Schwarz waveform relaxation algorithms : theory and ... · }For a given problem, split the domain : domain decomposition.}For a given problem, di erent numerical methods in di erent

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

Schwarz waveform relaxation algorithms : theoryand applications

Laurence HALPERN

LAGA - Universite Paris 13

DD17. July 2006

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Outline

1 Introduction

2 The SWR algorithm for advection diffusion equationDescriptionNumerical experimentsBack to the theoretical problem

3 The two-dimensional wave equationDirichlet transmission conditionsOptimized algorithms with overlap

4 Conclusion und perspectives

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Coupling process

Issues

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

3 / 29


Coupling process

Issues


♦ For a given problem, different numerical methods in different zones :FEM/FD, SM/FEM, AMR.

3 / 29


Coupling process

Issues



♦ Couple two different models in different zones.

3 / 29


Coupling process

Issues



♦ Couple two different models in different zones.

♦ Furthermore the codes can be on distant sites.

3 / 29


DDM for evolution problems

Usual methods

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

4 / 29



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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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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The goals





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..

4 / 29



The goals





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

4 / 29


Outline

1 Introduction




5 / 29


Outline

1 Introduction




6 / 29


The Schwarz algorithm

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

ν > 0.

t

Ω1 Γ21

Ω2Γ12

8<:

Luk+11 = f in Ω1 × (0,T )

uk+11 (·, 0) = u0 in Ω1

B1uk+11 = B1uk

2 on Γ12 × (0,T )8<:

Luk+12 = f in Ω2 × (0,T )

uk+12 (·, 0) = u0 in Ω2

B2uk+12 = B2uk

1 on Γ21 × (0,T )

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How to choose the transmission operators ?

Transmission conditions

B1uk+11 = B1uk

2 on Γ12 × (0,T ), B2uk+12 = B2uk

1 on Γ21 × (0,T )

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B1uk+11 = B1uk

2 on Γ12 × (0,T ), B2uk+12 = B2uk

1 on Γ21 × (0,T )

Classical Schwarz

Bj ≡ I AND overlap.

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Classical Schwarz


1D Numerical experiment

a = 1, ν = 0.2,Ω = (0, 6),T = 2.5, L = 0.08.

u11(·,T ), u2

2(·,T )

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5uu1

1

u22

PSfrag replacements

u31(·,T ), u4

2(·,T )

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5uu1

3

u24

PSfrag replacements

u51(·,T ), u6

2(·,T )

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5uu1

5

u26

PSfrag replacements

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B1uk+11 = B1uk

2 on Γ12 × (0,T ), B2uk+12 = B2uk

1 on Γ21 × (0,T )

Classical Schwarz


Optimized Schwarz Waveform relaxation

Bj ≡ absorbing boundary operator+optimization WITH OR WITHOUToverlap

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Comparison

u11(·,T ), u2

2(·,T )

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5uu1

1

u22

PSfrag replacements

u31(·,T ), u4

2(·,T )

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5uu1

3

u24

PSfrag replacements

u51(·,T ), u6

2(·,T )

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5uu1

5

u26

PSfrag replacements

u11(·,T ), u2

2(·,T )

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5uu1

1

u22

PSfrag replacements

u31(·,T ), u4

2(·,T )

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5uu1

3

u24

PSfrag replacements

u51(·,T ), u6

2(·,T )

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5uu1

5

u26

PSfrag replacements

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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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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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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 := ∂x u − a− p

2νu + q(∂t + b · ∇u − ν∆y u)

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Well-posedness and convergence


B1u := ∂x u − a − p

2νu + q(∂t + b · ∇u − ν∆y u)

B2u := ∂x u − a + p

2νu − q(∂t + b · ∇u − ν∆y u)

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B1u := ∂x u − a − p

2νu + q(∂t + b · ∇u − ν∆y u)

B2u := ∂x u − a + p

2νu − q(∂t + b · ∇u − ν∆y u)

