)∑

η=x,y,z

(|vη(q, r , s, t)|2

)and the corresponding semi-discrete energy

E (t) =∑l=1

∑m=1

∑n=1

P+1∑i=1

P+1∑j=1

P+1∑k=1

dE (qi , rj , sk , t)Jhi hj hk , J =∆xl

2∆ym

2∆zn

2.

TheoremConsider the semi-discrete approximation. When all the damping vanish, dη = 0, the solutionof the semi-discrete approximation satisfies the energy identity

d

dtE (t) = −

L∑l=1

M∑m=1

N∑n=1

∑η=q,r,s

N+1∑i=1

N+1∑j=1

(J√η2

x + η2y + η2

z

( 1

Z|Fη|2 +

1

Z|Gη|2 + BT(η)

))ij

hi hj

lmn

≤ 0.

The numerical approximation is asymptotically stable!

Result does not translate to the PML when dη > 0.


Stability of the discrete PML problem

Laplace transform + Eliminating the PML auxiliary variables:

1κ

sp + ∇D · v =1κ

fp −∑

η=x,y,z

1Sη

H−1η

(eη(−1)

ZFη −

eη(1)

ZGη)

+∑η=y,z

(1− ωη) (dη − dx )

sSηSxH−1η

(eη(−1)

ZFη −

eη(1)

ZGη),

ρsv + ∇D p = ρf−∑

η=x,y,z

1Sη

H−1η

(eη(−1)

ZnFη −

eη(1)

ZnGη

), ∇D = (1/Sx Dx , 1/Sy Dy , 1/Sz Dz )T

.

Consider a single element and introduce the modified discrete operators

Dη =1

Sη

(Dη +

1 + rη2

H−1η (Bη (−1,−1)− Bη (1, 1))

),

Hη = H(

I +(1− rη)c

2sSηH−1η (Bη(−1,−1) + Bη(1, 1))

)−1

.

Eliminate the velocity fields:

s∗Hsκ−1sp +∑η

(1

SηDη)†( (s∗S∗η)

ρSηHη

)(1

SηDη)

p

+ |s|2∑η

1 + rη2ZSη

HHη−1 (Bη(−1,−1) + Bη(1, 1)) p = s∗Hκ−1sFp −

∑η

s∗H(

1Sη

D0η

)fη.




1κ

sp + ∇D · v =1κ

fp −∑

η=x,y,z

1Sη

H−1η

(eη(−1)

ZFη −

eη(1)

ZGη)

+∑η=y,z

(1− ωη) (dη − dx )

sSηSxH−1η

(eη(−1)

ZFη −

eη(1)

ZGη),


η=x,y,z

1Sη

H−1η

(eη(−1)

ZnFη −

eη(1)

ZnGη

), ∇D = (1/Sx Dx , 1/Sy Dy , 1/Sz Dz )T

.


Dη =1

Sη

(Dη +

1 + rη2

H−1η (Bη (−1,−1)− Bη (1, 1))

),

Hη = H(

I +(1− rη)c

2sSηH−1η (Bη(−1,−1) + Bη(1, 1))

)−1

.



(1


ρSηHη

)(1

SηDη)

p

+ |s|2∑η

1 + rη2ZSη

HHη−1 (Bη(−1,−1) + Bη(1, 1)) p = s∗Hκ−1sFp −

∑η

s∗H(

1Sη

D0η

)fη.




1κ

sp + ∇D · v =1κ

fp −∑

η=x,y,z

1Sη

H−1η

(eη(−1)

ZFη −

eη(1)

ZGη)

+∑η=y,z

(1− ωη) (dη − dx )

sSηSxH−1η

(eη(−1)

ZFη −

eη(1)

ZGη),


η=x,y,z

1Sη

H−1η

(eη(−1)

ZnFη −

eη(1)

ZnGη

), ∇D = (1/Sx Dx , 1/Sy Dy , 1/Sz Dz )T

.


Dη =1

Sη

(Dη +

1 + rη2

H−1η (Bη (−1,−1)− Bη (1, 1))

),

Hη = H(

I +(1− rη)c

2sSηH−1η (Bη(−1,−1) + Bη(1, 1))

)−1

.



(1


ρSηHη

)(1

SηDη)

p

+ |s|2∑η

1 + rη2ZSη

HHη−1 (Bη(−1,−1) + Bη(1, 1)) p = s∗Hκ−1sFp −

∑η

s∗H(

1Sη

D0η

)fη.Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications24 / 34

Discrete energy estimate in the Laplace space

Introduce the discrete scalar product ⟨u, v⟩

H= v†Hu, (2)

and the energy norms

E 2p (s) =

⟨sp, sp

⟩H/κ

+∑

η=x,y,z

⟨ 1Sη

Dηp,1

SηDηp

⟩Hη/ρ

> 0, (3)

E 2f (s) =

⟨sFp, sFp

⟩H/κ

+∑

η=x,y,z

⟨ 1Sη

D0ηfη , 1

SηD0ηfη⟩

κH> 0. (4)

TheoremConsider the one element dG approximation of the PML in the Laplace space withRes = a > 0 and constant damping dη ≥ 0. If ωη = 1, then we have

aE 2p (s) +

∑η=x,y,z

⟨ 1Sη

Dη p,1

SηDη p

⟩εη Hη/ρ

+∑η

1− rηρ

BT(η)num + BT(s) ≤ 2Ep(s)Ef (s),

BT(s) = |s|2∑η

Re(

1Sη

)1 + rη

2Zp†[HH−1

η (Bη(−1,−1) + Bη(1, 1))]

p ≥ 0.


