Schwarz waveform relaxation algorithms : theory and applications - - PowerPoint PPT Presentation

Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Schwarz waveform relaxation algorithms : theory and applications

Laurence HALPERN

LAGA - Universit´ e Paris 13

• DD17. July 2006

1 / 29

Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Outline

1

Introduction

2

The SWR algorithm for advection diffusion equation Description Numerical experiments Back to the theoretical problem

3

The two-dimensional wave equation Dirichlet transmission conditions Optimized algorithms with overlap

4

Conclusion und perspectives

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Coupling process

Issues ♦ For a given problem, split the domain : domain decomposition.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Coupling process

Issues ♦ For a given problem, split the domain : domain decomposition. ♦ For a given problem, different numerical methods in different zones : FEM/FD, SM/FEM, AMR.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Coupling process

Issues ♦ For a given problem, split the domain : domain decomposition. ♦ For a given problem, different numerical methods in different zones : FEM/FD, SM/FEM, AMR. ♦ Couple two different models in different zones.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Coupling process

Issues ♦ For a given problem, split the domain : domain decomposition. ♦ For a given problem, different numerical methods in different zones : FEM/FD, SM/FEM, AMR. ♦ Couple two different models in different zones. ♦ Furthermore the codes can be on distant sites.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

DDM for evolution problems

Usual methods ⋄ Explicit + interpolation − > exchange of information every time step − > time consuming, possibly unstable for hyperbolic problems.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

DDM for evolution problems

Usual methods ⋄ Explicit + interpolation − > exchange of information every time step − > time consuming, possibly unstable for hyperbolic problems. ⋄ Implicit − > uniform time step.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

DDM for evolution problems

The goals ⋄ Different time and space steps in different subdomains, ⋄ Different models in different subdomains, ⋄ Different computing sites, ⋄ Easy to use, fast and cheap.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

DDM for evolution problems

The goals ⋄ Different time and space steps in different subdomains, ⋄ Different models in different subdomains, ⋄ Different computing sites, ⋄ Easy to use, fast and cheap. The means ⋄ Work on the PDE level, globally in time, ⋄ Split the space domain, ⋄ Use time windows, ⋄ Use the physical transmission conditions, transmit with improved (optimal/optimized) transmission conditions. ⋄ Then discretize separately..

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

DDM for evolution problems

The goals ⋄ Different time and space steps in different subdomains, ⋄ Different models in different subdomains, ⋄ Different computing sites, ⋄ Easy to use, fast and cheap. The means ⋄ Work on the PDE level, globally in time, ⋄ Split the space domain, ⋄ Use time windows, ⋄ Use the physical transmission conditions, transmit with improved (optimal/optimized) transmission conditions. ⋄ Then discretize separately.. Optimized Schwarz Waveform Relaxation

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Outline

1

Introduction

2

The SWR algorithm for advection diffusion equation Description Numerical experiments Back to the theoretical problem

3

The two-dimensional wave equation Dirichlet transmission conditions Optimized algorithms with overlap

4

Conclusion und perspectives

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Outline

1

Introduction

2

The SWR algorithm for advection diffusion equation Description Numerical experiments Back to the theoretical problem

3

The two-dimensional wave equation Dirichlet transmission conditions Optimized algorithms with overlap

4

Conclusion und perspectives

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

The Schwarz algorithm

Lu := ∂tu + a∂xu + (b · ∇)u − ν∆u + cu in Ω × (0, T) ν > 0.

t Ω1 Γ21 Ω2 Γ12

8 < : Luk+1

1

= f in Ω1 × (0, T) uk+1

1

(·, 0) = u0 in Ω1 B1uk+1

1

= B1uk

2

• n Γ12 × (0, T)

8 < : Luk+1

2

= f in Ω2 × (0, T) uk+1

2

(·, 0) = u0 in Ω2 B2uk+1

2

= B2uk

1

• n Γ21 × (0, T)

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

How to choose the transmission operators ?

Transmission conditions B1uk+1

1

= B1uk

2 on Γ12 × (0, T),

B2uk+1

2

= B2uk

1 on Γ21 × (0, T)

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

How to choose the transmission operators ?

Transmission conditions B1uk+1

1

= B1uk

2 on Γ12 × (0, T),

B2uk+1

2

= B2uk

1 on Γ21 × (0, T)

Classical Schwarz Bj ≡ I AND overlap.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

How to choose the transmission operators ?

