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Some Aspects in the Numerics of Nonlinear Acoustics: Time Integration and Open Domain Problems

Barbara Kaltenbacher

Alpen-Adria-Universit¨ at Klagenfurt RICAM Special Semester on Computational Methods in Science and Engineering Workshop 1: Analysis and Numerics of Acoustic and Electromagnetic Problems

Some Aspects in the Numerics of Nonlinear Acoustics: Time Integration and Open Domain Problems

Barbara Kaltenbacher

Alpen-Adria-Universit¨ at Klagenfurt RICAM Special Semester on Computational Methods in Science and Engineering Workshop 1: Analysis and Numerics of Acoustic and Electromagnetic Problems joint work with: Rainer Brunnhuber, AAU, Vanja Nikoli´ c, AAU, Christian Clason, U Duisburg-Essen, Manfred Kaltenbacher, TU Vienna, Irena Lasiecka, U Memphis, Richard Marchand, Slippery Rock U, Gunther Peichl, U Graz, Maria K. Pospieszalska, La Jolla Institute, Petronela Radu, U Nebraska at Lincoln, Igor Shevchenko, UCL, Mechthild Thalhammer, U Innsbruck Mathematics of Nonlinear Acoustics

Nonlinear Acoustic Wave Propagation

1

Nonlinear Acoustic Wave Propagation

1

Applications of High Intensity Focused Ultrasound HIFU

lithotripsy cleaning thermotherapy welding

2

Outline

models time integration nonreflecting boundary conditions

3

modeling

4

Physical Principles

main physical quantities: acoustic particle velocity v; acoustic pressure p; mass density ̺; decomposition into mean and fluctuating part:

• v =

v0 + v∼ = v∼ , p = p0 + p∼ , ̺ = ̺0 + ̺∼

5

Physical Principles

main physical quantities: acoustic particle velocity v; acoustic pressure p; mass density ̺; decomposition into mean and fluctuating part:

• v =

v0 + v∼ = v∼ , p = p0 + p∼ , ̺ = ̺0 + ̺∼ governing equations: Navier Stokes equation (under the assumption ∇ × v = 0) ̺

• vt + ∇(

v · v)

• + ∇p =

4µV 3 + ζV

• ∆

v equation of continuity ̺t + ∇ · (̺ v) = 0 state equation ̺∼ = 1 c2 p∼ − 1 ̺0c4 B 2Ap∼2 − κ ̺0c4 1 cV − 1 cp

• p∼t

5

Derivation of Wave Equation

main physical quantities:

• v =

v∼ , p = p0 + p∼ , ∇p0 = 0 , ̺ = ̺0 + ̺∼ , ̺0t = 0

6

Derivation of Wave Equation

main physical quantities:

• v =

v∼ , p = p0 + p∼ , ∇p0 = 0 , ̺ = ̺0 + ̺∼ , ̺0t = 0 governing equations: ̺

• vt + ∇(

v · v)

• + ∇p =

4µV 3 + ζV

• ∆

v ̺t + ∇ · (̺ v) = 0 ̺∼ = p∼ c2 − 1 ̺0c4 B 2Ap2

∼ −

κ ̺0c4 1 cV − 1 cp

• p∼t

6

Derivation of Wave Equation

main physical quantities: c2 and ̺0 are known parameters

• v =

v∼ , p = p0 + p∼ , ∇p0 = 0 , ̺ = ̺0 + ̺∼ , ̺0t = 0 governing equations: nht . . . nonlinear and higher order terms ̺

• vt + ∇(

v · v)

• + ∇p =

4µV 3 + ζV

• ∆

v ̺t + ∇ · (̺ v) = 0 ̺∼ = p∼ c2 − 1 ̺0c4 B 2Ap2

∼ −

κ ̺0c4 1 cV − 1 cp

• p∼t

6

Derivation of Wave Equation

main physical quantities: c2 and ̺0 are known parameters

• v =

v∼ , p = p0 + p∼ , ∇p0 = 0 , ̺ = ̺0 + ̺∼ , ̺0t = 0 governing equations: nht . . . nonlinear and higher order terms ̺

• vt + ∇(

v · v)

