Download Simultaneous Space-Time Adaptive Wavelet Collocation Method for

Simultaneous Space-Time Adaptive
Wavelet Collocation Method for
Intermittent Problems in Fluid Dynamics
Jahrul Alam
Department of Mathematics and Statistics
McMaster University, Canada
and
Nicholas Kevlahan
Department of Mathematics and Statistics
McMaster University, Canada
SHARCNET seminar series on HPC:
July 21, 2004
Department of Mathematics and Statistics
McMaster University, Canada
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Collaborators
• O. V. Vasilyev (University of Colorado at Boulder)
• D. Goldstein (University of Colorado at Boulder)
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Outline
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Outline
• Motivation
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Outline
• Motivation
• Adaptive wavelet collocation method
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Outline
• Motivation
• Adaptive wavelet collocation method
• Space-time discretization of PDEs
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Outline
• Motivation
• Adaptive wavelet collocation method
• Space-time discretization of PDEs
• Results and Discussion
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Outline
• Motivation
• Adaptive wavelet collocation method
• Space-time discretization of PDEs
• Results and Discussion
• Conclusion and Future direction
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Outline
• Motivation
• Adaptive wavelet collocation method
• Space-time discretization of PDEs
• Results and Discussion
• Conclusion and Future direction
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Motivation
• Basic equations of Fluid Dynamics
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Motivation
• Basic equations of Fluid Dynamics
– Newtonian, incompressible
– Conservation of mass and momentum
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Motivation
• Basic equations of Fluid Dynamics
– Newtonian, incompressible
– Conservation of mass and momentum
∇·u=0
∂u
1
+ u∇ · u = − ∇P + ν∇2u
∂t
ρ
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Motivation
• Basic equations of Fluid Dynamics
– Newtonian, incompressible
– Conservation of mass and momentum
∇·u=0
∂u
1 2
+ u∇ · u = −∇P +
∇u
∂t
Re
• What are the typical values of Re?
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Motivation
• Basic equations of Fluid Dynamics
– Newtonian, incompressible
– Conservation of mass and momentum
∇·u=0
∂u
1 2
+ u∇ · u = −∇P +
∇u
∂t
Re
• What are the typical values of Re?
For turbulent flows Re ∼ 105 − 106.
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Motivation
• Basic equations of Fluid Dynamics
– Newtonian, incompressible
– Conservation of mass and momentum
∇·u=0
∂u
1 2
+ u∇ · u = −∇P +
∇u
∂t
Re
• What are the typical values of Re?
For turbulent flows Re ∼ 105 − 106.
• Let us simplify the Navier-Stokes equation
∂u
∂u
∂ 2u
+u
=ν 2
∂t
∂x
∂x
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Motivation
• Computational problems of turbulence!
– DOF
∼
Re3 in space-time domain
• Aim of our study
– Adapt the # of DOF according to intermittency
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Motivation
• Computational problems of turbulence!
– DOF
∼
Re3 in space-time domain
• Aim of our study
– Adapt the # of DOF according to intermittency
Classical grid
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Motivation
• Computational problems of turbulence!
– DOF
∼
Re3 in space-time domain
• Aim of our study
– Adapt the # of DOF according to intermittency
t
x
Classical grid
We propose
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Adaptive wavelet collocation method
• What are wavelets?
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Adaptive wavelet collocation method
• What are wavelets?
A set of basis functions that are localized in space and scale
• Represent a function in terms of wavelet basis:
u(x) =
∞ X
X
djk ψkj (x)
j=0 k∈Kj
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Adaptive wavelet collocation method
• What are wavelets?
