Normal forms and geometric numerical integration of ...Normal forms and geometric numerical...
Transcript of Normal forms and geometric numerical integration of ...Normal forms and geometric numerical...
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