• + ∇p =

4µV 3 + ζV

• ∆

v ̺0 vt + ∇p∼ = nht ̺t + ∇ · (̺ v) = 0 ̺∼ = p∼ c2 − 1 ̺0c4 B 2Ap2

∼ −

κ ̺0c4 1 cV − 1 cp

• p∼t

6

Derivation of Wave Equation

main physical quantities: c2 and ̺0 are known parameters

• v =

v∼ , p = p0 + p∼ , ∇p0 = 0 , ̺ = ̺0 + ̺∼ , ̺0t = 0 governing equations: nht . . . nonlinear and higher order terms ̺

• vt + ∇(

v · v)

• + ∇p =

4µV 3 + ζV

• ∆

v ̺0 vt + ∇p∼ = nht ̺t + ∇ · (̺ v) = 0 ̺∼t + ̺0∇ · v = nht ̺∼ = p∼ c2 − 1 ̺0c4 B 2Ap2

∼ −

κ ̺0c4 1 cV − 1 cp

• p∼t

6

Derivation of Wave Equation

main physical quantities: c2 and ̺0 are known parameters

• v =

v∼ , p = p0 + p∼ , ∇p0 = 0 , ̺ = ̺0 + ̺∼ , ̺0t = 0 governing equations: nht . . . nonlinear and higher order terms ̺

• vt + ∇(

v · v)

• + ∇p =

4µV 3 + ζV

• ∆

v ̺0 vt + ∇p∼ = nht ̺t + ∇ · (̺ v) = 0 ̺∼t + ̺0∇ · v = nht ̺∼ = p∼ c2 − 1 ̺0c4 B 2Ap2

∼ −

κ ̺0c4 1 cV − 1 cp

• p∼t

̺∼ = 1 c2 p∼ + nht

6

Derivation of Wave Equation

̺0 vt + ∇p∼ = nht ̺∼t + ̺0∇ · v = nht ̺∼ = 1

c2 p∼ + nht

7

Derivation of Wave Equation

̺0 vt + ∇p∼ = nht ̺∼t + ̺0∇ · v = nht ̺∼ = 1

c2 p∼ + nht

insert line 3 into line 2 to eliminate ̺∼. . .

7

Derivation of Wave Equation

̺0 vt + ∇p∼ = nht

1 c2 p∼t + ̺0∇ ·

v = nht

8

Derivation of Wave Equation

̺0 vt + ∇p∼ = nht

1 c2 p∼t + ̺0∇ ·

v = nht (This is an evolution with a nice skew-symmetric structure, since ∇· = −∇∗

0!)

8

Derivation of Wave Equation

̺0 vt + ∇p∼ = nht

1 c2 p∼t + ̺0∇ ·

v = nht

9

Derivation of Wave Equation

− ∇ · ̺0 vt + ∇p∼ = nht ∂ ∂t

1 c2 p∼t + ̺0∇ ·

v = nht

9

Derivation of Wave Equation

− ∇ · ̺0 vt + ∇p∼ = nht ∂ ∂t

1 c2 p∼t + ̺0∇ ·

v = nht —————————

1 c2 p∼tt − ∆p∼ = nht

9

Classical Models of Nonlinear Acoustics I

Kuznetsov’s equation [Lesser & Seebass 1968, Kuznetsov 1971] p∼tt − c2∆p∼ − b∆p∼t = −

• B

2A̺0c2 p2

∼ + ̺0|

v|2

• tt

where ̺0 vt = −∇p for the particle velocity v and the pressure p, i.e., ψtt − c2∆ψ − b∆ψt = −

• B

2A c2 (ψt)2 + |∇ψ|2

t

since ∇ × v = 0 hence v = −∇ψ for a velocity potential ψ Westervelt equation [Westervelt 1963] p∼tt − c2∆p∼ − b∆p∼t = − 1 ̺0c2

• 1 + B

2A

• p2

∼tt

via ̺0| v|2 ≈ 1

c2 (p∼t)2

10

Classical Models of Nonlinear Acoustics II

Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation [Zabolotskaya & Khokhlov 1969] 2cp∼xt − c2∆yzp∼ − b c2 p∼ttt = βa ̺0c2 p2

∼tt

• x. . . direction of sound propagation

Burgers’ equation [Burgers 1974] p∼t − b 2c2 p∼ττ = βa ̺0c3 p∼ p∼τ τ = t − x

c . . . retarded time

11

Advanced Models of Nonlinear Acoustics (Examples)

