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Normal forms and geometric numericalintegration of Hamiltonian PDEs
Part I: Linear equations
Erwan Faou
INRIA & ENS Cachan Bretagne
Beijing, 21 May 2009
Joint works with Guillaume Dujardin (Cambridge)Arnaud Debussche (ENS Cachan Bretagne)
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 1 / 35
Time-dependent Schrodinger equation
i∂
∂tϕ(x , t) = −∆ϕ(x , t) + V (x)ϕ(x , t), ϕ(x , 0) = ϕ0(x).
x ∈ Td (d = 1). Laplace operator ∆ = ∂xx .
V (x) ∈ R analytic function.
Conservation properties : if H = −∆ + V
∀ s > 0, 〈ϕ(t)|Hs |ϕ(t)〉 = 〈ϕ(0)|Hs |ϕ(0)〉.
s = 1 : energy. s = 0 : L2 norm.
This implies the conservation of the regularity over long time.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 2 / 35
Splitting methods
Th = exp(ih∆) exp(−ihV )
where
e it∆ϕ0 :
{i∂tϕ(t, x) = −∆ϕ(t, x) (t, x) ∈ R× Tϕ(0, x) = ϕ0(x) x ∈ T
and
e−itVϕ0 :
{i∂tϕ(t, x) = V (x)ϕ(t, x) (t, x) ∈ R× Tϕ(0, x) = ϕ0(x) x ∈ T
Stepsize : h > 0
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 3 / 35
Splitting methods
Properties :
Easy to compute using FFT.
Order 1, effective order 2 (Jahnke & Lubich, 2000)
L2 norm conservation.
Practical computations : Resonances.
Long time behaviour in the infinite dimensional case ?
Same long time behavior as the Strang splitting
ϕ1 = exp(−ihV /2) exp(ih∆) exp(−ihV /2)ϕ0
.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 4 / 35
Numerical test
V (x) =0.03
5− 4 cos(x)and ψ0(x) = sin(x)
Stepsizes :
resonant : h =2π
62 − 22= 0.196 . . .
non resonant : h = 0.2
We plot the energies errors∣∣|ϕn|2k − |ϕ0|2k
∣∣ where
|ϕ|2k = |ϕk |2 + |ϕ−k |2.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 5 / 35
Conservation of energies
0 2 4 6 8 10x 104
!10
!8
!6
!4
!2
0
Iterations
log
of e
nerg
ies
0 2 4 6 8 10x 104
!10
!8
!6
!4
!2
0
Iterations
log
of e
nerg
ies
Fig.: Energies error. Non resonant stepsize (left) and resonant stepsize (right)
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 6 / 35
Modified energy ?
BCH formula :
Th = exp(ih∆) exp(−ihV )
' exp(ih(−∆ + V − 12 (ih)[−∆,V ] + · · ·+ (ih)kHk · · · ))
For all k, Hk is an operator of order k.
Does not converge for h > 0.No standard backward error analysis available.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 7 / 35
Analytic functions and operatorsWe identify a function ϕ(x) with its Fourier coefficientsϕn = 1
2π
∫T e−inxϕ(x)dx
Analytical norm for functions : ‖ϕ‖ρ
= supn∈Z
(eρ|n||ϕk |
)Operators : S = (Sij)i ,j∈Z. Action : (Sϕ)i =
∑j∈Z Sijϕj .
Product of two operators : (AB)ij =∑
k∈Z AikBkj
Analytical norm for operators : ‖S‖ρ
= supk,`∈Z
(eρ|k−`||Sk`|
)‖AB‖
ρ≤ C
δ‖A‖
ρ‖B‖
ρ+δ
Hypothesis on V : ‖V ‖ρV<∞. As operator : Vij = Vi−j .
V real implies V symmetric (Sij = S∗ji )
Aρ = { S | ‖S‖ρ<∞}
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 8 / 35
A family of schemesThe idea is consider V as a small perturbation of −∆.We embed Th into the family of unitary propagators :
L(λ) = exp(ih∆) exp(−ihλV ), λ > 0.
λ = 0 : Free Schrodinger operator.e ih∆ diagonal operator with entries e−ihk2
.Conservation of |ψn| for all n ∈ Z.
