Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Two-Scale Convergence and Two-Scale
Numerical Methods
Emmanuel Frénod1
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
May 14, 17 and 21, 2013
Algorithms
Implementation
1 LMBA (UMR 6205), Université de Bretagne-Sud, F-56017, Vannes, France.
[email protected]
http://web.univ-ubs.fr/lmam/frenod/index.html
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Two-Scale Convergence
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Two-Scale Convergence and
Homogenization
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Convergence first statements
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
G. Nguetseng.
A general convergence result for a functional related to the
theory of homogenization.
SIAM Journal on Mathematical Analysis, 20(3):608–623, 1989.
G. Nguetseng.
Asymptotic analysis for a stiff variational problem arising in
mechanics.
SIAM Journal on Mathematical Analysis, 21(6):1394–1414,
1990.
G. Allaire.
Homogenization and Two-scale Convergence.
SIAM Journal on Mathematical Analysis, 23(6):1482–1518,
1992.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
The simplest example I know to introduce
Homogenization
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Material shape
Microstructure
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Figure : Composite material - macroscopic shape and a microstructure Ratio size of the microstructure on the size of the material is ε.
u ε : Temperature field
h
i
x
∇ · aε (x, )∇u ε = 0 within the material,
ε
u ε given on the boundary of the material,
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
A slight digression to explain aε (x, xε ) (and even
more) - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
1.8
1.8
1.8
1.6
1.6
1.6
1.4
1.4
1.4
1.2
1.2
1.2
1
1
1
0.8
0.8
0.8
0.6
0.4
−4
0.6
−3
−2
−1
0
1
2
3
4
0.4
−4
0.6
−3
−2
−1
0
1
2
3
4
0.4
−4
−3
−2
−1
0
1
2
1
x
Figure : Graph of sin(x) + 1 + ε cos( ) for ε = 1/20 (left), 1/40
2
ε
(center) and 1/80 (right) between −π and π.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
3
4
A slight digression to explain aε (x, xε ) (and even
more) - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
2
2
2
1.8
1.8
1.8
1.6
1.6
1.6
1.4
1.4
1.4
1.2
1.2
1.2
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0
−4
−3
−2
−1
0
1
2
3
4
0
−4
0.2
−3
−2
−1
0
1
2
3
4
0
−4
−3
−2
−1
0
1
2
1
1
x
Figure : Graph of sin(x) + 1 + cos( ) for ε = 1/20 (left), 1/40
2
2
ε
(center) and 1/80 (right) between −π and π.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
3
4
A slight digression to explain aε (x, xε ) (and even
more) - 3
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
80
80
80
60
60
60
40
40
40
20
20
20
0
0
0
−20
−20
−20
−40
−40
−40
−60
−60
−80
−4
−3
−2
−1
0
1
2
3
4
−80
−4
−60
−3
−2
−1
0
1
2
3
4
−80
−4
−3
−2
−1
0
1
2
1
x
Figure : Graph of 5 sin(x) + 1 +
cos( ) for ε = 1/20 (left), 1/40
2ε
ε
(center) and 1/80 (right) between −π and π.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
3
4
A slight digression to explain aε (x, xε ) (and even
more) - 4
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.8
−0.8
−1
−4
−3
−2
−1
0
1
2
3
4
−1
−4
−0.8
−3
−2
−1
0
1
2
3
4
−1
−4
−3
−2
−1
0
1
2
3
1
x
Figure : Graph of (sin(x) + 1) cos( ) for ε = 1/20 (left), 1/40 (center)
2
ε
and 1/80 (right) between −π and π.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
4
A slight digression to explain aε (x, xε ) (and even
more) - 5
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
80
80
80
60
60
60
40
40
40
20
20
20
0
0
0
−20
−20
−20
−40
−40
−40
−60
−60
−80
−4
−3
−2
−1
0
1
2
3
4
−80
−4
−60
−3
−2
−1
0
1
2
3
4
−80
−4
−3
−2
−1
0
1
2
3
1
x
Figure : Graph of
(sin(x) + 1) cos( ) for ε = 1/20 (left), 1/40 (center)
4ε
ε
and 1/80 (right) between −π and π.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
4
A slight digression to explain aε (x, xε ) (and even
more) - 6
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
2.5
2.5
2
2
2.5
2
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
−0.5
−4
0
−3
−2
−1
0
1
2
3
4
−0.5
−4
0
−3
−2
−1
0
1
2
3
4
−0.5
−4
−3
−2
−1
0
1
2
3
1
1
x
Figure : Graph of cos(x) + 1 + (sin(x) + 1) cos( ) for ε = 1/20 (left),
2
2
ε
1/40 (center) and 1/80 (right) between −π and π.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
4
A slight digression to explain aε (x, xε ) (and even
more) - 7
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
80
80
80
60
60
60
40
40
40
20
20
20
0
0
0
−20
−20
−20
−40
−40
−40
−60
−60
−80
−4
−3
−2
−1
0
1
2
3
4
−80
−4
−60
−3
−2
−1
0
1
2
3
4
−80
−4
−3
−2
−1
0
1
2
1
x
Figure : Graph of 10 cos(x) + 1 + (sin(x) + 1) cos( ) for ε = 1/20
2ε
ε
(left), 1/40 (center) and 1/80 (right) between −π and π.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
3
4
A slight digression to explain aε (x, xε ) (and even
more) - 8
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
15
10
5
0
3
3
2
2
1
1
0
0
Figure : Graph of x 2 + y 2 +
Emmanuel Frénod
1
y
x
(sin( ) + 1) + (sin( ) + 1) .
2
ε
ε
Two-Scale Convergence and Two-Scale Numerical Methods
A slight digression to explain aε (x, xε ) (and even
more) - 9
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
15
10
5
0
3
3
2
2
1
1
0
0
y
x
Figure : Graph of x 2 + y 2 + sin(2x)(sin( ) + 1) + (sin( ) + 1) for
ε
ε
ε = 1/20 on [0, 3]2 .
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
A slight digression to explain aε (x, xε ) (and even
more) - 10
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
x
aε (x, ) can model a wide range of microscopic oscillations or
ε
heterogeneities.
This is why we use it in the model.
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Remark
Two-Scale Convergence is based on this capability
Remark
IF ξ 7→ aε (x, ξ) THEN periodic microscopic scale variations are
qualified of high frequency periodic oscillations.
Two-Scale Convergence is essentially designed for this
context.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Back to : the simplest example I know to introduce
Homogenization
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Material shape
Microstructure
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
u ε : Temperature field
ε
1
h
i
x
∇ · aε (x, )∇u ε = 0 within the material,
ε
ε
u given on the boundary of the material,
IF Solved with a numerical method INDUCES : ∆x << ε
IF interested in the tiny variation of u ε , WHY NOT (?)
OTHERWISE: Clearly NOT REASONNABLE
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Homogenization Goal
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Find an operator H (that neither contains nor generates oscillations
of size ε)
Such that u
Emmanuel
Frénod
Hu = 0
u = uGiven
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
within the material,
on the boundary of the material,
ε
close to u (in some sense)
h
i
x
∇ · aε (x, )∇u ε = 0 within the material,
ε
u ε = uGiven on the boundary of the material,
INDEPENDENTLY of uGiven
This means
H must induce average effect of oscillations in u
x
In some sense: H = lim ∇ · aε (x, )∇
ε→0
ε
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Homogenization Theory
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Homogenization Theory gathers a collection of methods that allow to
build operators H satisfying the required constraint for every problem
- containing or generating oscillations or heterogeneities - we can
imagine.
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Asymptotic Expansion: First Homogenization
method set out by Engineers in the 1970s
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
h
i
x
In the case of ∇ · aε (x, )∇u ε = 0:
ε
x
x
x
u ε (x) = U(x, ) + εU1 (x, ) + ε2 U2 (x, ) + . . . ,
ε
ε
ε
U(x, ξ), U1 (x, ξ), U2 (x, ξ), . . . periodic with respect to ξ.
Gathering terms in factor of ε−2 , ε−1 , ε0 , ε, ε2 , . . . :
H−2 U = 0, H−1 U1 = I(U), H0 U2 = I 0 (U, U1 ), . . . .
Algorithms
Implementation
Get well-posed equations for U, U1 , U2 , . . . .
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Mathematical justification of Asymptotic Expansion
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Needed:
x ε
u (x) − U(x, ) → 0,
ε ?
or in a weaker sense:
x u ε (x) − U(x, ) * 0.
ε
For higher orders, needed:


x
u ε (x) − U(x, )
ε − U (x, x ) → 0,

1
ε
ε
Algorithms
Implementation
1 1 ε
x
x
x u (x) − U(x, ) − U1 (x, ) − U2 (x, ) → 0,
ε ε
ε
ε
ε
and so on, in some sense.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Tools for mathematical justification of Asymptotic
Expansion - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
For Heat Equation with Dirichlet boundary conditions:
h
i
x
∇ · aε (x, )∇u ε = 0 within the material,
ε
u ε = uGiven on the boundary of the material,
Maximum Principle and boundary estimates WORKS.
SEE
A. Bensoussan, J. L. Lions, and G. Papanicolaou.
Asymptotic analysis for periodic structures.
Studies in Mathematics and its Applications, Vol. 5. North
Holland, 1978.
For any all problem: DOES NOT WORK.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Tools for mathematical justification of Asymptotic
Expansion - 2 : "Oscillating Test Function Method"
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
L. Tartar.
Cours Peccot.
Collège de France, 1977.
F. Murat.
H-convergence.
Séminaire d’Analyse Fonctionnelle et Numérique d’Alger, 1977.
L. Tartar.
The General Theory of Homogenization. A Personalized
Introduction.
Springer Verlag, dec 2009.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Brief overview of Oscillating Test Function Method
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Weak Formulation with Oscillating Test Functions (WFWOTF).
Z
i
h
x
x
∇ · aε (x, )∇u ε (x) ϕ(x, ) dx = 0,
ε
ε
Material
By the Stokes Formula:
Z
Z
h
x
x i
ε
ε
a (x, )∇u (x) · ∇ ϕ(x, ) dx = Something,
ε
ε
Material
Boundary
or
Z
Z
x
x
1
x
ε
a (x, )∇u (x) · ∇x ϕ(x, ) + ∇ξ ϕ(x, ) dx = Something.
ε
ε
ε
ε
Material
Boundary
ε
Difficulty: ∇u ε , aε (x, εx ), ∇x ϕ(x, εx ) and ∇ξ ϕ(x, εx ) converges in a
weak sense only.
Passing to the limit involves relatively sophisticated analytical
methods.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Tools for mathematical justification of Asymptotic
Expansion - 3 : Two-Scale Convergence
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Two-Scale Convergence offers an efficient framework to pass to the
limit in such terms, in the case when oscillations are periodic.
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Link Homogenization - Two-Scale Convergence:
Conclusion
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Two-Scale Convergence emerged as an efficient tools to justify
Asymptotic Expansion
Yet, it is more that this: It is a constructive Homogenization
Method very well adapted to Singularly Perturbed Hyperbolic
Equations.
Well adapted for problems with oscillations at one frequency:
1
.
ε
Can be improved to the case of oscillations with several
1
1
frequencies, if scale separation, for instance :
and 2 .
ε
ε
Cannot be improved to the case of several frequencies if no scale
separation.
Cannot be improved to the case of a variable frequency.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Two proofs which are typical
in Two-Scale Convergence
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
The Riemann-Lebesgue Lemma
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
The Lemma
x
ε
ε
If ψ ∈ L∞
# (R). Defining [ψ] by [ψ] (x) = ψ( ), then
ε
Z 1
[ψ]ε *
ψ(ξ) dξ in L∞ (R) weak-*.
0
This means: for any function ϕ ∈ L1 (R) (or ∈ D(R) by density)
Z
ε
Z
[ψ] (x) ϕ(x) dx →
Emmanuel Frénod
Z
ψ(ξ) dξ
0
R
1
ϕ(x) dx.
R
Two-Scale Convergence and Two-Scale Numerical Methods
The Riemann-Lebesgue Lemma proof - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Fix ϕ ∈ D(R)
Choose M s.t. supp(ϕ) ⊂ [−M, M]
Set
{−M, −M + ε, . . . , −M + E(2M/ε)ε, −M + (E(2M/ε) + 1)ε}
(E: integer part)
Z
E(2M/ε)+1 Z −M+iε
X
x
Split [ψ]ε (x) ϕ(x) dx =
ψ( ) ϕ(x) dx
ε
−M+(i−1)ε
R
i=1
Use Taylor formula: ∀x ∈ [−M + (i − 1)ε, −M + iε],
∃ci (x) ∈ [−M + (i − 1)ε, x] such that
ϕ(x) = ϕ(−M + (i − 1)ε) + (x + M − (i − 1)ε)ϕ0 (ci (x))
Z
E(2M/ε)+1 Z −M+iε
X
x
[ψ]ε (x) ϕ(x) dx =
ψ( ) dx ϕ(−M(i − 1)ε)
R
−M+(i−1)ε ε
i=1
E(2M/ε)+1 Z −M+iε
X
x
.
+
ψ( ) (x + M − (i − 1)ε)ϕ0 (ci (x)) dx
−M+(i−1)ε ε
i=1
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
The Riemann-Lebesgue Lemma proof - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Z
[ψ] (x) ϕ(x) dx =
R
.
+
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
E(2M/ε)+1 Z −M+iε
X
ε
i=1
E(2M/ε)+1 Z −M+iε
X
i=1
x
ψ( ) dx ϕ(−M(i − 1)ε)
−M+(i−1)ε ε
x
ψ( ) (x + M − (i − 1)ε)ϕ0 (ci (x)) dx
−M+(i−1)ε ε
E(2M/ε)+1
X Z −M+iε
x
ψ( ) dx ϕ(−M(i − 1)ε) =
−M+(i−1)ε ε
i=1
Z 1
Z 1
Z
E(2M/ε)+1
X
ε→0
ψ(ξ) dξ
ψ(ξ) dξ ε
ϕ(−M(i − 1)ε) −→
ϕ(x) dx
0
0
R
i=1
E(2M/ε)+1
Z
−M+iε
X
x
0
ψ( ) (x + M − (i − 1)ε)ϕ (ci (x)) dx ε
−M+(i−1)ε
i=1
Z 1
2M + 1
ε→0
.
≤
εkϕ0 k∞ −→ 0
|ψ(ξ)| εdξ
ε
0
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
The Riemann-Lebesgue Lemma generalization
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
The Lemma
0
0
If ψ = ψ(x, ξ) ∈ C 0 (R; C#
(R)) (or ∈ L∞ (R; C#
(R)) but it is more
x
ε
ε
technical). Defining [ψ] by [ψ] (x) = ψ(x, ), then
ε
Z 1
[ψ]ε *
ψ(x, ξ) dξ in L∞ (R) weak-*.
0
This means: for any function ϕ ∈ L1 (R) (or ∈ D(R) by density)
Z
Z Z 1
ε
[ψ] (x) ϕ(x) dx →
ψ(x, ξ) dξ ϕ(x) dx.
R
0
R
i.e.: ∀δ > 0, ∃ε0 >0, s.t. ∀ε ≤ ε0 ,
Z
Z Z
[ψ]ε (x) ϕ(x) dx −
R
R
Emmanuel Frénod
0
1
ψ(x, ξ) dξ ϕ(x) dx ≤ δ.
Two-Scale Convergence and Two-Scale Numerical Methods
The Riemann-Lebesgue Lemma generalization proof
-1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
step 1:
∀m ∈ N: partition of [0, 1] with m intervals of length 1/m
χm
i : characteristic functions of i-th interval, for i = 1 . . . , m
extended by periodicity over R. ξim : center of the i-th interval
m
X
m→∞
ψ̃m (x, ξ) =
ψ(x, ξim ) χm
i (ξ) −→ ψ(x, ξ) uniformally
Z
i=1
1
1
in L∞ (R) weak-*.
m
0
Z 1
m
X
1
ε→0
Hence [ψ̃m ]ε *
ψ(x, ξim ) =
ψ̃m (x, ξ) dξ
m
0
ε ε→0
[χm
i ] *
χm
i (ξ) dξ =
i=1
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
The Riemann-Lebesgue Lemma generalization proof
-2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
step 2:
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Z
Z Z 1
[ψ]ε (x) ϕ(x) dx −
ψ(x, ξ) dξ ϕ(x) dx ≤
R
R
0
Z ε
[ψ] (x) − [ψ̃m ]ε (x) |ϕ(x)| dx
R
Z Z 1
ε
+
[ψ̃m ] (x) −
ψ̃m (x, ξ) dξ ϕ(x) dx 0
R
Z Z
+
R
0
1
ψ̃m (x, ξ) − ψ(x, ξ) dξ |ϕ(x)| dx