A set of basis functions that are localized in space and scale
• Represent a function in terms of wavelet basis:
u(x) =
∞ X
X
djk ψkj (x)
j=0 k∈Kj
• Wavelets:
– follow intermittency in position and scale
– provide automatic grid adaptation
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Adaptive wavelet collocation method
• Adaptive wavelet grid:
u(x) =
∞ X
X
djk ψkj (x)
j=0 k∈Kj
Gj = {xjk ∈ R : xjk = 2−j k, k ∈ Z, j ∈ Z}
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Adaptive wavelet collocation method
• Adaptive wavelet grid:
u(x) =
∞ X
X
djk ψkj (x)
j=0 k∈Kj
Gja = {xjk ∈ R : xjk = 2−j k, k ∈ Z, j ∈ Z, |djk | ≥ }
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Adaptive wavelet collocation method
• ALGORITHM: General grid adaptation strategy
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Adaptive wavelet collocation method
• ALGORITHM: General grid adaptation strategy
Start with an initial grid
Solve the PDE
Determine the region of intermittency
Refine the grid
Continue until sequence of grids are converged
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Adaptive wavelet collocation method
• ALGORITHM: General grid adaptation strategy
Start with an initial grid
Solve the PDE
Determine the region of intermittency
Refine the grid
Continue until sequence of grids are converged
6
5
j
4
3
2
1
0
−1
−0.5
0
x
0.5
1
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Adaptive wavelet collocation method
• ALGORITHM: General grid adaptation strategy
Start with an initial grid
Solve the PDE
Determine the region of intermittency
Refine the grid
Continue until sequence of grids are converged
6
Adjacent zone:
5
j
4
3
k
j+1
j
j−1
2
1
0
−1
−0.5
0
x
0.5
1
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Adaptive wavelet collocation method
• ALGORITHM: General grid adaptation strategy
Start with an initial grid
Solve the PDE
Determine the region of intermittency
Refine the grid
Continue until sequence of grids are converged
6
Adjacent zone:
5
j
4
3
k
j+1
j
j−1
2
1
0
−1
−0.5
0
x
0.5
1
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Space-time discretization of PDEs
• Discretization stencils
• Classical approach: sequence of algebraic problem
• Present study: single algebraic problem
Explicit
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Space-time discretization of PDEs
• Discretization stencils
• Classical approach: sequence of algebraic problem
• Present study: single algebraic problem
Explicit
Implicit
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Space-time discretization of PDEs
• Discretization stencils
• Classical approach: sequence of algebraic problem
• Present study: single algebraic problem
Explicit
Implicit
Crank-Nicholson
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Results and discussion
• Burgers equation
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Results and discussion
• Burgers equation
∂u
∂ 2u
∂u
+u
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω ⊂ R × (0, tmax], Ω = (−1, 1)
u(−1, t) = u(1, t),
u(x, 0) = sin(πx)
ν = 10−2/π
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Results and discussion
• Burgers equation
∂u
∂ 2u
∂u
+u
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω ⊂ R × (0, tmax], Ω = (−1, 1)
u(−1, t) = u(1, t),
u(x, 0) = sin(πx)
ν = 10−2/π
Initial condition
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Results and discussion
• Burgers equation
∂u
∂ 2u
∂u
+u
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω ⊂ R × (0, tmax], Ω = (−1, 1)
u(−1, t) = u(1, t),
u(x, 0) = sin(πx)
ν = 10−2/π
Grid
0.3
0.25
y
0.2
0.15
0.1
0.05
0
−1
Initial condition
−0.8
−0.6
−0.4
−0.2
0
x
0.2
0.4
0.6
0.8
1
Grid
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Results and discussion
• Burgers equation
∂u
∂ 2u
∂u
+u
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω ⊂ R × (0, tmax], Ω = (−1, 1)
u(−1, t) = u(1, t),
u(x, 0) = sin(πx)
ν = 10−2/π
Solution of Burgers equation at fixed time
1
Initial
Computed
0.8
0.6
0.4
u
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
x
0.2
0.4
0.6
0.8
1
Computed solution
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Results and discussion
• Burgers equation
∂u
∂ 2u
∂u
+u
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω ⊂ R × (0, tmax], Ω = (−1, 1)
u(−1, t) = u(1, t),
u(x, 0) = sin(πx)
ν = 10−2/π
Solution of Burgers equation at fixed time
1
Initial
Computed
0.8
0.6
0.4
u
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
x
0.2
0.4
0.6
Computed solution
0.8
1
Adapted grid
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Results and discussion
• Burgers equation
∂u
∂ 2u
∂u
+u
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω ⊂ R × (0, tmax], Ω = (−1, 1)
u(−1, t) = u(1, t),
u(x, 0) = sin(πx)
ν = 10−2/π
Solution surface
Adapted grid
(Space-time domain)
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Results and discussion
• Burgers equation
∂u
∂ 2u
∂u
+u
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω ⊂ R × (0, tmax], Ω = (−1, 1)
u(−1, t) = u(1, t),
u(x, 0) = sin(πx)
ν = 10−2/π
Grid
Grid
0.3
0.25
y
0.2
0.15
0.1
0.05
0
−1
−0.8
−0.6
−0.4
−0.2
0
x
0.2
0.4
0.6
0.8
1
Solution
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Results and discussion
• Burgers equation
∂u
∂ 2u
∂u
+u
= ν 2,
∂t
∂x
∂x
x ∈ (−π, π)
u(−π, t) = u(π, t)
u(x, 0) = sin(x)
Compare wavelet solution with a spectral code.