2D: Stable approximations

-60 -40 -20 0 20 40 60

x[km]

0

10

20

30

40

50

y[k

m]

t=500 s

0

1

2

3×10 -5

GLL.

-60 -40 -20 0 20 40 60

x[km]

0

10

20

30

40

50y[k

m]

t=500 s

0

1

2

3×10 -5

GL.

-60 -40 -20 0 20 40 60

x[km]

0

10

20

30

40

50

y[k

m]

t=500 s

0

1

2

3×10 -5

GLR.


2D: Stable approximations

0 100 200 300 400 500

t[s]

10-4

10-3

10-2

10-1

100

101

102

103

104

105

L∞-norm

ω = 0

ω = 1

GLL.

0 100 200 300 400 500

t[s]

10-4

10-3

10-2

10-1

100

101

102

103

104

105

L∞-norm

ω = 0

ω = 1

GL.

0 100 200 300 400 500

t[s]

10-4

10-3

10-2

10-1

100

101

102

103

104

105

L∞-norm

ω = 0

ω = 1

GLR.


h- and p-convergence

2 4 6 8 10

∆x

10-10

10-8

10-6

10-4

10-2

error

GLL

GL

GLR

C∆x5

h-convergence.

2 4 6 8

polynomial degree

10-10

10-8

10-6

10-4

10-2

error

GLL

GL

GLR

p-convergence.

dx (x) =

0 if |x | ≤ 50 km,

d0

(|x|−50δ

)3if |x | ≥ 50 km,

(5)

d0 =4c2δ

ln1tol

, tol = C0

[1δ

∆xP + 1

]P+1. (6)


Stable PML boundaries

1

2

3

4

X Axis

4Y Axis

4

Y Axis

1

2

3

4

Z Axis

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Z Axis

1 2 3 4

X Axis

t = 1.7 s

1

2

3

4

X Axis

4Y Axis

4

Y Axis

1

2

3

4

Z Axis

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Z Axis

1 2 3 4

X Axis

t = 2.5 s

1

2

3

4

X Axis

4Y Axis

4

Y Axis

1

2

3

4

Z Axis

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Z Axis

1 2 3 4

X Axis

t = 3.0 s

0 2 4 6 8 10t[s]

-0.4

-0.2

0

0.2

0.4

p[M

Pa]

∆x = 5/9∆x = 5/27

Analytical

Absorbing boundary condition

0 2 4 6 8 10t[s]

-0.4

-0.2

0

0.2

0.4

p[M

Pa]

∆x = 5/9∆x = 5/27

Analytical

PML


Extensions to linear elasticity

0 1 2 3

time [s]

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

velo

city [m

/s]

PML

ABC

analytical


LOH1: A Seismology benchmark problem

time [s]15

10

5

0

5

10

15

v x [m

/s]

analyticalABCPML

15

10

5

0

5

10

15

v y [m

/s]

0 1 2 3 4 5 6 7 8 9time [s]

7.5

5.0

2.5

0.0

2.5

5.0

7.5

v z [m

/s]

Receiver 4

time [s]

4

2

0

2

4

v x [m

/s]

analyticalABCPML

4

2

0

2

4

v y [m

/s]

0 1 2 3 4 5 6 7 8 9time [s]

2

1

0

1

2

v z [m

/s]

Receiver 5

time [s]

0.5

0.0

0.5

v x [m

/s]

analyticalABCPML

0.5

0.0

0.5v y

[m/s

]

0 1 2 3 4 5 6 7 8 9time [s]

1.0

0.5

0.0

0.5

1.0

v z [m

/s]

Receiver 6

time [s]1.5

1.0

0.5

0.0

0.5

1.0

1.5

v x [m

/s]

analyticalABCPML

1.5

1.0

0.5

0.0

0.5

1.0

1.5

v y [m

/s]

0 1 2 3 4 5 6 7 8 9time [s]

1.0

0.5

0.0

0.5

1.0

v z [m

/s]

Receiver 9Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications31 / 34

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Summary

PML for acoustics equation is well-posed and asymptotically stable.

Numerical flux procedures can introduce instability.

Stable numerical flux procedures can be designed by mimicking continuous energyestimate.

Ideas has been extended to linear elasticity.

Initial ideas (2D SBP FDM): K. Duru SIAM J. Sci. Comput., 38(2016), A1171-A1194.

K. Duru, A.-A. Gabriel and G. Kreiss, Computer Methods in Applied Mechanics andEngineering, 350(2019), 898–937.

K. Duru, et al., Extensions to linear elastodynamics, Submitted to Numerische Mathematik(2019).


Support & Acknowledgements

Prof. Gunilla Kreiss, Uppsala University Sweden.

Prof. Eric M. Dunham, Stanford University, CA.

Dr. Alice-Agnes Gabriel, LMU Munich.

ExaHyPE team:

Heinz-Otto Kreiss


Documents

On Energy Stable dG Approximation of the PML for the Wave ...users.monash.edu/~jdroniou/MWNDEA/slides/slides-duru.pdf · -2 -1 0 1 2 3-1-0.5 0 0.5 1 U PML x PML! Kenneth Duru: On