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
The Riemann-Lebesgue Lemma generalization proof
-2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
step 2:
Fix m s.t. :
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Z
Z Z 1
[ψ]ε (x) ϕ(x) dx −
ψ(x, ξ) dξ ϕ(x) dx ≤
R
R
0
Z δ
ε
[ψ] (x) − [ψ̃m ]ε (x) |ϕ(x)| dx ≤ , ∀ε > 0
3
R
Z Z 1
ε
ψ̃m (x, ξ) dξ ϕ(x) dx +
[ψ̃m ] (x) −
0
R
Z Z
+
R
0
1
ψ̃m (x, ξ) − ψ(x, ξ) dξ |ϕ(x)| dx
Emmanuel Frénod
≤
δ
3
Two-Scale Convergence and Two-Scale Numerical Methods
The Riemann-Lebesgue Lemma generalization proof
-2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
step 2:
Fix m and ε0 s.t. :
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Z
Z Z 1
[ψ]ε (x) ϕ(x) dx −
ψ(x, ξ) dξ ϕ(x) dx ≤
R
R
0
Z δ
ε
[ψ] (x) − [ψ̃m ]ε (x) |ϕ(x)| dx ≤ , ∀ε > 0
3
R
Z Z 1
δ
ε
ψ̃m (x, ξ) dξ ϕ(x) dx ≤ , ∀ε ≤ ε0
+
[ψ̃m ] (x) −
3
R
0
Z Z 1 δ
+
ψ̃m (x, ξ) − ψ(x, ξ) dξ |ϕ(x)| dx ≤
3
0
R
≤ δ, ∀ε ≤ ε0
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Two-Scale Convergence:
definitions and results
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Key Points of the Theory - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Several variants of the Two-Scale Convergence theory, for various
targeted applications and involving various functional spaces.
Very close to each other. All follow the same routine based :
Emmanuel
Frénod
A continuous injection Lemma
A compactness Theorem
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
See
G. Nguetseng.
A general convergence result for a functional related to the theory of
homogenization.
SIAM Journal on Mathematical Analysis, 20(3):608–623, 1989.
G. Nguetseng.
Asymptotic analysis for a stiff variational problem arising in mechanics.
SIAM Journal on Mathematical Analysis, 21(6):1394–1414, 1990.
G. Allaire.
Homogenization and Two-scale Convergence.
SIAM Journal on Mathematical Analysis, 23(6):1482–1518, 1992.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Key Points of the Theory - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
M. Amar.
Two-scale convergence and homogenization on BV(ω).
Asymptotic Analysis, 65(1):65–84, 1998.
J. Casado-Díaz and I. Gayte.
The two-scale convergence method applied to generalized Besicovitch spaces.
Proceedings of the Royal Society of London. Series A: Mathematical, Physical and
Engineering Sciences, 458(2028):2925–2946, 2002.
E. Frénod, P. A. Raviart, and E. Sonnendrücker.
Asymptotic expansion of the Vlasov equation in a large external magnetic field.
J. Math. Pures et Appl., 80(8):815–843, 2001.
G. Nguetseng and N. Svanstedt.
Σ−convergence.
Banach Journal of Mathematical Analysis, 5(1):101–135, 2011.
G. Nguetseng and J.-L. Woukeng.
Σ−Convergence of nonlinear parabolic operators.
Nonlinear Analysis: Theory, Methods & Applications, 66(4):968–1004, feb 2007.
E. Frénod.
Two-Scale Convergence.
ESAIM: Proceedings, 38:1–35, December 2012.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Key Points of the Theory - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
H. E. Pak.
Geometric two-scale convergence on forms and its applications to maxwell’s
equations.
Proceedings of the Royal Society of Edinburgh, 135A:133–147, 2005.
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Definitions
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Definitions
Notations
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Ω: a regular domain in Rn
L a usual functional Banach space: L0 its topological dual
space. L0 h., .iL : duality bracket. |.| L , |.| L0 : norms
q ∈ [1, +∞) and p ∈ (1, +∞] s.t. 1/q + 1/p = 1
0
C#
(Rn ; L): continuous functions Rn → L, periodic of period 1
with respect to every variable
Lp (Ω, L0 ): functions f : Ω → L0
s.t. |f |pL0 is integrable if p < ∞
s.t. |f |L0 is essentially bounded if p = ∞
Lp# (Rn ; L0 ): functions f : Rn → L0
s.t. |f |pL0 is locally integrable if p < ∞
s.t. |f |L0 is locally essentially bounded if p = ∞
and periodic of period 1.
Lp# (Rn ; L0 ) = (Lq# (Rn ; L))0 (because of the separability of L)
0
Lq (Ω; Lq# (Rn , L)), Lq (Ω; C#
(Rn ; L)) and Lp (Ω; Lp# (Rn , L0 ))
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Definitions
Two-Scale Convergence definition
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Definition
(u ε ) = (u ε (x)) ⊂ Lp (Ω; L0 ) Two-Scale converges to
U = U(x, ξ) ∈ Lp (Ω; Lp# (Rn , L0 ))
0
if, for any function φ = φ(x, ξ) ∈ Lq (Ω; C#
(Rn ; L)),
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Z
lim
ε→0
L0
Ω
x
h u ε (x), φ(x, )iL dx =
ε
Z Z
L0 h U(x, ξ), φ(x, ξ)iL
Ω
dxdξ,
[0,1]n
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Definitions
Strong Two-Scale Convergence definition
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Definition
IF p = q = 2, L is a Hilbert space,
IF
(u ε ) = (u ε (x)) ⊂ L2 (Ω; L0 ) Two-Scale converges to U = U(x, ξ)
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
0
and IF U ∈ L2 (Ω; C#
(Rn ; L0 )).
THEN we say
Order 0
Order 1
(u ε ) = (u ε (x)) Strongly Two-Scale converges to U = U(x, ξ)
Two-Scale
Numerics
Algorithms
Implementation
if
Z x 2
ε
u (x) − U(x, ) 0 dx = 0
ε→0 Ω
ε L
lim
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Link with weak-∗ convergence
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Link with weak-∗ convergence
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
Proposition
If (u ε ) ⊂ Lp (Ω; L0 ) Two-Scale converges to U ∈ Lp (Ω; Lp# (Rn ; L0 )),
then
Z
ε
u *
U(., ξ) dξ weak-* in Lp (Ω; L0 ).
[0,1]n
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
In the definition of Two-Scale Convergence: φ(x, ξ) = φ(x).
Z
L0 h u
lim
ε→0
ε
Z Z
L0 h U(x, ξ), φ(x)iL
(x), φ(x)iL dx =
Ω
Ω
Z
Z
L0 h
Ω
Emmanuel Frénod
dxdξ =
[0,1]n
!
U(x, ξ)dξ , φ(x)iL dx.
[0,1]n
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Two-Scale Convergence criterion
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Convergence criterion
Injection Lemma - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Injection Lemma
0
If φ ∈ Lq (Ω; C#
(Rn ; L)), then for all ε > 0, function [φ]ε : Ω → L
defined by
x
[φ]ε (x) = φ(x, )
ε
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
satisfies
Hyperbolic