Blackstock-Crighton equation [Brunnhuber & Jordan 2016], [Blackstock 1963], [Crighton 1979]

ψttt −

• b +

ν

Pr

• ∆ψtt + (1+B/(2A))ν

Pr

• b −

νB 2APr

• ∆2ψt − c2∆ψt + c2 ν

Pr∆2ψ = −

• B

2Ac2 (ψ2 t ) + |∇ψ|2 tt 12

Advanced Models of Nonlinear Acoustics (Examples)

Blackstock-Crighton equation [Brunnhuber & Jordan 2016], [Blackstock 1963], [Crighton 1979]

ψttt −

• b +

ν

Pr

• ∆ψtt + (1+B/(2A))ν

Pr

• b −

νB 2APr

• ∆2ψt − c2∆ψt + c2 ν

Pr∆2ψ = −

• B

2Ac2 (ψ2 t ) + |∇ψ|2 tt

(∂t − a∆)

• ψtt − c2∆ψ − b∆ψt
• − r∆ψt = −

B 2Ac2 (ψ2

t ) + |∇ψ|2

• tt

a =

ν

• Pr. . . thermal conductivity

12

Advanced Models of Nonlinear Acoustics (Examples)

Blackstock-Crighton equation [Brunnhuber & Jordan 2016], [Blackstock 1963], [Crighton 1979]

ψttt −

• b +

ν

Pr

• ∆ψtt + (1+B/(2A))ν

Pr

• b −

νB 2APr

• ∆2ψt − c2∆ψt + c2 ν

Pr∆2ψ = −

• B

2Ac2 (ψ2 t ) + |∇ψ|2 tt

(∂t − a∆)

• ψtt − c2∆ψ − b∆ψt
• − r∆ψt = −

B 2Ac2 (ψ2

t ) + |∇ψ|2

• tt

a =

ν

• Pr. . . thermal conductivity

Jordan-Moore-Gibson-Thompson equation [Jordan 2009, 2014], [Christov 2009], [Straughan 2010] τψttt + ψtt − c2∆ψ − b∆ψt = − B 2Ac2 (ψt)2 + |∇ψ|2

• t

τ. . . relaxation time

12

Advanced Models of Nonlinear Acoustics (Examples)

Blackstock-Crighton equation [Brunnhuber & Jordan 2016], [Blackstock 1963], [Crighton 1979]

ψttt −

• b +

ν

Pr

• ∆ψtt + (1+B/(2A))ν

Pr

• b −

νB 2APr

• ∆2ψt − c2∆ψt + c2 ν

Pr∆2ψ = −

• B

2Ac2 (ψ2 t ) + |∇ψ|2 tt

(∂t − a∆)

• ψtt − c2∆ψ − b∆ψt
• − r∆ψt = −

B 2Ac2 (ψ2

t ) + |∇ψ|2

• tt

a =

ν

• Pr. . . thermal conductivity

Jordan-Moore-Gibson-Thompson equation [Jordan 2009, 2014], [Christov 2009], [Straughan 2010] τψttt + ψtt − c2∆ψ − b∆ψt = − B 2Ac2 (ψt)2 + |∇ψ|2

• t

τ. . . relaxation time

• cf. Kuznetsov’s equation:

ψtt − c2∆ψ − b∆ψt = − B 2Ac2 (ψ2

t ) + |∇ψ|2

• t

12

Some Asymptotics

Blackstock-Crighton equation:

• ψa

tt−c2∆ψa−(a+b)∆ψa t

• t+ad∆2ψa

t +ac2∆2ψa = −

• B

2Ac2 (ψa t 2) + |∇ψa|2 tt

Kuznetsov’s equation: ψtt − c2∆ψ − b∆ψt = −

• B

2Ac2 (ψ2 t ) + |∇ψ|2 t

13

Some Asymptotics

Blackstock-Crighton equation:

• ψa

tt−c2∆ψa−(a+b)∆ψa t

• t+ad∆2ψa

t +ac2∆2ψa = −

• B

2Ac2 (ψa t 2) + |∇ψa|2 tt

Kuznetsov’s equation: ψtt − c2∆ψ − b∆ψt = −

• B

2Ac2 (ψ2 t ) + |∇ψ|2 t

Existence of a limit ψ0 of ψa as a ց 0?