Classical Schwarz Bj ≡ I AND overlap. 1D Numerical experiment a = 1, ν = 0.2, Ω = (0, 6), T = 2.5, L = 0.08. u1

1(·, T), u2 2(·, T)

1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 u u1 1 u2 2

u3

1(·, T), u4 2(·, T)

1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 u u1 3 u2 4

u5

1(·, T), u6 2(·, T)

1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 u u1 5 u2 6

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

How to choose the transmission operators ?

Transmission conditions B1uk+1

1

= B1uk

2 on Γ12 × (0, T),

B2uk+1

2

= B2uk

1 on Γ21 × (0, T)

Classical Schwarz Bj ≡ I AND overlap. Optimized Schwarz Waveform relaxation Bj ≡ absorbing boundary operator+optimization WITH OR WITHOUT

• verlap

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

How to choose the transmission operators ?

Comparison u1

1(·, T), u2 2(·, T)

1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 u u1 1 u2 2

u3

1(·, T), u4 2(·, T)

1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 u u1 3 u2 4

u5

1(·, T), u6 2(·, T)

1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 u u1 5 u2 6

u1

1(·, T), u2 2(·, T)

1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 u u1 1 u2 2

u3

1(·, T), u4 2(·, T)

1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 u u1 3 u2 4

u5

1(·, T), u6 2(·, T)

1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 u u1 5 u2 6

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

The optimal SWR algorithm

Ω1 = (−∞, L) × Rn, Ω2 = (0, ∞) × Rn. Bj ≡ ∂x + Sj(∂t, ∂y) a > 0, Fourier transform t ↔ ω, y ↔ κ S1(iω, iκ) = δ1/2 − a 2ν , S2(iω, iκ) = δ1/2 + a 2ν . δ(ω, k) = a2 + 4ν((i(ω + b · k) + ν|k|2 + c) Convergence in 2 iterations (I if I subdomains). Two options : Use the optimal transmission condition (easier in 1D)

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

The optimal SWR algorithm

Ω1 = (−∞, L) × Rn, Ω2 = (0, ∞) × Rn. Bj ≡ ∂x + Sj(∂t, ∂y) a > 0, Fourier transform t ↔ ω, y ↔ κ S1(iω, iκ) = δ1/2 − a 2ν , S2(iω, iκ) = δ1/2 + a 2ν . δ(ω, k) = a2 + 4ν((i(ω + b · k) + ν|k|2 + c) Convergence in 2 iterations (I if I subdomains). Two options : Use the optimal transmission condition (easier in 1D) Approximate the optimal − > optimized transmission conditions

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Design of approximate SWR algorithms

boundary operators δ(ω, k) := a2 + 4ν((i(ω + b · k) + ν|k|2 + c) S1(iω, iκ) = δ1/2 − a 2ν , δ(ω, k) = a2 + 4ν((i(ω + b · k) + ν|k|2 + c) ˜ S1(iω, iκ) = P − a 2ν , P(ω, k) = p + q(i(ω + b · k) + ν|k|2), (p, q) ∈ R2. B1u := ∂xu − a − p 2ν u + q(∂t + b · ∇u − ν∆yu)

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Well-posedness and convergence

Transmission conditions B1u := ∂xu − a − p 2ν u + q(∂t + b · ∇u − ν∆yu) B2u := ∂xu − a + p 2ν u − q(∂t + b · ∇u − ν∆yu)

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Well-posedness and convergence

Transmission conditions B1u := ∂xu − a − p 2ν u + q(∂t + b · ∇u − ν∆yu) B2u := ∂xu − a + p 2ν u − q(∂t + b · ∇u − ν∆yu) Convergence factor ρ(ω, k, P, L) = P − δ1/2 P + δ1/2 2 e−2δ1/2L/ν

• ek+2

j

(ω, 0, k) = ρ(ω, k, P, L) ek

j (ω, 0, k)

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Well-posedness and convergence

Transmission conditions B1u := ∂xu − a − p 2ν u + q(∂t + b · ∇u − ν∆yu) B2u := ∂xu − a + p 2ν u − q(∂t + b · ∇u − ν∆yu) Convergence factor ρ(ω, k, P, L) = P − δ1/2 P + δ1/2 2 e−2δ1/2L/ν

