schwarz waveform relaxation algorithms : theory and ... · }for a given problem, split the domain :...
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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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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 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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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 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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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 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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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives
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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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+11 = B1uk
2 on Γ12 × (0,T ), B2uk+12 = 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+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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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.
u11(·,T ), u2
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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+11 = B1uk
2 on Γ12 × (0,T ), B2uk+12 = B2uk
1 on Γ21 × (0,T )
Classical Schwarz
Bj ≡ I AND overlap.
Optimized Schwarz Waveform relaxation
Bj ≡ absorbing boundary operator+optimization WITH OR WITHOUToverlap
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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
u11(·,T ), u2
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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 := ∂x u − a− p
2νu + q(∂t + b · ∇u − ν∆y u)
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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 := ∂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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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives
Well-posedness and convergence
Transmission conditions
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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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives
Well-posedness and convergence
Transmission conditions
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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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives
One dimension : influence of the parameters
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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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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives
One dimension : comparison
0 2 4 6 8 10
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erro
r
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CLASSICALOPTIMIZED ORDER 1
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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 0 1 2 3 4 5−2
−1
0
1
2
3
4
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0.4
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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 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
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1
0 10 20 30 40 50−6
−5
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Er
Nu=0.1
Taylor Ordre0Ordre0 OptimiséTaylor Ordre1Ordre1 Optimisé
interface 0.3
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Er
Nu=0.1
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interface 0.4
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−5
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−2
−1
0
1
2
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Iterations
Er
Nu=0.1
Taylor Ordre0Ordre0 OptimiséTaylor Ordre1Ordre1 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
infP∈Pn
supz∈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
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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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−15 , q ≈ Cq∆x
35 , |ρ| ≈ 1− O(∆x
15 ).
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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 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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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
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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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, 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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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, 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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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
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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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 = 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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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 = ckω .
|ρ| =J∏
j=1
∣∣∣∣∣αj −
√1−
(ckω
)2
αj +
√1−
(ckω
)2
∣∣∣∣∣ =J∏
j=1
∣∣∣∣αj − cos θ
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
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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 = ckω .
|ρ| =J∏
j=1
∣∣∣∣∣αj −
√1−
(ckω
)2
αj +√
1−(
ckω
)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 = ckω .
|ρ| =J∏
j=1
∣∣∣∣∣αj −
√1−
(ckω
)2
αj +√
1−(
ckω
)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( nLcT ),
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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 = ckω .
|ρ| =J∏
j=1
∣∣∣∣∣αj −
√1−
(ckω
)2
αj +√
1−(
ckω
)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( nLcT ),
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 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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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 + energyestimates) 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 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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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.27 / 29
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Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives
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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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
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
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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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