13

Some Asymptotics

Blackstock-Crighton equation:

• ψa

tt−c2∆ψa−(a+b)∆ψa t

• t+ad∆2ψa

t +ac2∆2ψa = −

• B

2Ac2 (ψa t 2) + |∇ψa|2 tt

Kuznetsov’s equation: ψtt − c2∆ψ − b∆ψt = −

• B

2Ac2 (ψ2 t ) + |∇ψ|2 t

Existence of a limit ψ0 of ψa as a ց 0? Does ψ0 solve Kuznetsov’s equation?

13

Theorem (BK&Thalhammer, 2016)

There exist ρT > 0, ρ0 = ρ0(ρT) > 0, such that for T and initial data satisfying ∆ψ0, ∆ψ1, ψ2 ∈ H1

0(Ω), ∆ψ0L2(Ω) + T ≤ ρT, and

∇ψ2L2(Ω) + ∇∆ψ1L2(Ω) + ∆ψ1L2(Ω) + ∇∆ψ0L2(Ω) ≤ ρ0, and ψ2 − c2∆ψ0 − b∆ψ1 = − B

Ac2 ψ2ψ1 − 2∇ψ1 · ∇ψ0, we get

ψa

∗

⇀ ψ0 in W 2

∞(0, T; H1(Ω))

ψa

∗

⇀ ψ0 in W 1

∞(0, T; H2(Ω))

ψa⇀ ψ0 in H2(0, T; H2(Ω)) ψa → ψ0 in C 1(0, T; C 0(Ω)) and in C 1(0, T; W 1

4 (Ω)) ,

14

Theorem (BK&Thalhammer, 2016)

There exist ρT > 0, ρ0 = ρ0(ρT) > 0, such that for T and initial data satisfying ∆ψ0, ∆ψ1, ψ2 ∈ H1

0(Ω), ∆ψ0L2(Ω) + T ≤ ρT, and

∇ψ2L2(Ω) + ∇∆ψ1L2(Ω) + ∆ψ1L2(Ω) + ∇∆ψ0L2(Ω) ≤ ρ0, and ψ2 − c2∆ψ0 − b∆ψ1 = − B

Ac2 ψ2ψ1 − 2∇ψ1 · ∇ψ0, we get

ψa

∗

⇀ ψ0 in W 2

∞(0, T; H1(Ω))

ψa

∗

⇀ ψ0 in W 1

∞(0, T; H2(Ω))

ψa⇀ ψ0 in H2(0, T; H2(Ω)) ψa → ψ0 in C 1(0, T; C 0(Ω)) and in C 1(0, T; W 1

4 (Ω)) ,

T

• Ω
• ψ0

ttv + b ∇ψ0 t · ∇v + c2 ∇ψ0 · ∇v

• dx dt

= − T

• Ω
• B

2Ac2 (ψ0 t 2) +

• ∇ψ0

2

t v dx dt

for all v ∈ L1(0, T; H1

0(Ω)) .

idea of proof:

14

Models and their Analysis

further models:[Angel & Aristegui 2014], [Christov & Christov & Jordan 2007], [Kudryashov & Sinelshchikov 2010], [Ockendon & Tayler 1983], [Makarov & Ochmann 1996], [Rend´

• n & Ezeta &P´

erez-L´

• pez 2013], [Rasmussen &

Sørensen & Christiansen 2008], [Soderholm 2006], . . . resonances, shock waves:[Ockendon & Ockendon & Peake & Chester 1993], [Ockendon & Ockendon 2001, 2004, 2016],. . . traveling waves solutions:[Jordan 2004], [Chen & Torres & Walsh 2009], [Keiffer & McNorton & Jordan & Christov, 2014], [Gaididei & Rasmussen & Christiansen & Sørensen, 2016],. . . well-posendness and asymptotic behaviour: for KZK: [Rozanova-Pierrat 2007, 2008, 2009, 2010] for Westervelt, Kuznetsov, Blackstock-Crighton, JMGT: based on semigroup theory and energy estimates:[BK & Lasiecka 2009, 2012], [BK & Lasiecka & Veljovi´ c 2011], [BK & Lasiecka & Marchand 2012], [BK & Lasiecka & Pospiezalska 2012], [Lasiecka & Wang 2015], [Liu & Triggiani 2013], [Marchand & McDevitt & Triggiani 2012], [Nikoli´ c 2015], [Nikoli´ c & BK 2016] based on maximal Lp regularity:[Meyer & Wilke 2011, 2013], [Meyer & Simonett 2016], [Brunnhuber & Meyer 2016], [BK 2016]