Convergence factor

ρ(ω, k,P, L) =

(P − δ1/2

P + δ1/2

)2

e−2δ1/2L/ν

ek+2j (ω, 0, k) = ρ(ω, k,P, L)ek

j (ω, 0, k)

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B1u := ∂x u − a − p

2νu + q(∂t + b · ∇u − ν∆y u)

B2u := ∂x u − a + p

2νu − q(∂t + b · ∇u − ν∆y u)

Convergence factor

ρ(ω, k,P, L) =

(P − δ1/2

P + δ1/2

)2

e−2δ1/2L/ν

ek+2j (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 Sobolevspaces and converges with and without overlap.

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One dimension : influence of the parameters

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

−10

−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

PSfrag replacements

error at iteration 5

p

q

∆x

iterations

Error obtained running the algorithm with first order transmissionconditions for 5 steps and various choices of p and q.

p∗, q∗ : theoretical values ,p∗, q∗ : Taylor approximations.

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One dimension : comparison

0 2 4 6 8 10

10−10

10−5

100

iteration

erro

r

2 SUBDOMAINS

CLASSICALOPTIMIZED ORDER 1

0 2 4 6 8 10

10−10

10−5

100

iteration

erro

r

4 SUBDOMAINS

0 2 4 6 8 10

10−10

10−5

100

iteration

erro

r

8 SUBDOMAINS

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Two dimensions : coupling different numerical methods

The heat bubble hitting an airfoil

−2 −1 0 1 2 3 4 5−2

−1

0

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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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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Robustness : rotating velocities

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

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50−6

−5

−4

−3

−2

−1

0

1

2

3

4

Iterations

Er

Nu=0.1

Taylor Ordre0Ordre0 OptimiséTaylor Ordre1Ordre1 Optimisé

interface 0.3

0 10 20 30 40 50−6

−5

−4

−3

−2

−1

0

1

2

3

4

Iterations

Er

Nu=0.1


interface 0.4

0 10 20 30 40 50−6

−5

−4

−3

−2

−1

0

1

2

3

4

Iterations

Er

Nu=0.1


interface 0.5

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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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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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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


√δ(0),

Best approximation

infP∈Pn

supz∈K|ρ(z ,P, L)|, K = (

π

T,π

∆t), kj ∈ (

π

Xj,π

∆xj)

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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


√δ(0),

Best approximation

infP∈Pn

supz∈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 uniquesolution characterized by an equioscillation property.

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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−15 , q ≈ Cq∆x

35 , |ρ| ≈ 1− O(∆x

15 ).

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Outline

1 Introduction




19 / 29


Schwarz Waveform relaxation algorithm

Lu := utt − c2∆u, x ∈ Ω ⊂ Rm

t

Ω1 Γ21

Ω2Γ12

8<:

Luk+11 = f in Ω1 × (0,T )

uk+11 (·, 0) = u0 in Ω1

B1uk+11 (L, ·) = B1uk

2 (L, ·) in (0,T )8<:

Luk+12 = f in Ω2 × (0,T )

uk+12 (·, 0) = u0 in Ω2

B2uk+12 (0, ·) = B2uk

1 (0, ·) in (0,T )

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A numerical experiment

Data

c = 1, T = 1,Ω = (0, 1)× (0, 1).

Two subdomains, overlapL = 0.08.

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

10

0.2

0.4

0.6

0.8

1

1.2

1.4

xy

u0

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A numerical experiment

Data

c = 1, T = 1,Ω = (0, 1)× (0, 1).

Two subdomains, overlapL = 0.08.

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

10

0.2

0.4

0.6

0.8

1

1.2

1.4

xy

u0Convergence history : Dirichlet transmission conditions with overlap

0 2 4 6 8 10 12 14 16 18 2010−12

10−10

10−8

10−6

10−4

10−2

100

102

erro

r

n

Convergence after n > cTL = 12 iterations

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Other transmission conditions

General transmission operators

B1 =J∏

j=1

(∂x + αj∂t),B2 =J∏

j=1

(∂x − αj∂t).