Is it possible to find a normal form for L(λ) (λ small) :
Q(λ)L(λ)Q(λ)∗ = Σ(λ)
Q(0) = Id, Q(λ) unitary : Q(λ)Q(λ)∗ = Id
Σ(0) = e ih∆, Σ(λ) unitary and “nice” ( ! ?) (with conservationproperties)
Q(λ) and Σ(λ) are in Aρ for some ρ
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 9 / 35
Formal series equations
To control the unitarity of Q(λ) and Σ(λ) we introduce
S(λ) = Q(λ)∗(i∂λQ(λ)) and X (λ) = Σ(λ)∗(i∂λΣ(λ))
S and X symmetric implies Q and Σ unitary.
Equation : S(λ)− L∗(λ)S(λ)L(λ) = hV − Q(λ)∗X (λ)Q(λ)
Formal series : S(λ) =∑
n≥0 λnSn, X (λ) =
∑n≥0 λ
nXn.
Recursive equations
Sn − e−ih∆Sneih∆ + Xn = Gn(V ,Si ,Xi | i = 1, . . . , n − 1)
Gn symmetric if Si and Xi symmetric.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 10 / 35
Homological equation
Given an operator G symmetric, is it possible to find S and Xsymmetric (and nice !) such that
S − e−ih∆Se ih∆ + X = G
In coordinates
∀ (k, `) ∈ Z2, (1− e ih(k2−`2))Sk` + Xk` = Gk`
Problems when h(k2 − `2) ' 2πm, m ∈ Z (resonances)
Non-resonance condition : for all k ∈ Z, k 6= 0,∣∣∣∣1− e ihk
h
∣∣∣∣ ≥ γ|k |−ν , γ > 0, ν > 1.
(see Shang 2000 ; Hairer, Lubich, Wanner (GNI) Chap. X.)Generic condition on h
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 11 / 35
X-shaped operators
(1− e ih(k2−`2))Sk` + Xk` = Gk`
Under the non-resonance condition, we can solve this equation by :
|k | = |`| :
{Sk` = 0,Xk` = Gk`,
|k | 6= |`| :
{Sk` = 1
1−e ih(k2−`2)Gk`,
Xk` = 0,
(S and X symmetric)X-shaped operators :
Xk` 6= 0 =⇒ |k| = |`|
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 12 / 35
Solution of the homological equation
Problem : using the diophantine condition, we do not stay in Aρ.
With the preceding definition, we have
|Sk`| ≤ γ−1h−1|k2 − `2|ν |Gk`|
G ∈ Aρ implies Gk` ≤ Ce−ρ|k−`| but this does not implies S ∈ Aρ.Possible unbounded k + `.
For a given K > 0 we define the set of indices
IK = {(k , `) ∈ Z | |k | ≤ K or |`| ≤ K}
(k , `) ∈ IK =⇒ |k + `| ≤ 2K + |k − `|
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 13 / 35
IK -solution of the homological equation
(1− e ih(k2−`2))Sk` + Xk` = Gk`
Under the non resonance condition, we can solve this equation by :
|k | = |`| or (k , `) /∈ IK :
{Sk` = 0,Xk` = Gk`,
(k , `) ∈ IK such that |k | 6= |`| :
{Sk` = 1
1−e ih(k2−`2)Gk`,
Xk` = 0,
Bounds
‖S‖ρ−δ ≤
K ν
γh
(4ν
eδ
)2ν
‖G‖ρ
and ‖X‖ρ≤ ‖G‖
ρ.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 14 / 35
Nekoroshev machinery
We can prove (α = 2ν and β = 4ν + 3).
‖SJ‖ρV /3+ ‖QJ‖ρV /3
≤(C0K
αJβ)J
and
‖XJ‖ρV /3≤ h
(C0K
αJβ)J,
Optimal truncations : S [N](λ) =∑N
j=0 λjSj , etc.
K ' λ−σ and N ' λ−µ
We have (C0λKαNβ
)N' exp(−cλ−σ)
Almost X-shaped operators : X ∈ Xρ if
Xk` 6= 0 =⇒(|k | = |`| or (k , `) /∈ IK
)E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 15 / 35
A normal form theorem
Theorem [Dujardin & Faou, 2007]∃Q(λ) ∈ AρV /4 and Σ(λ) ∈ XK
ρV /4 with K = λ−σ where
σ = 1/16(ν + 1) < 1 satisfying for λ ∈ (0, λ0)
‖Q(λ)− Id‖ρV /4
≤ C1λ1/2 and ‖Σ(λ)− e ih∆‖
ρV /4≤ C2hλ
1/2
and such that the following relations hold :
Q(λ)∗Q(λ) = Id, and Σ(λ)∗Σ(λ) = Id,
andQ(λ)L(λ)Q(λ)∗ = Σ(λ) + R(λ)
with‖R(λ)‖
ρV /5≤ C3 exp(−cλ−σ).