15

Potential Degeneracy

e.g., for Westervelt ptt − c2∆p − b∆pt = − k

2(p2)tt

16

Potential Degeneracy

e.g., for Westervelt ptt − c2∆p − b∆pt = − k

2(p2)tt = −kp ptt − k(pt)2

16

Potential Degeneracy

e.g., for Westervelt ptt − c2∆p − b∆pt = − k

2(p2)tt = −kp ptt − k(pt)2

(1 + kp)ptt − c2∆p − b∆pt = −k(pt)2

16

Potential Degeneracy

e.g., for Westervelt ptt − c2∆p − b∆pt = − k

2(p2)tt = −kp ptt − k(pt)2

(1 + kp)ptt − c2∆p − b∆pt = −k(pt)2 ⇒ degeneracy for u ≤ − 1

k

16

Potential Degeneracy

e.g., for Westervelt ptt − c2∆p − b∆pt = − k

2(p2)tt = −kp ptt − k(pt)2

(1 + kp)ptt − c2∆p − b∆pt = −k(pt)2 ⇒ degeneracy for u ≤ − 1

k

similarly for Kuznetsov, Jordan-Moore-Gibson-Thompson, Blackstock-Crighton.

16

Potential Degeneracy

e.g., for Westervelt ptt − c2∆p − b∆pt = − k

2(p2)tt = −kp ptt − k(pt)2

(1 + kp)ptt − c2∆p − b∆pt = −k(pt)2 ⇒ degeneracy for u ≤ − 1

k

similarly for Kuznetsov, Jordan-Moore-Gibson-Thompson, Blackstock-Crighton. employ energy estimates to obtain bound on p in C(0, T; H2(Ω)) use smallness of p in C(0, T; H2(Ω)) and H2(Ω) → L∞(Ω) embedding to guarantee 1 + kp ≥ α > 0

16

Potential Degeneracy

e.g., for Westervelt ptt − c2∆p − b∆pt = − k

2(p2)tt = −kp ptt − k(pt)2

(1 + kp)ptt − c2∆p − b∆pt = −k(pt)2 ⇒ degeneracy for u ≤ − 1

k

similarly for Kuznetsov, Jordan-Moore-Gibson-Thompson, Blackstock-Crighton. employ energy estimates to obtain bound on p in C(0, T; H2(Ω)) use smallness of p in C(0, T; H2(Ω)) and H2(Ω) → L∞(Ω) embedding to guarantee 1 + kp ≥ α > 0 fixed point argument

16

time integration

17

Westervelt Equation as a Nonlinear Evolutionary System

ψtt − b 1 + kψt ∆ ψt − c2 1 + kψt ∆ ψ = 0 + homogeneous Dirichlet boundary conditions + initial conditions

18

Westervelt Equation as a Nonlinear Evolutionary System

ψtt − b 1 + kψt ∆ ψt − c2 1 + kψt ∆ ψ = 0 + homogeneous Dirichlet boundary conditions + initial conditions abstract Cauchy problem for u : [0, T] → X : t → u(t) = ψ(·, t) ψt(·, t)

• :
• ut(t) = F
• u(t)
• ,

t ∈ (0, T] u(0) = u0 , where F : D(F) → X (quasilinear)

b(v2) =

b 1+k v2 ,

c2(v2) =

c2 1+k v2

F(v) =

• v2
• b(v2) ∆v2 +

c2(v2) ∆v1

• =
• I
• c2(v2) ∆
• b(v2) ∆

v1 v2

• 18

Time-Splitting Methods

ut(t) = F

• u(t)
• ,

t ∈ (0, T] , (1) u : [0, T] → X splitting: F = A + B efficient numerical solvers available for subproblems vt(t) = A

