on energy stable dg approximation of the pml for the wave...
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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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications2 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications3 / 34
Truncated Domain
Which boundary conditions ensure that numerical simulations converge to the infinite domainproblem? (old but relevant: Engquist and Majda (1977)).
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications4 / 34
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
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications5 / 34
Absorbing Layer
Equations must be perfectly matched: J. P. Berenger (1994).
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications6 / 34
Anechoic chamber
”non-reflective”, ”non-echoing”, ”echo-free”.
A room designed to completely absorb reflections of either sound or electromagnetic waves.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications7 / 34
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
!
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications8 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications9 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications9 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications9 / 34
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 .
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications10 / 34
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 .
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications10 / 34
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 .
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications10 / 34
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 .
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications10 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications11 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications11 / 34
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).
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications12 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications13 / 34
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
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications14 / 34
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
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications15 / 34
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
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications15 / 34
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
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications15 / 34
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
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications15 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications16 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications16 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications16 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications16 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications17 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications17 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications17 / 34
Energy estimate in the Laplace space
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications18 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications19 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications19 / 34
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
f (x , y , z)dxdydz =K∑
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications20 / 34
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
f (x , y , z)dxdydz =K∑
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications20 / 34
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
dη︸ ︷︷ ︸PML stabilizing flux fluctuation
.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications21 / 34
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
dη︸ ︷︷ ︸PML stabilizing flux fluctuation
.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications21 / 34
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
dη︸ ︷︷ ︸PML stabilizing flux fluctuation
.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications21 / 34
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
dη︸ ︷︷ ︸PML stabilizing flux fluctuation
.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications21 / 34
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
)︸ ︷︷ ︸
PML stabilizing flux fluctuation
.
∇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 .
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications22 / 34
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
)︸ ︷︷ ︸
PML stabilizing flux fluctuation
.
∇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 .
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications22 / 34
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
)︸ ︷︷ ︸
PML stabilizing flux fluctuation
.
∇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 .
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications22 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications23 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications23 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications24 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications24 / 34
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.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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications25 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications26 / 34
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.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications27 / 34
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)
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications28 / 34
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
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications29 / 34
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
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications30 / 34
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
ExaHyPE: Exa-scale Hyperbolic PDE simulation Engine
ExascaleSpacetreeADER-DG
TUM TRE
DUR
Seismic
Astro
Software for next generation super computers (1018 flops/sec),Exascale HPC scalability,Energy efficiency, etc.
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications32 / 34
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).
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications33 / 34
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
Kenneth Duru: On Energy Stable dG Approximation of the PML for the Wave Equation— Monash Workshop on Numerical Differential Equations and Applications34 / 34