PDEs
k[φ]ε kLq (Ω;L) ≤ kφkLq (Ω;C 0 (Rn ;L))
#
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
q
kφkLq (Ω;C 0 (Rn ;L))
#
ε
k[φ] kLq (Ω;L)
!q
Z
=
Ω
sup |φ(x, ξ)|L
dx
ξ∈[0,1]n
Z
Z x q
=
φ(x, ) dx ≤
ε L
Ω
Ω
Emmanuel Frénod
!q
sup |φ(x, ξ)|L
dx
ξ∈[0,1]n
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Convergence criterion
Injection Lemma - 2: Supplementary Proposition
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Supplementary Proposition
0
If φ ∈ Lq (Ω; C#
(Rn ; L)), then
Z x q
q
lim k[φ]ε kLq (Ω;L) = lim
φ(x, ) dx
ε→0 Ω
ε→0
ε L
Z Z
q
q
=
|φ(x, ξ)|L dxdξ = kφkLq (Ω;Lq (Rn ;L))
Ω
[0,1]n
#
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Convergence criterion
Injection Lemma - 3: Suppl. Proposition proof
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
step 1:
∀m ∈ N: partition of [0, 1]n with m hypercubes of measure 1/m
m
ξm
i : center of the i-th hypercube χi : characteristic function of
the i-th Z
hypercube extended by periodicity to Rn
1
in L∞ (Ω; R) weak-*
[χi ]ε *
χi (ξ) dξ =
m
n
[0,1]
Z
1
in L∞ (Ω; R) weak-*
([χi ]ε )q *
χqi (ξ) dξ =
m
n
[0,1]
m
X
φ̃m (x, ξ) =
φ(x, ξ i ) χi (ξ) s.t.
i=1
Z X
Z m
x q
q
|φ(x, ξ i )|L ([χi ]ε )q dx =
lim
φ̃m (x, ) dx = lim
ε→0 Ω
ε→0 Ω
ε L
i=1
Z
Z Z
m
q
X
1
q
|φ(x, ξ i )|L dx =
φ̃m (x, ξ) dxdξ
m Ω
L
n
Ω [0,1]
i=1
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Convergence criterion
Injection Lemma - 4: Suppl. Proposition proof
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
step 2: WeZhave
Z Z x q
x q
x q
k[φ]ε k.qLq (Ω;L) =
φ(x, ) dx =
φ̃m (x, ) dx +
φ(x, ) dx −
ε L
ε L
ε L
Ω
Ω
Ω
!
!
Z Z
Z Z
Z q
q
q
x . φ̃m (x, ) dx −
φ̃m (x, ξ) dxdξ +
φ̃m (x, ξ) dxdξ
ε L
L
L
Ω [0,1]n
Ω [0,1]n
Ω
And
Z Z
Ω
[0,1]n
q
φ̃m (x, ξ) dxdξ
!
L
!
Z Z
→
Ω
[0,1]n
|φ(x, ξ)|qL
dxdξ
as m → +∞
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Z Z q
x q
φ̃m (x, ξ) dxdξ → 0
φ̃m (x, ) dx −
ε L
L
Ω
Ω
as ε → 0
Z Z Z q
x q
x q
x q ≤
φ(x, x ) dx −
φ̃
(x,
)
dx
)
−
φ̃
(x,
)
dx
φ(x,
m
m
ε L
ε L ε L
ε L
Ω
Ω
Ω
Z
q
≤
sup φ(x, ξ) − φ̃m (x, ξ) dx → 0 as m → +∞
Ω ξ∈[0,1]n
Emmanuel Frénod
L
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Convergence criterion
The criterion - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Theorem
If a sequence (u ε ) is bounded in Lp (Ω; L0 ), i.e. if
Z
ε
ku kLp (Ω;L0 ) =
|u
Ω
ε
p
(x)|L0
1p
dx
≤ c,
for a constant c independent of ε, then, there exists a profile
U ∈ Lp (Ω; Lp# (Rn ; L0 )) such that, up to a subsequence,
(u ε ) Two-Scale converges to U.
Two-Scale
Numerics
Algorithms
Implementation
Two ingredients for the proof
the sequential Banach-Alaoglu Theorem
the Riesz Representation Theorem.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Convergence criterion
Proof of the Theorem - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Injection Lemma and assumption of the Theorem →
0
∀φ = φ(x, ξ) ∈ Lq (Ω; C#
(Rn ; L)) ((1/p) + (1/q) = 1)
Z
x
ε
ε
L0 h u (x), φ(x, )iL dx ≤ c k[φ] kLq (Ω,L)
ε
Ω
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
≤ ckφkLq (Ω;C#0 (Rn ;L))
Hence (thanks to the second inequality)
0
µε : Lq (Ω; C#
(Rn ; L))
Algorithms
Implementation
R
Z
Order 0
Order 1
Two-Scale
Numerics
→
7→
φ
L0 h u
Ω
q
ε
x
(x), φ(x, )iL dx
ε
0
(Ω; C#
(Rn ; L)))0
n
0
bounded in (L
0
0
As (Lq (Ω; C#
(R ; L))) dual of separable space Lq (Ω; C#
(Rn ; L))
0
µε * µ in (Lq (Ω; C#
(Rn ; L)))0 weak-* (up to a subsequence)
0
In particular: hµε , φi → hµ, φi, ∀φ ∈ Lq (Ω; C#
(Rn ; L))
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Convergence criterion
Proof of the Theorem - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
0
We have: ∀φ = φ(x, ξ) ∈ Lq (Ω; C#
(Rn ; L)) ((1/p) + (1/q) = 1)
Z
x
ε
ε
0
0 (Rn ;L))
L h u (x), φ(x, )iL dx ≤ c k[φ] kLq (Ω,L) ≤ ckφkLq (Ω;C#
ε
Ω
Making ε → 0 →
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
0
(Rn ; L))
|hµ, φi| ≤ c kφkLq (Ω;Lq (Rn ;L)) ∀φ ∈ Lq (Ω; C#
#
0
Since Lq (Ω; C#
(Rn ; L)) is dense in Lq (Ω; Lq# (Rn ; L))
(whose dual is Lp (Ω; Lp# (Rn ; L0 )))
Riez Representation Theorem → ∃U ∈ Lp (Ω; Lp# (Rn ; L0 )) s.t.
Z Z
hµ, φi =
L0 h U(x, ξ), φ(x, ξ)iL dxdξ,
Ω
Z
L0
Ω
[0,1]n
x
h u (x), φ(x, )iL dx →
ε
ε
Z Z
L0 h U(x, ξ), φ(x, ξ)iL
Ω
dxdξ
[0,1]n
as ε → 0
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Strong Two-Scale Convergence criterion
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Strong Two-Scale Convergence criterion
Preliminary results -1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Lemma
0
IF ψ = ψ(x, ξ) ∈ L2 (Ω; C#
(Rn ; L))
([ψ]ε ) Strongly Two-Scale converges to ψ
x
(recall: [ψ]ε (x) = ψ(x, ))
ε
step 1: Two-Scale convergence
Consequence of the Riemann-Lebesgue generalization
Z
Z Z
x
x
L h ψ(x, ), φ(x, )iL dx →
L h ψ(x, ξ), φ(x, ξ)iL dxdξ
ε
ε
Ω
Ω [0,1]n
0
∀φ ∈ L2 (Ω; C#
(Rn ; L)), i.e.
([ψ]ε ) Two-Scale converges to ψ
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Strong Two-Scale Convergence criterion
Preliminary results - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
step 2: Strong Two-Scale convergence
Z x 2
ε
[ψ] (x) − ψ(x, ) 0 dx → 0,
ε L
Ω
x
Completely obvious: [ψ]ε (x) = ψ(x, )
ε
Hence:
([ψ]ε ) Strongly Two-Scale converges to ψ
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Strong Two-Scale Convergence criterion
Preliminary results - 3
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Also easy to prove:
Lemma
0
IF ψ = ψ(x, ξ) ∈ L2 (Ω; C#
(Rn ; L))
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
21
x 2
k[ψ] kL2 (Ω;L) =
ψ(x, ) dx =
ε L
Ω
!12
Z
12
Z Z
x
x
. Lh ψ(x, ), ψ(x, )iL dx →
L h ψ(x, ξ), ψ(x, ξ)iL dxdξ
ε
ε
Ω
Ω [0,1]n
!12
Z Z
Z
ε
2
=
Ω
[0,1]n
Emmanuel Frénod
|ψ(x, ξ)|L dx
= kψkL2 (Ω;L2 (Rn ;L)) .
#
Two-Scale Convergence and Two-Scale Numerical Methods
Strong Two-Scale Convergence criterion
The Criterion
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Theorem
IF(u ε ) ⊂ L2 (Ω; L) Two-Scale converges to U
0
IF U ∈ L2 (Ω; C#
(Rn ; L))
IF
lim ku ε kL2 (Ω;L) = kUkL2 (Ω;L2 ([0,1]n ;L) ,
ε→0
THEN
(u ε ) Strongly Two-Scale converges to U,
and, ∀(v ε ) ⊂ L2 (Ω; L) Two-Scale converging towards V ,
Z
ε
ε
h
u
,
v
i
*
in D0 (Ω).
L
L
L h U(., ξ), V (., ξ)iL dξ,
[0,1]n
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Strong Two-Scale Convergence criterion