• v(t)
• ,

wt(t) = B

• w(t)
• ,

t ∈ (0, T] . (2) e.g., first-order Lie –Trotter splitting method: u(tn+1) ≈ uh(tn+1) = w(tn+1) where vt(t) = A(t, v(t)) , t ∈ (tn, tn+1) , v(tn) = uh(tn) wt(t) = B(t, w(t)) , t ∈ (tn, tn+1) , w(tn) = v(tn+1)

19

Time-Splitting Methods

ut(t) = F

• u(t)
• ,

t ∈ (0, T] , (1) u : [0, T] → X splitting: F = A + B efficient numerical solvers available for subproblems vt(t) = A

• v(t)
• ,

wt(t) = B

• w(t)
• ,

t ∈ (0, T] . (2) e.g., first-order Lie –Trotter splitting method: u(tn+1) ≈ uh(tn+1) = w(tn+1) where vt(t) = A(t, v(t)) , t ∈ (tn, tn+1) , v(tn) = uh(tn) wt(t) = B(t, w(t)) , t ∈ (tn, tn+1) , w(tn) = v(tn+1) also higher order splitting methods, e.g., Strang

19

Decompositions for Westervelt

F(v) =

• v2
• b(v2) ∆v2 +

c2(v2) ∆v1

• F = A + B

Decomposition I: Decomposition II: A(v) =

• v2
• b(v2) ∆v2
• A(v) =
• 1

2 v2

• b(v2) ∆v2
• B(v) =
• c2(v2) ∆v1
• B(v) =
• 1

2 v2

• c2(v2) ∆v1
• Decomposition III:

Decomposition VI: A(v) =

• b(v2) ∆v2
• A(v) =
• b(v2) ∆v2 − k v2

c2(v2) ∆v1

• B(v) =
• v2
• c2(v2) ∆v1
• B(v) =
• v2

c2 ∆v1

• 20

Decompositions for Westervelt

F(v) =

• v2
• b(v2) ∆v2 +

c2(v2) ∆v1

• F = A + B

Decomposition I: Decomposition II: A(v) =

• v2
• b(v2) ∆v2
• A(v) =
• 1

2 v2

• b(v2) ∆v2
• B(v) =
• c2(v2) ∆v1
• B(v) =
• 1

2 v2

• c2(v2) ∆v1
• Decomposition III:

Decomposition VI: A(v) =

• b(v2) ∆v2
• A(v) =
• b(v2) ∆v2 − k v2

c2(v2) ∆v1

• B(v) =
• v2
• c2(v2) ∆v1
• B(v) =
• v2

c2 ∆v1

• [BK & Nikoli´

c & Thalhammer 2014]

20

Decomposition I:

subproblems to be solved: A :

• Ψ1t(x, t) = Ψ2(x, t) ,

Ψ2t(x, t) =

b 1+k Ψ2(x,t)∆Ψ2(x, t) ,

nonlinear diffusion equation for second component Ψ2; plain time integration to obtain first component Ψ1 from Ψ2 B :

• Ψ1t(x, t) = 0 ,

Ψ2t(x, t) =

c2 1+k Ψ2(x,t)∆Ψ1(x, t) ,

first component Ψ1 remains constant; explicit representation for second component:

Ψ2(x, t) = − 1

k

• 1 −
• 1 + k Ψ2(x, 0)

2 + 2 c2k t ∆Ψ1(x, 0)

• 21

Westervelt Equation: Well-Posedness and Regularity

Theorem (nondegeneracy, well-posedness, exp. decay; BK&Lasiecka’09)

There exist constant ρ, M, ω > 0 such that if E[ψ](0) < ρ, then for all t > 0 the solution to the Westervelt equation exists, is unique, stays bounded E[ψ](t) < M, as well as nondegenerate 0 < α ≤ 1 + k ψt(x, t) ≤ α < ∞ , (x, t) ∈ Ω × [0, T] , and decays exponentially E[ψ](t) < C e−ωtE[ψ](0) , where E[ψ](t) := 1 2

• ψttt2

L2(Ω) + ∇ψtt2 L2(Ω) + ∆ψt2 L2(Ω)

• Higher regularity [BK& Nikoli´

c&Thalhammer’14]

• ψ0, ψ1
• ∈ H2(m+1)(Ω)×H2m+1(Ω) , ∆lψtt(·, 0) ∈ H1

0(Ω) , l ∈ {1, . . . m} ,

⇒

• ψ, ψt
• ∈ C
• [0, T], H2(m+1)(Ω) × H2m+1(Ω)
• 22

Convergence of Lie-Trotter Splitting

Theorem (convergence Lie –Trotter, decomposition I, KNT’14)

Assume that

• (ψ0, ψ1)
• ∈ H5(Ω) × H3(Ω).