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Other transmission conditions

General transmission operators

B1 =J∏

j=1

(∂x + αj∂t),B2 =J∏

j=1

(∂x − αj∂t).

Plane waves analysis

ek1 = ak

1 (ω, k)eσ(x−L), ek2 = ak

2 (ω, k)eσx .

σ =

8<:|ω|

c

q`ckω

´2 − 1, evanescent waves,

iωc

q1−

`ckω

´2, propagating waves.

|ρ| =

8>>>><>>>>:

e−L|ω|

c

r`ckω

´2−1, evanescent waves,

JY

j=1

˛˛˛αj −

q1−

`ckω

´2

αj +q

1−`

ckω

´2

˛˛˛ propagating waves.

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Plane wave analysis : continue

Convergence factor, propagating case

θ angle of incidence on the interface, sin θ = ckω .

|ρ| =J∏

j=1

∣∣∣∣∣αj −

√1−

(ckω

)2

αj +

√1−

(ckω

)2

∣∣∣∣∣ =J∏

j=1

∣∣∣∣αj − cos θ

αj + cos θ

∣∣∣∣.

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|ρ| =J∏

j=1

∣∣∣∣∣αj −

√1−

(ckω

)2

αj +

√1−

(ckω

)2

∣∣∣∣∣ =J∏

j=1


αj + cos θ

∣∣∣∣.

Strategy 1 : orthogonal absorption α = 1

0 5 10 1510−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

102

iteration

Linf

erro

r at T

Classical SchwarzOrthogonal 1st OrderOrthogonal 2nd Order

PSfrag replacements

nerror

23 / 29





|ρ| =J∏

j=1

∣∣∣∣∣αj −

√1−

(ckω

)2

αj +√

1−(

ckω

)2

∣∣∣∣∣ =J∏

j=1


αj + cos θ

∣∣∣∣.

Strategy 2 : optimization

Given eps, find n and α(n) such that

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|ρ| =J∏

j=1

∣∣∣∣∣αj −

√1−

(ckω

)2

αj +√

1−(

ckω

)2

∣∣∣∣∣ =J∏

j=1


αj + cos θ

∣∣∣∣.



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

23 / 29





|ρ| =J∏

j=1

∣∣∣∣∣αj −

√1−

(ckω

)2

αj +√

1−(

ckω

)2

∣∣∣∣∣ =J∏

j=1


αj + cos θ

∣∣∣∣.



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

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

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Comparison

Example : eps = 10−2

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

Iteration 0 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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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 + energyestimates) also nonconforming discretization in time. 1D case.

Error estimates for non conforming grids in time.

25 / 29


Outline

1 Introduction




26 / 29


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.27 / 29


Collaborators

Mostly : M. Gander (Universite Geneve).

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).

Schrodinger equation and non linear models : J. Szeftel.

Application to micromagnetism : S. Labbe (P11) et K.Santugini(Geneve)

http ://www.math.univ-paris13.fr/ halpern See MS M04 today at 4pm.

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Two applications

H2 Bubble – Shock Interaction

5

4

3

4

1 2

Uniformsupersonic flowEuler N2-O2

non-reactiveCoarse mesh

Non-interactingacoustic waves

Euler N2-O2 non-reactive

Coarse mesh

2

Shock- bubbleinteractionNavier-Stokes multi-

species reactiveFine mesh

3 Interacting acousticwavesEuler N2-O2 reactiveFine mesh

4 Vortex and flame front Navier-Stokes multi-

species reactiveVery fine mesh

Combustion

29 / 29


Two applications

Ocean and ocean-atmosphere computations

29 / 29

Documents

Schwarz waveform relaxation algorithms : theory and ... · }For a given problem, split the domain : domain decomposition.}For a given problem, di erent numerical methods in di erent