• ek+2

j

(ω, 0, k) = ρ(ω, k, P, L) ek

j (ω, 0, k)

theorem For p, q > 0, p > a2

4ν q, the algorithm is well-posed in suited Sobolev

spaces and converges with and without overlap.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

One dimension : influence of the parameters

0.5 1 1.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 − 1 − 9 −9 −8 − 8 −7 −7 −7 − 7 −6 − 6 −6 −6 −6 − 5 −5 −5 −5 − 5 − 5 − 4 − 4 −4 −4 −3 − 2

p q

Error obtained running the algorithm with first order transmission conditions for 5 steps and various choices of p and q. p∗, q∗ : theoretical values , p∗, q∗ : Taylor approximations.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

One dimension : comparison

2 4 6 8 10 10

−10

10

−5

10

iteration error

2 SUBDOMAINS CLASSICAL OPTIMIZED ORDER 1 2 4 6 8 10 10

−10

10

−5

10

iteration error

4 SUBDOMAINS 2 4 6 8 10 10

−10

10

−5

10

iteration error

8 SUBDOMAINS

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Two dimensions : coupling different numerical methods

The heat bubble hitting an airfoil

−2 −1 1 2 3 4 5 −2 −1 1 2 3 4 5 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Evolution of a heat bubble around an airfoil. Coupling through Corba, “Common Object Request Broker Architecture”.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Two dimensions : coupling different numerical methods

Programming F.E in Ω1, F.D in Ω2, Write the interface problem, solve by Krylov, Results for a time window=10 timesteps the steady algorithm is : do time iterations 1 :N do Krylov iterations with preconditioning residual vectors = size of interface 15 iterations ×10. the unsteady algorithm is : do Krylov iterations do time iterations 1 :N residual vectors = size of interface x N 100 iterations.

P.d’Anfray, J. Ryan,L.H. M2AN 2002

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Robustness : rotating velocities

a(x, y) = 0.32π sin(4πx) sin(4πy), b(x, y) = 0.32π cos(4πy) cos(4πx).

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 10 20 30 40 50 −6 −5 −4 −3 −2 −1 1 2 3 4 Iterations Er Nu=0.1 Taylor Ordre0 Ordre0 Optimisé Taylor Ordre1 Ordre1 Optimisé

interface 0.3

10 20 30 40 50 −6 −5 −4 −3 −2 −1 1 2 3 4 Iterations Er Nu=0.1 Taylor Ordre0 Ordre0 Optimisé Taylor Ordre1 Ordre1 Optimisé

interface 0.4

10 20 30 40 50 −6 −5 −4 −3 −2 −1 1 2 3 4 Iterations Er Nu=0.1 Taylor Ordre0 Ordre0 Optimisé Taylor Ordre1 Ordre1 Optimisé

interface 0.5

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Optimization of the convergence factor

δ(z) = a2 + 4νc + 4νz, z = i(ω + b · k) + ν|k|2 ρ(z, P, L) = P(z) − δ1/2(z) P(z) + δ1/2(z) 2 e−2δ1/2L

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Optimization of the convergence factor

δ(z) = a2 + 4νc + 4νz, z = i(ω + b · k) + ν|k|2 ρ(z, P, L) = P(z) − δ1/2(z) P(z) + δ1/2(z) 2 e−2δ1/2L Taylor expansion,P(z) =

• δ(0) + 2νz/
• δ(0),

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Optimization of the convergence factor

δ(z) = a2 + 4νc + 4νz, z = i(ω + b · k) + ν|k|2 ρ(z, P, L) = P(z) − δ1/2(z) P(z) + δ1/2(z) 2 e−2δ1/2L Taylor expansion,P(z) =

• δ(0) + 2νz/
• δ(0),

Best approximation inf

P∈Pn sup z∈K

|ρ(z, P, L)|, K = ( π T , π ∆t ), kj ∈ ( π Xj , π ∆xj )

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Optimization of the convergence factor

δ(z) = a2 + 4νc + 4νz, z = i(ω + b · k) + ν|k|2 ρ(z, P, L) = P(z) − δ1/2(z) P(z) + δ1/2(z) 2 e−2δ1/2L Taylor expansion,P(z) =