Then the global error estimate ((ψN, ψtN) − (ψ(tN), ψt(tN))H3(Ω)×H1(Ω) ≤ C

• (ψ0, ψ1) − (ψ(0), ψt(0))H3(Ω)×H1(Ω) + h
• , 0 ≤ tN ≤ T

holds with constant depending on bounds for

• (ψ(0), ψt(0))
• H6(Ω)×H5(Ω) as well as
• (ψ0, ψ1)
• H5(Ω)×H3(Ω)

and the final time T.

23

Numerical Experiments

1-d Test problem: unit parameters: b = 1 , c2 = 1 , k = −1 , homogeneous Dirichlet conditions : ψ(−a, t) = 0 = ψ(a, t) , t ∈ [0, T] , regular initial conditions: ψ(x, 0) = exp(−x2) , ψt(x, 0) = − x exp(−x2) , x ∈ [−a, a] . a = 8, T = 1

24

Local (left) and global (right) errors for the Lie–Trotter and Strang splitting methods with respect to the L2 × L2-norm obtained for Decomposition I. Comparison of different time integration methods for the numerical solution of the subproblems.

25

Local (left) and global (right) errors for the Lie–Trotter and Strang splitting methods with respect to the H3 × H1-norm obtained for Decomposition I. Comparison of different time integration methods for the numerical solution of the subproblems.

26

nonreflecting boundary conditions

27

Sound-Hard versus Absorbing Boundary Conditions

28

Sound-Hard versus Absorbing Boundary Conditions

29

Sound-Hard versus Absorbing Boundary Conditions

30

Sound-Hard versus Absorbing Boundary Conditions

31

Sound-Hard versus Absorbing Boundary Conditions

32

Sound-Hard versus Absorbing Boundary Conditions

33

Sound-Hard versus Absorbing Boundary Conditions

34

Sound-Hard versus Absorbing Boundary Conditions

35

Absorbing Boundary Conditions: a selection of references

Engquist& Majda 1977 Givoli, 1991, 2004 Ha-Duong & Joly 1994 Hagstrom, Acta Numerica 1999

• E. B´

ecache, D. Givoli, T. Hagstrom 2010

• J. Szeftel 2006 (semilinear wave equation)

. . .

36

Absorbing Boundary Conditions for the Wave Equation

1-dim. linear case: utt − c2uxx = 0 on [−1, 1] (∂t − c∂x)(∂t + c∂x)u = (∂t + c∂x)(∂t − c∂x)u = 0

37

Absorbing Boundary Conditions for the Wave Equation

1-dim. linear case: utt − c2uxx = 0 on [−1, 1] (∂t − c∂x)(∂t + c∂x)u = (∂t + c∂x)(∂t − c∂x)u = 0 annihilate reflected waves: (∂t − c∂x)u = 0 at x = −1 (∂t + c∂x)u = 0 at x = 1

37

Absorbing Boundary Conditions for the Wave Equation

1-dim. linear case: utt − c2uxx = 0 on [−1, 1] (∂t − c∂x)(∂t + c∂x)u = (∂t + c∂x)(∂t − c∂x)u = 0 annihilate reflected waves: (∂t − c∂x)u = 0 at x = −1 (∂t + c∂x)u = 0 at x = 1 d-dim. linear case: utt − c2∆u = 0 in Ω ⊆ Rd un = − 1

c ut on ∂Ω

37

Absorbing Boundary Conditions for the Wave Equation

1-dim. linear case: utt − c2uxx = 0 on [−1, 1] (∂t − c∂x)(∂t + c∂x)u = (∂t + c∂x)(∂t − c∂x)u = 0 annihilate reflected waves: (∂t − c∂x)u = 0 at x = −1 (∂t + c∂x)u = 0 at x = 1 d-dim. linear case: utt − c2∆u = 0 in Ω ⊆ Rd un = − 1