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Results and discussion
• Time evolution of error in spatial grid
−4
3.5
x 10
3
L∞ error on spatial grid
2.5
2
1.5
1
0.5
0
0
500
1000
1500
Number of time steps (with t∞ = 0.35)
2000
2500
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Results and discussion
• Efficiency of the method
E ∼ N ()−(p−m)/n,
–
→0
Error vs. # of wlt
0
10
with p=4
log E = −1.5log N
−2
10
−4
10
−6
10
2
10
3
4
10
5
10
6
10
10
0
E = ||u(x)−uex(x)||∞
10
with p=6
log E = −2.5log N
−2
10
−4
10
−6
10
2
10
3
4
10
5
10
10
5
10
with p=8
log E = −3.5log N
0
10
−5
10
−10
10
2
10
3
10
4
N(ε)
10
5
10
Error vs N ()
(Space-time domain)
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Results and discussion
• Efficiency of the method
E ∼ N ()−(p−m)/n,
–
N () ∼ −n/p,
→ 0,
Error vs. # of wlt
0
10
ε vs. # of wlt
0
10
with p=4
log E = −1.5log N
−2
10
→0
with p=4
log(ε) = −2log N
−5
10
−4
10
−6
10
−10
2
10
3
4
10
5
10
10
6
10
10
0
3
4
10
5
10
6
10
10
with p=6
log E = −2.5log N
−2
10
ε
E = ||u(x)−uex(x)||∞
2
10
10
0
10
with p=6
log(ε) = −3log N
−5
10
−4
10
−6
10
−10
2
10
3
4
10
5
10
10
5
2
10
3
4
10
5
10
10
5
10
10
with p=8
log E = −3.5log N
0
10
with p=8
log(ε) = −4log N
0
10
−5
−5
10
10
−10
10
10
−10
2
10
3
10
4
N(ε)
10
Error vs N ()
5
10
10
2
10
3
10
4
N(ε)
10
5
10
vs N ()
(Space-time domain)
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Results and discussion
• Efficiency of the method
– Define Compression ratio: CJ = N /N J × 100
– Number of collocation points at smallest scale,
N J = 4194304
– Number of collocation actual points in computation,
N = 22982
120
100
Cj
80
60
40
20
0
0
2
4
6
Iteration
8
10
12
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Result and discussion: Cont’d...
• Moving shock
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Result and discussion: Cont’d...
• Moving shock
∂u
∂ 2u
∂u
+ (u + v)
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω × (0, tmax], Ω = (0, 2)
u(0, t) = 1, u(2, t) = −1,
ν = 10−2,
x − x0
u(x, 0) = − tanh
2ν
x0 = 0.5
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Result and discussion: Cont’d...
• Moving shock
∂u
∂ 2u
∂u
+ (u + v)
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω × (0, tmax], Ω = (0, 2)
u(0, t) = 1, u(2, t) = −1,
u
↑→ x
ν = 10−2,
x − x0
u(x, 0) = − tanh
2ν
x0 = 0.5
Initial condition
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Result and discussion: Cont’d...
• Moving shock
∂u
∂ 2u
∂u
+ (u + v)
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω × (0, tmax], Ω = (0, 2)
u(0, t) = 1, u(2, t) = −1,
u
↑→ x
ν = 10−2,
x − x0
u(x, 0) = − tanh
2ν
x0 = 0.5
Solution at t = 1.0
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Result and discussion: Cont’d...
• Moving shock
∂u
∂ 2u
∂u
+ (u + v)
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω × (0, tmax], Ω = (0, 2)
u(0, t) = 1, u(2, t) = −1,
u
↑→ x
ν = 10−2,
Solution at t = 1.0
x − x0
u(x, 0) = − tanh
2ν
x0 = 0.5
Adapted grid
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Result and discussion: Cont’d...
• Moving shock
∂u
∂ 2u
∂u
+ (u + v)
= ν 2,
∂t
∂x
∂x
(x, t) ∈ Ω × (0, tmax], Ω = (0, 2)
u(0, t) = 1, u(2, t) = −1,
u
↑→ x
Solution
ν = 10−2,
x − x0
u(x, 0) = − tanh
2ν
x0 = 0.5
Adapted grid
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Conclusion and Future direction
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Conclusion and Future direction
• Conclusion
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Conclusion and Future direction
• Conclusion
– An adaptive numerical method is developed
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Conclusion and Future direction
• Conclusion
– An adaptive numerical method is developed
– An evolution type boundary condition is proposed
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Conclusion and Future direction
• Conclusion
– An adaptive numerical method is developed
– An evolution type boundary condition is proposed
– A multilevel adaptive solver has been developed
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Conclusion and Future direction
• Conclusion
– An adaptive numerical method is developed
– An evolution type boundary condition is proposed
– A multilevel adaptive solver has been developed
• Future plan
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Conclusion and Future direction
• Conclusion
– An adaptive numerical method is developed
– An evolution type boundary condition is proposed
– A multilevel adaptive solver has been developed
• Future plan
– Solve more complicated problems in Fluid Dynamics
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Conclusion and Future direction
• Conclusion
– An adaptive numerical method is developed
– An evolution type boundary condition is proposed
– A multilevel adaptive solver has been developed
• Future plan
– Solve more complicated problems in Fluid Dynamics
∗ Fluid-structure interaction
∗ Turbulent transport
∗ Passive scalar advection
∗ Nano-technology
∗ Meteorology
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Thank You
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