The constants depend only on V , h0, γ and ν.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 16 / 35
Long time behavior ?
New variables : ψ = Q(λ)ϕ.
Action of Σ(λ) : if ψ1 = Σ(λ)ψ0, then for |k | ≤ λ−σ,(ψ1
k
ψ1−k
)=
(ak(λ) bk(λ)ck(λ) dk(λ)
)(ψ0
k
ψ0−k
)The 2× 2 matrix in this equation is unitary . This implies
∀ |k | ≤ λ−σ |ψ1k |2 + |ψ1
−k |2 = |ψ0k |2 + |ψ0
−k |2,
Notation : For k ≥ 0,
|ψ|2k := |ψk |2 + |ψ−k |2
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 17 / 35
Corollaries
For n ∈ N, let ϕn = L(λ)nϕ0 (in the new variables : ψn = (Σ + R)nψ0).(i) Assume that ϕ0 is in L2 . With the notations of the previous theorem,we have for all n ≤ exp(cλ−σ/2) and all λ ∈ (0, λ0),
∀ |k| ≤ λ−σ∣∣ |ϕn|k − |ϕ0|k
∣∣ ≤ Cλ1/2‖ϕ0‖ ,
for a constant C that depend only on V , h0, γ and ν.Key estimate : (global L2 preservation)∣∣ |ψn+1|k − |ψn|k
∣∣ ≤ C‖R(λ)‖ρV /5‖ψn‖
≤ C exp(−cλ−σ)‖ϕ0‖ .
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 18 / 35
Conservation of regularity
For s > 0, we introduce the norm :
‖ϕ‖s,∞ = sup
k≥0((1 + k)s |ϕ|k)
For all λ ∈ (0, λ0), all n ≤ exp(cλ−σ/2) we have(ii) Let s > 1/2 be given, and let s ′ be such that s − s ′ ≥ 1/2.
sup0≤k≤λ−σ
((1 + k)s′
∣∣ |ϕn|k − |ϕ0|k∣∣ ) ≤ Csλ
1/2‖ϕ0‖s,∞ ,
(iii) Let ρ ∈ (0, ρV ), there exists µ0 ∈ (0, ρ) such that for all µ < µ0,
sup0≤k≤λ−σ
(eµk
∣∣ |ϕn|k − |ϕ0|k∣∣ ) ≤ Cρ,µλ
1/2‖ϕ0‖ρ
The constant depend only on V , h0, γ and ν.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 19 / 35
Numerical test
V (x) =3
5− 4 cos(x)and ψ0(x) = sin(x)
Stepsizes :
bad : h =2π
62 − 22= 0.196 . . .
good : h = 0.2
We plot the energies errors∣∣|ϕn|2k − |ϕ0|2k
∣∣.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 20 / 35
Conservation of energies
0 2 4 6 8 10x 105
!12
!10
!8
!6
!4
!2
0
Iterations
Log(
erro
r)
0 2 4 6 8 10x 105
!20
!15
!10
!5
0
Iterations
Log(
erro
r)
Fig.: Energies error for the 5 first modes, λ = 0.1. Non resonant stepsize (left)and resonant stepsize (right)
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 21 / 35
Conservation of energies
0 2 4 6 8 10x 105
!10
!8
!6
!4
!2
0
Iterations
Log(
erro
r)
0 2 4 6 8 10x 105
!20
!15
!10
!5
0
Iterations
Log(
erro
r)
Fig.: Energies error for the 5 first modes, λ = 0.01. Non resonant stepsize (left)and resonant stepsize (right)
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 22 / 35
Implicit-explicit integrators
Can we find numerical schemes without resonances ?
Can we do backward error analysis for PDEs in some cases ?
Positive answer in the linear case.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 23 / 35
Implicit-explicit integrators
Linear Schrodinger equation on the torus
i∂tu(x , t) = −∆u(x , t) + V (x)u(x , t)
Mid-split scheme :
exp(−ih(−∆ + V )) ' R(ih∆) exp(−ihV )
where
R(z) =1 + z/2
1− z/2' exp(z)
Next figures : plot of the maximal oscillations of the H1 normsbetween t = 0 and t = 50.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 24 / 35
Implicit-explicit integrators
0.02 0.04 0.06 0.08 0.10.30.40.50.60.70.80.9
h0.02 0.04 0.06 0.08 0.11
1.2
1.4
1.6
1.8
2
h
Left : mid-split. Right : classical splitting scheme with resonances.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 25 / 35
Implicit-explicit integrators
Operator exp(−ih∆). Resonances reflect the control of
exp(ih(|k |2 − |`|2)) 6= 1
for all k and ` in Zd , |k | 6= |`|.Mid-split integrators :
R(ih∆) =1 + ih∆/2
1− ih∆/2= exp(2i arctan(h∆/2))
Control of
exp(2i arctan(h|k |2/2)− 2i arctan(h|`|2/2)) 6= 1
Always satisfied !