c ut on ∂Ω

decreases energy

37

Absorbing Boundary Conditions for the Wave Equation

1-dim. linear case: utt − c2uxx = 0 on [−1, 1] (∂t − c∂x)(∂t + c∂x)u = (∂t + c∂x)(∂t − c∂x)u = 0 annihilate reflected waves: (∂t − c∂x)u = 0 at x = −1 (∂t + c∂x)u = 0 at x = 1 d-dim. linear case: utt − c2∆u = 0 in Ω ⊆ Rd un = − 1

c ut on ∂Ω

decreases energy 2-dim. nonlinear case: (1 + ku)utt − c2∆u − b∆ut = −k(ut)2 in Ω ⊆ R2 zero order: un = − 1

c

√ 1 + ku ut on ∂Ω first order: un tt = − 1

c

√ 1 + ku uttt + 1

2c

√ 1 + kuuϑϑt + . . . on ∂Ω

37

Absorbing Boundary Conditions for the Wave Equation

1-dim. linear case: utt − c2uxx = 0 on [−1, 1] (∂t − c∂x)(∂t + c∂x)u = (∂t + c∂x)(∂t − c∂x)u = 0 annihilate reflected waves: (∂t − c∂x)u = 0 at x = −1 (∂t + c∂x)u = 0 at x = 1 d-dim. linear case: utt − c2∆u = 0 in Ω ⊆ Rd un = − 1

c ut on ∂Ω

decreases energy 2-dim. nonlinear case: (1 + ku)utt − c2∆u − b∆ut = −k(ut)2 in Ω ⊆ R2 zero order: un = − 1

c

√ 1 + ku ut on ∂Ω first order: un tt = − 1

c

√ 1 + ku uttt + 1

2c

√ 1 + kuuϑϑt + . . . on ∂Ω

37

Absorbing Boundary Conditions for the Wave Equation

1-dim. linear case: utt − c2uxx = 0 on [−1, 1] (∂t − c∂x)(∂t + c∂x)u = (∂t + c∂x)(∂t − c∂x)u = 0 annihilate reflected waves: (∂t − c∂x)u = 0 at x = −1 (∂t + c∂x)u = 0 at x = 1 d-dim. linear case: utt − c2∆u = 0 in Ω ⊆ Rd un = − 1

c ut on ∂Ω

decreases energy 2-dim. nonlinear case: (1 + ku)utt − c2∆u − b∆ut = −k(ut)2 in Ω ⊆ R2 zero order: un = − 1

c

√ 1 + ku ut on ∂Ω first order: un tt = − 1

c

√ 1 + ku uttt + 1

2c

√ 1 + kuuϑϑt + . . . on ∂Ω (formal) pseudodifferential calculus to decompose differential operator + energy estimates well-posedness

37

Absorbing Boundary Conditions for the Wave Equation

1-dim. linear case: utt − c2uxx = 0 on [−1, 1] (∂t − c∂x)(∂t + c∂x)u = (∂t + c∂x)(∂t − c∂x)u = 0 annihilate reflected waves: (∂t − c∂x)u = 0 at x = −1 (∂t + c∂x)u = 0 at x = 1 d-dim. linear case: utt − c2∆u = 0 in Ω ⊆ Rd un = − 1

c ut on ∂Ω

decreases energy 2-dim. nonlinear case: (1 + ku)utt − c2∆u − b∆ut = −k(ut)2 in Ω ⊆ R2 zero order: un = − 1

c

√ 1 + ku ut on ∂Ω first order: un tt = − 1

c

√ 1 + ku uttt + 1

2c

√ 1 + kuuϑϑt + . . . on ∂Ω (formal) pseudodifferential calculus to decompose differential operator + energy estimates well-posedness [BK&Shevchenko 2015,’16]

37

Numerical Experiments

Ω ΓA ΓA ΓA ΓN geometric setup for the high-intensity focused ultrasound problem.

38

(a) (b) (d) (e) (a) reference solution (b) ABC2,1

lin , (d) ABC2,0 nl , (e) ABC2,1 nl .

39

Outlook

coupling of nonlinearly acoustic fluid with (linearly) elastic lens

• ptimal boundary control (excitation)

shape optimization of focusing lens: shape derivatives → implementation with isogeometric FE → modeling and analysis: temperature coupling, cavitation, fractional order damping → analysis and numerics for first order evolutionary system

40

Thank you for your attention!

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