• δ(0) + 2νz/
• δ(0),

Best approximation inf

P∈Pn sup z∈K

|ρ(z, P, L)|, K = ( π T , π ∆t ), kj ∈ ( π Xj , π ∆xj ) theorem For any n, for L = 0 or sufficiently small, the problem has a unique solution characterized by an equioscillation property.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Asymptotic results

Example : overlapping case, L ≈ C∆x Dirichlet transmission conditions : |ρ| ≈ 1 − α∆x, Taylor approximation : |ρ| ≈ 1 − β √ ∆x, Optimization : p ≈ Cp∆x− 1

5 , q ≈ Cq∆x 3 5 , |ρ| ≈ 1 − O(∆x 1 5 ). 18 / 29

Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Outline

1

Introduction

2

The SWR algorithm for advection diffusion equation Description Numerical experiments Back to the theoretical problem

3

The two-dimensional wave equation Dirichlet transmission conditions Optimized algorithms with overlap

4

Conclusion und perspectives

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Schwarz Waveform relaxation algorithm

Lu := utt − c2∆u, x ∈ Ω ⊂ Rm

t Ω1 Γ21 Ω2 Γ12

8 < : Luk+1

1

= f in Ω1 × (0, T) uk+1

1

(·, 0) = u0 in Ω1 B1uk+1

1

(L, ·) = B1uk

2(L, ·)

in (0, T) 8 < : Luk+1

2

= f in Ω2 × (0, T) uk+1

2

(·, 0) = u0 in Ω2 B2uk+1

2

(0, ·) = B2uk

1(0, ·)

in (0, T)

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

A numerical experiment

Data c = 1, T = 1, Ω = (0, 1) × (0, 1). Two subdomains, overlap L = 0.08.

0.2 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 1.2 1.4 x y u0

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

A numerical experiment

Data c = 1, T = 1, Ω = (0, 1) × (0, 1). Two subdomains, overlap L = 0.08.

0.2 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 1.2 1.4 x y u0

Convergence history : Dirichlet transmission conditions with overlap

2 4 6 8 10 12 14 16 18 20 10

10

10

10

10

10

10 10

error n

Convergence after n > cT

L = 12 iterations

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Other transmission conditions

General transmission operators B1 =

J

• j=1

(∂x + αj∂t), B2 =

J

• j=1

(∂x − αj∂t).

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Other transmission conditions

General transmission operators B1 =

J

• j=1

(∂x + αj∂t), B2 =

J

• j=1

(∂x − αj∂t). Plane waves analysis

ek

1 = ak 1(ω, k)eσ(x−L), ek 2 = ak 2(ω, k)eσx.

σ = 8 < :

|ω| c

q` ck

ω

´2 − 1, evanescent waves,

iω c

q 1 − ` ck

ω

´2, propagating waves. |ρ| = 8 > > > > < > > > > : e

−L |ω|

c

r`

ck ω

´2

−1,

evanescent waves,

J

Y

j=1

˛ ˛ ˛ ˛ ˛ αj − q 1 − ` ck

ω

´2 αj + q 1 − ` ck

ω

´2 ˛ ˛ ˛ ˛ ˛ propagating waves.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Plane wave analysis : continue

Convergence factor, propagating case θ angle of incidence on the interface, sin θ = ck

ω .

|ρ| =

J

• j=1
• αj −
• 1 −

ck

ω

2 αj +

• 1 −

ck

ω

2

• =

J

• j=1
• αj − cos θ

αj + cos θ

• .

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Plane wave analysis : continue

Convergence factor, propagating case θ angle of incidence on the interface, sin θ = ck

ω .

|ρ| =

J

• j=1
• αj −
• 1 −

ck

ω

2 αj +

• 1 −

ck

ω

2

• =

J

• j=1
• αj − cos θ

αj + cos θ

• .

Strategy 1 : orthogonal absorption α = 1

5 10 15 10

−16

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10 10

2

iteration Linf error at T Classical Schwarz Orthogonal 1st Order Orthogonal 2nd Order

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Plane wave analysis : continue

Convergence factor, propagating case θ angle of incidence on the interface, sin θ = ck

ω .

|ρ| =

J

• j=1
• αj −
• 1 −

ck

ω

2 αj +

• 1 −

ck

ω

2

• =

J

• j=1
• αj − cos θ

αj + cos θ

• .