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 26 / 35
Search for a modified energy
Framework : for operators S = (Sk`)k,`∈Zd ,
‖S‖α
= supk,`|Sk`(1 + |k − `|α)|
We have (α > d)‖AB‖
α≤ Cα‖A‖α ‖B‖α
Search for a function t 7→ Z (t), t ∈ [0, h] such that
exp(−itV )R(ih∆) = exp(iZ (t))
Z (0) = Z0 = 2 arctan(h∆/2), k , ` ∈ Zd ,
(Z0)k` = −δk`2 arctan(h|k |2/2).
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 27 / 35
Search for a modified energy
We differentiate exp(−itV )R(ih∆) = exp(iZ (t)) with respect to t :
iV exp(−itV )R(ih∆) = i(d expiZ(t) Z ′(t)) exp(iZ (t))
equivalent to
Z ′(t) = (d expiZ(t))−1V =∑k≥0
Bk
k!adk
iZ(t)(V )
adA(B) = [A,B] = AB − BA
Bk Bernouilli numbers.
∀ |z | < 2π,∑k≥0
Bk
k!zk =
z
ez − 1.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 28 / 35
Search for a modified energy
Formal series : Z (t) =∑
`≥0 t`Z`Plugging into the previous one :∑
`≥1
`t`−1Z` =∑k≥0
Bk
k!
(i∑`≥0
t`adZ`
)k(V )
=∑`≥0
t`∑k≥0
Bk
k!ik
∑`1+···+`k=`
adZ`1· · · adZ`k
(V ).
Identifying the coefficients in the formal series, we obtain
∀ ` ≥ 1, (`+ 1)Z`+1 =∑k≥0
Bk
k!ik
∑`1+···+`k=`
adZ`1· · · adZ`k
(V ).
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 29 / 35
Search for a modified energy
For ` = 0, this equation yields
Z1 =∑k≥0
Bk
k!ikadk
Z0(V ).
Crucial estimate :‖adZ0W ‖α ≤ π‖W ‖α
Proof :
(adZ0W )k` = −(2 arctan(h|k |2/2)− 2 arctan(h|`|2/2)
)Wk`
and ∣∣2 arctan(h|k |2/2)− 2 arctan(h|`|2/2)∣∣ ≤ π!!
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 30 / 35
Search for a modified energy
Hence we have
‖Z1‖α ≤∑k≥0
Bk
k!πk‖V ‖
α< +∞
Convergent series !
By induction we can show
‖Z`‖α ≤ (C‖V ‖α
)`
Z (h) well defined as a convergent series for
|h| < 1
C‖V ‖α
.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 31 / 35
Modified energy
Theorem [Debussche & Faou, 2009]There exists a symmetric operator S(h) such that for all h ≤ h0
R(ih∆) exp(−ihV ) = exp(ihS(h))
Moreover
S(h) =2
harctan(h∆/2) + V (h)
V (h) modified potential
〈u|S(h)|u〉 invariant of the numerical scheme
No residual term.
Backward error analysis result.
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 32 / 35
Long time behavior
S(h) =2
harctan(h∆/2) + V (h)
We have for the numerical solution un :
〈un|S(h)|un〉 = 〈u0|S(h)|u0〉
Control of the H1 norm for low modes and L2 norm for high modes .
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 33 / 35
Control of the solution
For low modes : |k |2 . 1/h, arctan(x) ' x
2
harctan(h|k |2/2)|uk |2 ' c |k |2|uk |2
For high modes : |k|2 & 1/h, arctan(x) ' π/2,
2
harctan(h|k |2/2)|uk |2 '
c
h|uk |2
Corollary : for all n we have∑|k|≤1/
√h
|k |2|unk |2 +
1
h
∑|k|>1/
√h
|unk |2 ≤ C0‖u0‖2
H1 .
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 34 / 35
Conclusions
In the linear case :
Resonances for pure splitting methodsGenerically no problem
Use of implicit integrator for the unbounded part :No resonancesBackward error analysis resultVery specific (midpoint rule) because of the constant π !
E. Faou (INRIA) Normal forms & numerics for PDEs Beijing, 21 May 2009 35 / 35
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