The Criterion proof - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
First part of the Theorem
Z x 2
ε
u (x) − U(x, ) dx =
ε L
Z ZΩ
Z
x
x 2
2
ε
|u (x)|L dx − 2 Lh u ε (x), U(x, )iL dx +
U(x, ) dx
ε
ε L
Ω
Ω
Ω
ε→0
Z
lim
ε→0
Ω
2
|u ε (x)|L dx − 2
Order 0
Order 1
Ω
L h U(x, ξ), U(x, ξ)iL
[0,1]n
Z Z
Ω
dxdξ
2
+
Two-Scale
Numerics
Algorithms
Implementation
−→
Z Z
[0,1]n
|U(x, ξ)|L dxdξ =
Z Z
Ω
Z Z
2
2
|U(x, ξ)|L dxdξ − 2
|U(x, ξ)|L dxdξ
[0,1]n
Ω [0,1]n
Z Z
2
+
|U(x, ξ)|L dxdξ = 0.
Ω
Emmanuel Frénod
[0,1]n
Two-Scale Convergence and Two-Scale Numerical Methods
Strong Two-Scale Convergence criterion
The Criterion proof - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Second part of the Theorem - ∀ϕ ∈ D(Ω)
Z
ε
ε
ε
ε
0
D hLh u , v iL , ϕiD =
L h u (x), v (x)iL ϕ(x) dx =
Ω
Z
Z
x
x
ε
ε
ε
),
v
(x)i
ϕ(x)
dx+
h
U(x,
L
L h u (x)−U(x, ), v (x)iL ϕ(x) dx.
L
ε
ε
Ω
Ω
Z
x
x
• u ε (x) − U(x, ) → 0 → Lh u ε (x) − U(x, ), v ε (x)iL ϕ(x) dx → 0
ε
ε
Ω
Z
Z
x
x
ε
• . Lh U(x, ), v ε (x)iL ϕ(x) dx =
Lh v (x), U(x, )iL ϕ(x) dx =
ε
ε
Ω
Ω
Z
Z Z
x
ε
Lh v (x), ϕ(x)U(x, )iL dx →
Lh V (x, ξ), ϕ(x)U(x, ξ)iL dxdξ
ε
Ω
Ω [0,1]n
Z Z
=
Lh V (x, ξ), U(x, ξ)iL ϕ(x) dxdξ
Ω
[0,1]n
Z Z
Lh U(x, ξ), V (x, ξ)iL
=
Ω
Emmanuel Frénod
ϕ(x) dxdξ
[0,1]n
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Homogenization of
singularly perturbed
Hyperbolic Partial
Differential Equations
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Motivation : Tokamaks and
Stellarators
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Equation of interest
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Some words on Tokamaks and Stellarators - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Some words on Tokamaks and Stellarators - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Edge instability Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Discharge simula5on Hyperbolic
PDEs
Turbulence Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
∂f ε
M
+ v · ∇x f ε + (Eε + v × (Bε +
)) · ∇v f ε = 0
∂t
ε
∂f ε
v⊥
M
+ vk · ∇x f ε +
· ∇x f ε + (Eε + v ×
) · ∇v f ε = 0
∂t
ε
ε
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Equation of interest and setting
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Equation of interest and setting
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
∂u ε
t
1
+ a(t, , x) · ∇u ε + b · ∇u ε = 0
∂t
ε
ε
u ε |t=0 = u0
u ε = u ε (t, x), x ∈ Rd , t ∈ [0, T ), for T > 0
Assumptions:
a is regular
∇·a=0
τ 7→ a(t, τ, x) periodic of period 1
b = b(x) = Mx, M matrix s.t.
trM = 0
τ 7→ e τ M periodic of period 1
⇒ ∇ · b = 0 and τ 7→ X(τ ) = e τ M x periodic of period 1
( ∂X
∂τ = MX = b(X), X(0) = x)
u0 ∈ L2 (Rd )
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
A priori estimate
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
A priori estimate
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Z
∂u ε
t
1
+ a(t, , x) · ∇u ε + b · ∇u ε = 0 ×u ε ,
dx →
ε Z
ε
Z ∂t ε
Z Rd
∂u ε
t
1
u dx +
a(t, , x) · ∇u ε u ε dx +
b · ∇u ε u ε dx = 0
εZ
ε
d
Rd ∂t
Rd
R
ε 2
ε
Z
d
|u
|
dx
d
ku
k
ε
2
d
L (R ))
∂u ε
1
1
Rd
u dx =
=
2
2Z
dt
ZRd ∂t
Z dt
a · ∇u ε u ε dx = −
d
RZ
−
a · ∇u ε u ε dx −
Rd
∇ · a u ε u ε dx =
Rd
a · ∇u ε u ε dx = 0
Rd
Same thing
for last term
d ku ε kL2 (Rd ))
= 0→ ku ε kL2 (Rd )) constant→ ku ε kL2 ([0,T );L2 (Rd )) bounded
.
dt
(u.ε ) Two-Scale Converges to U = U(t, τ, x) ∈ L2 ([0, T ); L2# ((R; L2 (Rd )))
up to a subsequence
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Order 0 Homogenization
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Weak Formulation with Oscillating Test Functions
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 0 Homogenization
Weak Formulation With Oscillating Test Functions
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
For
t
φ = φ(t, τ, x) regular: [φ]ε (t, x) = φ(t, , x)
ε
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
ε
ε
∂φ
1 ∂φ
∂[φ]ε
=
+
∂t
∂t
ε ∂τ
ε
Z
∂u
t
1
[φ]ε ×
+ a(t, , x) · ∇u ε + b · ∇u ε ,
, IBP ⇒
∂t
ε
ε
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Z
T
Z
.
0
u
Rd
ε
∂φ
∂t
ε
ε
1 ∂φ
t
1
ε
ε
+
+ a(t, , x) · [∇φ] + b · [∇φ] dxdt
ε ∂τ
ε
ε
Z
+
u0 φ(0, 0, .) dx = 0
Rd
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Order 0 Homogenization - Constraint
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 0 Homogenization
Constraint
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
WFOTF:
ε
ε
Z TZ
1 ∂φ
t
1
∂φ
ε
ε
ε
+
+ a(t, , x) · [∇φ] + b · [∇φ] dxdt
.
u
∂t
ε ∂τ
ε
ε
0
Rd
Z
+
u0 φ(0, 0, .) dx = 0
Rd
×ε, ε → 0 →
∂U
+ b · ∇U = 0
∂τ
→
∃V (t, y) ∈ L2 ([0, T ); L2 (Rd )) s.t. U(t, τ, x) = V (t, e −τ M x)
Two-Scale
Numerics
Algorithms
Implementation
(Recall:
∂(e τ M x)
= M(e τ M x) = b(e τ M x)
∂τ
∂(V (t, e −τ M x))
+ b · ∇(V (t, e −τ M x)) =
∂τ
∇V (t, e −τ M x))·((−e −τ M )Mx)+((e −τ M )Mx)·∇V (t, e −τ M x)) = 0)
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Order 0 Homogenization - Equation for V
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 0 Homogenization
Equation for V - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
γ = γ(t, y) regular: φ(t, τ, x) = γ(t, e −τ M x) s.t.
For
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
∂φ
+ b · ∇φ = 0
∂τ
In WFOTF →
ε
Z TZ
Z
∂φ
t
ε
uε
+ a(t, , x) · [∇φ] dxdt +
u0 φ(0, 0, .) dx = 0
∂t
ε
0
Rd
Rd
ε→0 →
Z
T
Z
1
Z
U(t, τ, x)
0
0
Rd
∂φ
(t, τ, x) + a(t, τ, x) · ∇φ(t, τ, x) dxdτ dt
∂t
Z
+
u0 φ(0, 0, .) dx = 0
Rd
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 0 Homogenization
Equation for V - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Z
T
Z
1
Z
U(t, τ, x)
0
0
Rd
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
∂φ
(t, τ, x) + a(t, τ, x) · ∇φ(t, τ, x) dxdτ dt
∂t
Z
+
u0 φ(0, 0, .) dx = 0
Rd
U in terms of V ; φ in terms of γ
∂φ
∂γ
(t, τ, x) =
(t, e −τ M x) and ∇φ(t, τ, x) = (e −τ M )T ∇γ(t, e −τ M x)
∂t
∂t
→
Z T Z 1Z
∂γ
V (t, y)
(t, y) + e −τ M a(t, τ, e τ M y) · ∇γ(t, y) dydτ dt
∂t
0
0
Rd
Z
+
u0 (y) γ(0, y) dy = 0
Rd
∂V
+
∂t
Z
1
e −σM a(t, σ, e σM y) dσ · ∇V = 0 V|t=0 = u0
0
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Order 1 Homogenization
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
To simplify computations :
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