Strategy 2 : optimization Given eps, find n and α(n) such that

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Plane wave analysis : continue

Convergence factor, propagating case θ angle of incidence on the interface, sin θ = ck

ω .

|ρ| =

J

• j=1
• αj −
• 1 −

ck

ω

2 αj +

• 1 −

ck

ω

2

• =

J

• j=1
• αj − cos θ

αj + cos θ

• .

Strategy 2 : optimization Given eps, find n and α(n) such that

1

The overlap takes care of the wide angles θ ≥ θmax(n) = arccos( nL

cT ),

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Plane wave analysis : continue

Convergence factor, propagating case θ angle of incidence on the interface, sin θ = ck

ω .

|ρ| =

J

• j=1
• αj −
• 1 −

ck

ω

2 αj +

• 1 −

ck

ω

2

• =

J

• j=1
• αj − cos θ

αj + cos θ

• .

Strategy 2 : optimization Given eps, find n and α(n) such that

1

The overlap takes care of the wide angles θ ≥ θmax(n) = arccos( nL

cT ),

2

the convergence rate ρ is optimized by ρ(θmax(n))n < eps.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Comparison

Example : eps = 10−2 First order : n = 3.7459 ≈ 3 − 4 ,θmax ≈ 73o. Second order : n = 1.9540 ≈ 2,θmax ≈ 81o.

Iteration 1 2 3 4 5 Dirichlet 0.7059 1.0555 0.8146 0.7340 0.7321 0.5760 Orthogonal O1 0.7059 0.5793 0.2035 0.0413 0.0061 0.0010 Optimized O1 0.7059 0.4403 0.1132 0.0216 0.0062 0.0018 Orthogonal O2 0.7059 0.5853 0.0701 0.0045 0.0003 0.0000 Optimized O2 0.7059 0.5847 0.0415 0.0099 0.0030 0.0004

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Theoretical results

Continuous level Well-posedness of the best approximation problems (explicit), Well-posedness of the subdomain problems (Kreiss theory), Convergence of the algorithm (Fourier analysis, “` a la” Engquist-Majda). Discrete level Discretization by finite volumes schemes, Well-posedness of the discrete algorithm,1D case. Convergence of the discrete algorithm (Fourier analysis + energy estimates) also nonconforming discretization in time. 1D case. Error estimates for non conforming grids in time.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Outline

1

Introduction

2

The SWR algorithm for advection diffusion equation Description Numerical experiments Back to the theoretical problem

3

The two-dimensional wave equation Dirichlet transmission conditions Optimized algorithms with overlap

4

Conclusion und perspectives

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Parabolic problems 1D theoretical analysis (M. Gander and L.H.) 2D with non constant velocity (V. Martin) Shallow water (V. Martin) Non conformal coupling (M.G., L.H., C. Japhet and M. Kern) Hyperbolic problems 1D heterogeneous (M. Gander and L.H.) optimal SWR. 2D homogeneous overlapping SWR (M. Gander and L.H.) 1D Mesh refinement, Nonoverlapping SWR in 2D (M. Gander and L.H.) Nonlinear waves in 1D (L.H and J. Szeftel), Mixed coupling a large scale oceanic model and a coastal model, coupling Euler and Navier-Stokes in an AMR frame. coupling ocean and atmosphere models.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Collaborators

Mostly : M. Gander (Universit´ e Gen` eve). 1D wave equation : F. Nataf (CNRS P6). 2D advection-diffusion : P. D’Anfray et J. Ryan (ONERA). V. Martin (Amiens). Heterogeneous problems (application to oceanography) : C. Japhet (P13), M. Kern (INRIA), E. Blayo (Grenoble). Schr¨

• dinger equation and non linear models : J. Szeftel.

Application to micromagnetism : S. Labb´ e (P11) et K. Santugini(Gen` eve) http ://www.math.univ-paris13.fr/ halpern See MS M04 today at 4pm.

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Two applications

H2 Bubble – Shock Interaction

5 4 3 4 1 2 Uniform supersonic flow Euler N2-O2 non-reactive Coarse mesh Non-interacting acoustic waves Euler N2-O2 non-reactive Coarse mesh 2 Shock- bubble interaction Navier-Stokes multi- species reactive Fine mesh 3 Interacting acoustic waves Euler N2-O2 reactive Fine mesh 4 Vortex and flame front Navier-Stokes multi- species reactive Very fine mesh

Combustion

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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives

Two applications

Ocean and ocean-atmosphere computations

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