From now: a(t, τ, x) = a(x)
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Order 1 Homogenization - Preparations: Equation for U
and u
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization
Equation for U and u - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Linearity → Equation for U → Equation for u (w-∗ limit of (u ε )):
WRITE
Z 1
∂V
−σM
σM
+
e
a(e y) dσ · ∇V = 0 in y = e −τ M x
∂t
0
USE: U(t, τ, x) = V (t, e −τ M x)
∇U(t, τ, x) = (e −τ M )T ∇V (t, e −τ M x) i.e.
∇V (t, e −τ M x) = (e τ M )T ∇U(t, τ, x) →
Z 1
∂ V (t, e −τ M x)
−σM
σM −τ M
0=
+
e
a(e e
x)dσ ·∇V (t, e −τ M x)
∂t
0
Z 1
∂U
=
+ eτM
e −σM a(e (σ−τ )M x)dσ · ∇U
∂t
0
Z 1
∂U
=
+
e (τ −σ)M a(e (σ−τ )M x)dσ · ∇U
∂t
0
Z 1
∂U
−σM
σM
=
+
e
a(e x)dσ · ∇U,
∂t
0
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization
Equation for U and u - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Z 1
∂ V (t, e −τ M x)
−σM
σM −τ M
0=
+
e
a(e e
x)dσ ·∇V (t, e −τ M x)
∂t
0
Z 1
∂U
(τ −σ)M
(σ−τ )M
=
+
e
a(e
x)dσ · ∇U
∂t
0
Z 1
∂U
=
+
e −σM a(e σM x)dσ · ∇U,
∂t
0
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
→
Z 1
∂U
+
e −σM a(e σM x)dσ · ∇U = 0, U|t=0 = u0 (e −τ M x)
∂t
0
Z 1
u=
U(., τ, .)dτ →
0
∂u
+
∂t
Z
1
e
−σM
a(e
σM
Z
x)dσ · ∇u = 0, u|t=0 =
0
1
u0 (e −τ M x) dτ
0
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Order 1 Homogenization - Strong Two-Scale convergence
of u ε
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization
Strong Two-Scale convergence of U - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
∂u ε
∂(u ε )2
= 2u ε
and ∇(u ε )2 = 2u ε ∇u ε
∂t
∂t
multiplying equation for u ε by 2u ε →
∂(u ε )2
1
+ a · ∇(u ε )2 + b · ∇(u ε )2 = 0 (u ε )2|t=0 = u02
∂t
ε
2
2
d
4
IF u0 ∈ L (R ), i.e. if u0 ∈ L (Rd ), doing the same →
(u ε )2 Two-Scale converges to Z solution to
Hyperbolic
PDEs
Order 0
Order 1
∂Z
+
∂t
Two-Scale
Numerics
Algorithms
Implementation
Z
Z|t=0 =
→Z =U
1
e
−σM
a(e
σM
x)dσ · ∇Z = 0
0
u02 (e −τ M x)
2
((u ε )2 ) Two-Scale Converges to U 2
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization
Strong Two-Scale convergence of U - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
((u ε )2 ) Two-Scale Converges to U 2
→
ku ε kL2 ([0,T );L2 (Rd )) → kUkL2 ([0,T );L2 ((R;L2 (Rd )))
#
0
d
Moreover: IF u0 ∈ C (R ) →
0
u ε ∈ C 0 ([0, T ); C 0 (Rd )), U ∈ C 0 ([0, T ); C#
((R; C 0 (Rd ))),
V ∈ C 0 ([0, T ); C 0 (Rd ))
HENCE: IF u0 ∈ (L2 ∩ L4 ∩ C 0 )(Rd ), THEN in addition to every
already stated results
(u ε ) Strongly Two-Scale Converges to U
(We have: (u ε − [U]ε ) → 0
Now: Get more: (u ε − [U]ε )/ε Two-Scale Converges)
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Order 1 Homogenization - Function W1
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization - Function W1 - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Step 1:
h ∂U iε 1 h ∂U iε h ∂U iε 1
∂U
∂[U]ε
+ b · ∇U = 0 →
=
+
=
− b · ∇[U]ε
∂τ
∂t
∂t
ε ∂τ
∂t
ε
Z 1
∂U
+
e −σM a(e σM x)dσ · ∇U = 0, U|t=0 = u0 (e −τ M x)
∂t
0
∂u ε
1
+ a(x) · ∇u ε + b · ∇u ε = 0, u ε |t=0 = u0
∂t
ε
→
ε
u − [U]ε
ε
ε
∂
u − [U]ε
u − [U]ε
1
ε
+a·∇
+ b·∇
∂t
ε
ε
ε
Z 1
1
−σM
σM
=−
a−
e
a(e x)dσ · ∇[U]ε
ε
0
ε
u − [U]ε
=0
ε
|t=0
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization - Function W1 - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Step 2: DEFINE: W1 = W1 (t, τ, y) s.t
W̃1 = W̃1 (t, τ, x) = W1 (t, τ, e −τ M x) solution to
Z 1
∂ W̃1
+ b · ∇W̃1 = − a −
e −σM a(e σM x)dσ · ∇U
∂τ
0
THEN: [W̃1 ]ε = [W̃1 ]ε (t, x) = W̃1 (t, t/ε, x):
∂[W̃1 ]ε
1
+ a · ∇[W̃1 ]ε + b · ∇[W̃1 ]ε
∂t "
#ε
" ε #ε
∂ W̃1
1 ∂ W̃1
1
=
+
+ a · ∇[W̃1 ]ε + b · ∇[W̃1 ]ε
∂t
ε ∂τ
ε
"
#ε
Z 1
1
∂ W̃1
=
+ a · ∇[W̃1 ]ε −
a−
e −σM a(e σM x)dσ · ∇[U]ε
∂t
ε
0
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization - Function W1 - 3
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
∂
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
ε
u ε − [U]ε
1
u − [U]ε
+ b·∇
ε
ε
ε
Z 1
1
−σM
σM
ε
a−
e
a(e x)dσ · ∇[U]
=−
ε
ε
0
∂[W̃1 ]
1
ε
ε
+ a · ∇[W̃1 ] + b · ∇[W̃1 ]
∂t
ε
#ε
"
Z 1
1
∂ W̃1
ε
−σM
σM
ε
+ a · ∇[W̃1 ] −
=
a−
e
a(e x)dσ · ∇[U]
∂t
ε
0
Emmanuel
Frénod
Hyperbolic
PDEs
u ε − [U]ε
ε
∂t
+a·∇
u ε − [U]ε
ε
− [W̃1 ]ε
u − [U]ε
ε
+a·∇
− [W̃1 ]ε
∂t
ε
"
#ε
ε
∂ W̃1
1
u − [U]ε
ε
+ b·∇
− [W̃1 ] = −
− a · ∇[W̃1 ]ε
ε
ε
∂t
ε
u − [U]ε
ε
− [W̃1 ]
= −[W̃1 ]ε |t=0
ε
|t=0
∂
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization - Function W1 - 4
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Step 3: expression of the function W1 :
W̃1 (t, τ, x) = W1 (t, τ, e −τ M x)
Z 1
∂ W̃1
+ b · ∇W̃1 = − a −
e −σM a(e σM x)dσ · ∇U
∂τ
0
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
→
Z 1
∂W1
τM
−σM
(σ+τ )M
= − a(e y) −
e
a(e
y)dσ · ∇U(t, τ, e τ M y)
∂τ
0
∇U(t, τ, e τ M y) = (e −τ M )T ∇ U(t, τ, e τ M y) = (e −τ M )T ∇V (t, y)
→
Z 1
∂W1
−τ M
τM
−(σ+τ )M
(σ+τ )M
=− e
a(e y) −
e
a(e
y) dσ ·∇V (t, y)
∂τ
0
Z 1
= − e −τ M a(e τ M y) −
e −σM a(e σM y) dσ · ∇V (t, y)
0
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization - Function W1 - 5
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Z 1
∂W1
= − e −τ M a(e τ M y) −
e −(σ+τ )M a(e (σ+τ )M y) dσ ·∇V (t, y)
∂τ
0
Z 1
−τ M
τM
−σM
σM
=− e
a(e y) −
e
a(e y) dσ · ∇V (t, y)
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
0
→
W1 (t, τ, y) =
Z τ
Z
e −σM a(e σM y) dσ − τ
−
1
e −σM a(e σM y) dσ · ∇V (t, y)
0
0
ε
By-product: [W̃1 ] |t=0 = 0
"
#ε
∂ W̃1
. −
− a · ∇[W̃1 ]ε ∂t
≤ C1
L∞ ([0,T );L2 (Rd ))
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Order 1 Homogenization - A priori estimate and
convergence
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization
A priori estimate and convergence - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
u ε − [U]ε
ε
ε
− [W̃1 ]
∂
u − [U]ε
ε
+a·∇
− [W̃1 ]ε
∂t
ε
"
#ε
ε
ε
∂ W̃1
1
u − [U]
ε
+ b·∇
− [W̃1 ] = −
− a · ∇[W̃1 ]ε
ε
ε
∂t
ε
u − [U]ε
ε
− [W̃1 ]
= −[W̃1 ]ε |t=0 = 0
ε
|t=0
Z
×((u ε − [U]ε )/ε − [W̃1 ]ε ),
dx, IBP →
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Rd
Algorithms
Implementation
Z
d
Rd
!
ε
2
u − [U]ε
− [W̃1 ]ε dx
ε
dt
Z
≤ C1
Rd
Emmanuel Frénod
!21
ε
2
u − [U]ε
− [W̃1 ]ε dx
ε
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization
A priori estimate and convergence - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
u ε − [U]ε
− [W̃1 ]ε
ε
and consequently
u ε − [U]ε
ε
bounded in L2 ([0, T ); L2 (Rd )). Then, up to subsequences,
ε
u − [U]ε
Two-Scale Converges to U1 = U1 (t, τ, x)
ε
ε
u − [U]ε
ε
− [W̃1 ]
Two-Scale Converges to U1 − W̃1
ε
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Order 1 Homogenization - Constraint
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization
Constraint
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
1
WFOTF : φ = φ(t, τ, x) ∈ C 1 ([0, T ); C#
((R; C 1 (Rd )))
T
Z
0
Z
Rd
u ε − [U]ε
− [W̃1 ]ε
ε
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
=
0
×ε, ε → 0
ε
ε
1
1 ∂φ
ε
ε
+
+ a · [∇φ] + b · [∇φ] dxdt
ε ∂τ
ε
!
ε
∂ W̃1
ε
−
− a · ∇[W̃1 ] [φ]ε dxdt
∂t
Rd
∂φ
∂t
Z TZ
→
∂(U1 − W̃1 )
+ b · ∇(U1 − W̃1 ) = 0
∂τ
∃V1 = V1 (t, y) ∈ L2 ([0, T ); L2 (Rd )) s.t.
U1 (t, τ, x) − W̃1 (t, τ, x) = V1 (t, e −τ M x) i.e.
U1 (t, τ, x) = V1 (t, e −τ M x) + W1 (t, τ, e −τ M x)
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Order 1 Homogenization - Equation for V1
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization
Equation for V1 - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
And also Homogenization
Typical proofs
Definitions and
results
T
Z
1
Z
Z
V1 (t, e
0
Two-Scale
Numerics
Algorithms
Implementation
∂γ
−τ M
−τ M
−τ M
x)
(t, e
x) + e
a(x) · ∇γ(t, e
x) dxdτ dt
∂t
Z 1Z ∂ W̃1
−
− a(x) · ∇W̃1 γ(t, e −τ M x)dxdτ dt
∂t
0
Rd
−τ M
Rd
0
T
Z
=
0
Hyperbolic
PDEs
Order 0
Order 1
∂φ
+ b · ∇φ = 0
∂τ
USE φ(t, τ, x) in WFOTF, ε → 0 →
Emmanuel
Frénod
Two-Scale
Convergence
γ = γ(t, y) regular: φ(t, τ, x) = γ(t, e −τ M x) s.t.
For
change of variables (t, τ, x) 7→ (t, τ, y = e −τ M x) gives
T
1
∂γ
−τ M
τM
V1 (t, y)
(t, y) + e
a(e y) · ∇γ(t, y) dydτ dt
∂t
0
0
Rd
Z T Z 1Z ∂W1
=
−
− e −τ M a(e τ M y) · ∇W1 γ(t, y)dydτ dt
∂t
0
0
Rd
Z
Z
Z
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization
Equation for V1 - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Z
Emmanuel
Frénod
T
Z
0
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Z
∂γ
(t, y) + e −τ M a(e τ M y) · ∇γ(t, y) dydτ dt
∂t
∂W1
−τ M
τM
−e
a(e y) · ∇W1 γ(t, y)dydτ dt
−
∂t
V1 (t, y)
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
1
Z
0
T
Z
Rd
1Z
=
0
0
Rd
Z
1
→
∂V1
+
∂t
e −σM a(e σM y)dσ · ∇V1 =
0
Z
Algorithms
Implementation
1
0
∂W1
−τ M
τM
−
−e
a(e y) · ∇W1 dτ
∂t
V1|t=0 = 0
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 1 Homogenization
Equation for V1 - 3
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Heavy computation to get:
Z 1
∂W1
− e −τ M a(e τ M y) · ∇W1 dτ
−
∂t
0
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Z 1
∂V1
−σM
σM
+
e
a(e y)dσ · ∇V1 =
∂t
0
Z τ
Z 1
−τ M
∇ e
a(e τ M y)
e −σM a(e σM y) dτ
0
0
Z 1
!
Z 1
1
−σM
σM
−σM
σM
+
e
a(e y) dσ
e
a(e y) dσ
· (∇V )
∇
2
0
0
V1|t=0 = 0.
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Two-Scale Numerical
Methods
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Motivation : Tokamaks and
Stellarators
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Long term target : 10 ms of a Tokamak working
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Edge instability Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Discharge simula5on Hyperbolic
PDEs
Turbulence Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
∂f ε
M
+ v · ∇x f ε + (Eε + v × (Bε +
)) · ∇v f ε = 0
∂t
ε
∂f ε
v⊥
M
+ vk · ∇x f ε +
· ∇x f ε + (Eε + v ×
) · ∇v f ε = 0
∂t
ε
ε
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Two-Scale Numerical
Method Algorithms
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Algorithm for order 0 Two-Scale Numerical Method
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
To compute u ε solution to
∂u ε
t
1
+ a(t, , x) · ∇u ε + b · ∇u ε = 0
∂t
ε
ε
u ε |t=0 = u0 .
for ε small:
Compute V solution to
Z 1
∂V
+
e −σM a(t, σ, e σM y) dσ · ∇V = 0 V|t=0 = u0
∂t
0
And use
t
u ε (t, x) ∼ U(t, , x)
ε
Emmanuel Frénod
t
t
U(t, , x) = V (t, e − ε M x)
ε
Two-Scale Convergence and Two-Scale Numerical Methods
Algorithm for order 1 Two-Scale Numerical Method
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
For ε small, to compute u ε solution to
∂u ε
1
+ a(x) · ∇u ε + b · ∇u ε = 0
∂t
ε
u ε |t=0 = u0 .
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Compute: W1 (t, τ, y) =
Z τ
Z
−
e −σM a(e σM y) dσ − τ
0
1
e −σM a(e σM y) dσ
· ∇V (t, y)
0
Compute: V and V1 solution to
Z 1
∂V
−σM
σM
+
e
a(e y) dσ · ∇V = 0 V|t=0 = u0
∂t
0
Z 1
∂V1
+
e −σM a(e σM y)dσ · ∇V1 = RHS(V )
∂t
0
And use
t
t
u ε (t, x) ∼ U(t, , x) + εU1 (t, , x)
ε
ε
t
t
t
t
= V (t, e − ε M x) + ε(V1 (t, e − ε M x) + W1 (t, , e − ε M x))
ε
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Two-Scale Numerical
Method implementation for
beam simulation
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
A beam in a focusing channel
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
A beam in a focusing channel
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
x
y
E~r
And also Homogenization
Typical proofs
Definitions and
results
vr
r
Hyperbolic
PDEs
z~t
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
A beam in a focusing channel - Simulation
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
A beam in a focusing channel - Simulation
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
A beam in a focusing channel - Simulation
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Digression on Pic Methods
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Pic Method explained - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Pic Method explained - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Pic Method explained - 3
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Pic Method explained - 4
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Pic Method explained - 5
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Pic Method explained - 6
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Beam in a focusing channel : PDE Model
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
PDE Model
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
fε = fε (t, r , vr ), t ∈ [0, T ), r ∈ R+ and vr ∈ R:

2
2

 ∂fε + 4π vr ∂fε + Er − 4π r ∂fε = 0

ε

 ∂t
ε
∂r
ε
∂vZr
1 ∂(r Er ε )
= ρε (t, r ),
ρε (t, r ) =
fε (t, r , vr ) dvr



r ∂r
R


fε (t = 0, r , vr ) = f0
∂u ε
1
+ aε · ∇u ε + b · ∇u ε = 0 with x replaced by (r , vr ) and
∂t
ε
2 0
4π vr
ε
a =
,
b=
Er ε (t, r )
−4π 2 r
M=
0
−2π
2π τ M
cos(2πτ ) sin(2πτ )
e
=
0
− sin(2πτ ) cos(2πτ )
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Order 0 Homogenization
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Assumptions: Zf0 ≥ 0, f0 ∈ (L1 ∩ Lp )(R2 ; rdrdvr ) for p ≥ 2
R2
(r 2 + vr2 )f0 rdrdvr < +∞
Then:
2
2
fε Two-Scale Converges to F ∈ L∞ ([0, T ); L∞
# (R; L (R ; rdrdvr )))
∞
∞
1,3/2
Er ε Two-Scale Converges to Er ∈ L ([0, T ); L# (R; W
(R; rdr )))
∞
2
2
∃G = G (t, q, ur ) ∈ L ([0, T ); L (R ; qdqdur )):
F (t, τ, r , vr ) = G (t, cos(2πτ )r − sin(2πτ )vr , sin(2πτ )r + cos(2πτ )vr )

Z 1
∂G
∂G


+
− sin(2πσ) Er (t, σ, cos(2πσ)q + sin(2πσ)ur ) dσ


 ∂t
∂q
0 Z
1
∂G

+
cos(2πσ) Er (t, σ, cos(2πσ)q + sin(2πσ)ur ) dσ
=0


∂u

r
0

G (t = 0) = f0
Er = Er (t, τ, r , vr ):
Z
1 ∂(r Er )
=
G t, cos(2πτ )r − sin(2πτ )vr , sin(2πτ )r + cos(2πτ )vr ) dvr
r ∂r
R
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Two-Scale Pic Method for a beam in a focusing channel
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Pic Method to compute G - 1
Introduction
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
G approximated by GN (q, u, t) =
N
X
wk δ(q − Qk (t))δ(u − Uk (t))
k=1
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
From (Qkl , Ukl ) at time tl , compute (Qkl+1 , Ukl+1 ) as an approximated
solution to
Z 1
dQk
=−
sin(2πσ) Er (t, σ, cos(2πσ)Qk + sin(2πσ)Uk ) dσ, Qk (tl ) = Qkl
dt
0
Z 1
dUk
cos(2πσ) Er (t, σ, cos(2πσ)Qk + sin(2πσ)Uk ) dσ, Uk (tl ) = Ukl
=
dt
0
at time tl+1 = tl + ∆t
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Pic Method to compute G - 1
Recall Runge-Kutta 4 Method
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
tl,1 = tl , y l,1 = y l
1
∆t
, y l,2 = y l + I 1 with I 1 = ∆t K (tl,1 , y l,1 ),
tl,2 = tl +
2
2
∆t
1 2
l,3
l
tl,3 = tl +
, y = y + I with I 2 = ∆t K (tl,2 , y l,2 ),
2
2
tl,4 = tl + ∆t, y l,4 = y l + I 3 , with I 3 = ∆t K (tl,3 , y l,3 )
1
1
1
1
y l+1 = y l + I 1 + I 2 + I 3 + I 4 with I 4 = ∆t K (tl,4 , y l,4 )
6
3
3
6
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Pic Method to compute G - 1
Implementation - 1
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
And also Homogenization
Typical proofs
Definitions and
results
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
In other words, we have to compute Qkl,2 as follows:
1
Qkl,2 = Qkl + I 1 with
2
p
X
I 1 = ∆t −
γm sin(2πσm ) Er (tl , σm , cos(2πσm )Qkl + sin(2πσm )Ukl )
m=1
1
Qkl,3 = Qkl + I 2 with
2
p
X
I 2 = ∆t −
γm sin(2πσm )
m=1
Er2 (tl +
Emmanuel Frénod
∆t
, σm , cos(2πσm )Qkl,2 + sin(2πσm )Ukl,2 )
2
Two-Scale Convergence and Two-Scale Numerical Methods
Two-Scale Pic Method to compute G - 1
Implementation - 2
Two-Scale
Convergence
and Two-Scale
Numerical
Methods
Emmanuel
Frénod
Two-Scale
Convergence
Qkl,4 = Qkl + I 3 , with
p
X
I 3 = ∆t −
γm sin(2πσm )
m=1
And also Homogenization
Typical proofs
Definitions and
results
Er3 (tl +
∆t
, σm , cos(2πσm )Qkl,3 + sin(2πσm )Ukl,3 )
2
Hyperbolic
PDEs
Order 0
Order 1
Two-Scale
Numerics
Algorithms
Implementation
1
1
1
1
Qkl+1 = Qkl + I 1 + I 2 + I 3 + I 4 , with
6
3
3
6
p
X
I 4 = ∆t −
γm sin(2πσm )
m=1
Er4 (tl + ∆t, σm , cos(2πσm )Qkl,4 + sin(2πσm )Ukl,4 )
Emmanuel Frénod
Two-Scale Convergence and Two-Scale Numerical Methods