6.3 Numerical scheme for the SPGPE
6.3.1 Definition of the numerical scheme
As we previously stated, we aim at solving Equation (6.4) in the small temperature limit, which means from a physical point of view to chooseεof order 10−3 to 10−5 (see the end of Section 2.4.3 for a justification of this order of magnitude). Thus, the dynamical system defined by (6.4) contains two time scales: a quick Hamiltonian dynamics and a slow stochastic fluctuation/dissipation dynamics. Moreover, we are interested in the long time effects of the slow dynamics on the Hamiltonian one. Thus, special care must be taken to design a numerical scheme taking into accounts both dynamics. To do so, we propose to split the system into the purely Hamiltonian part,
dφt=−i∇EK(φt) dt, (6.8)
and the reversible stochastic part,
dφt=−ε∇EK(φt) dt+εp
2γdWt. (6.9)
Note that using the change of variablet←tε, that is to say setting ψ(t) :=φ(ε−1t), the process (ψ(t))t≥0 is equal in law to the solution of the following dynamics,
dψt=−∇EK(ψt) dt+p
2γεdWt, (6.10)
for any Brownian motion (Wt)t≥0 taking values inK, since (ε1/2Wε−1t)t≥0 is also a Brow-nian motion.
It appears that the drift over a time step h for Equation (6.9) is ε times smaller than the drift for Equation (6.8). Then, the idea is to alternate one step of resolution of Equation (6.10) withε−1 steps of resolution of Equation (6.8). More precisely, fors, t∈R+
and t≥s, we denote by ΦH(t, s) the flow of Equation (6.8), and by Φ(t, s), ΦL(t, s) and ΦL,ε(t, s) respectively the stochastic flows of equations (6.4), (6.9) and (6.10) between timessand t[110]. Then for any time steph and any n∈N∗, we approximate the law of Φ(nh,0) by
Φ(nh,0)≈
(n−1)ε
Y
k=0
ΦL,ε((k+ 1)h, kh)ΦH(ε−1(k+ 1)h, ε−1kh). (6.11) The approximation (6.11) makes clear that the integration time of Equation (6.8) is ε−1 times longer than the integration time of Equation (6.10). This remark motivates our choice to use a more precise integration method for Equation (6.8) than for (6.10). We
6.3. Numerical scheme for the SPGPE 187 propose to use a simple explicit exponential Euler scheme to approximate the flow ΦL,ε. Because of the additive noise, we classically expect it to be of strong order one, which is the best order we can reach without computing iterated Itˆo integrals. We refer to Section 6.3 [102] for a general presentation of exponential Euler schemes for SPDEs. We propose to use a symplectic Lawson method of high order to approximate the Hamiltonian dynamics.
Furthermore, since it conserves the energy E, and only the Langevin dynamics enables to cross through the levels of energy, we are keen on having a Hamiltonian integrator to conserve the energy well enough between all the numerical integrations of the Langevin dynamics.
We now present the two numerical integrators to approximate the flows ΦL,ε and ΦH. We denote them respectively by ΦL,ε,hand ΦH,h, wherehstands for the time step. In each case, let T be the time horizon of integration. Let (tn)0≤n≤N be a uniform subdivision of [0, T], that is to say for alln∈N,n≤N,tn=nhwithh=T /N (which denotes the time step). Both of these integrators suppose that the nonlinearityPK|φ|2φcan be computed exactly for allφ∈K. We explain in Appendix6.6.1how this can be done in practice.
Numerical integration of the Langevin dynamics
We begin by presenting the numerical integrator for the Langevin dynamics (6.10).
The drift∇E(ψ(t)) can be split into a linear part−PKAand a nonlinear part N(ψ(t)) = PK
µ−g|ψ(t)|2
ψ(t). Thus, a mild formulation of Equation (6.10) is given for alls, t >0 witht≥sby,
ψ(t) = ΦL,ε(t, s)ψ(s)
=S(t−s)ψ(s) + Z t
s
S(t−σ)N(ψ(σ))dσ+p (2γε)
Z t s
S(t−σ)PKdWσ, whereS(·) denotes the semigroup of generator−A. The explicit exponential Euler scheme can be seen as a discretisation of this formulation. It consists in approximating the flow ΦL,ε between times tn and tn+1 by the discrete flow φL,ε,h defined for n ≤ N and any ψ∈L2(R2) by
ΦL,ε,h(tn+1, tn)ψ=S(h)ψ+A−1(S(h)−Id)N(ψ) +p
(2γε) Z tn+1
tn
S(tn+1−σ)PKdWσ.
The first term can be trivially computed and is diagonal in the Gauss-Laguerre basis. The third term, called stochastic convolution is equal in law to a Gaussian random variable with a diagonal covariance matrix, and can be trivially simulated as well. The main diffi-culty consists in computing the non-linearity PK|ψ|2ψ in the Gauss-Laguerre basis. The same difficulty appears in the numerical integration of the Hamiltonian dynamics, and is postponed to Appendix6.6.1.
This scheme is expected to converge strongly at order 1, which is the best order we can obtain without discretising the iterated Itˆo integrals. The use of an exponential Euler scheme rather than a classical Euler scheme comes from classical results from the SPDE literature. In infinite dimension the exponential Euler scheme involves only bounded oper-ators, and thus it is usually more stable and does not require a CFL condition to hold. This property is especially interesting for us since even though the problem is finite dimensional, its dimension may be large.
Integration of the Hamiltonian dynamics
The numerical integrator for the Hamiltonian dynamics (6.8) uses techniques similar to the ones used for the Langevin dynamics. The idea follows from [22], where the authors propose several numerical high order integrators for nonlinear Schr¨odinger equations. We propose to use a Lawson method, as described in [22, Section 3]. Yet, our scheme is slightly different, since it amounts to considering a differentspatial discretisation that coincides with the projection PK as explained above. Thus the main difference comes from the way we split the linear and nonlinear parts of Equation (6.8). We include the harmonic potential and the rotation operator inside the linear part, contrarily to [22]. Moreover, this discretisation seems to be of practical interest thanks to its good approximation properties with few degrees of freedom (see Remark6.9).
The idea consists in approximating Equation (6.8) after a change of unknown that transforms this equation into a nonstiff one. More precisely, suppose that
u(t,r) =S(t,0)−1φ(t,r), (6.12)
whereS is the semigroup of generator−iA. Then for anyφ0 ∈L2(R2),φ(t,r) is solution of Equation (6.8) with initial conditionPKφ0 if and only if u(t,r) is the solution of
(du(t) =S(t,0)−1N(S(t,0)u(t)) dt,
u(0) =PKφ0, (6.13)
with this timeN(v) =i
µ−gPK|v|2
v for all v∈L2(R2).
The Lawson method consists now in applying a Runge-Kutta scheme to Equation (6.13).
For the sake of completeness, we reproduce the clear explanations given in [22]. We consider as-stage Runge-Kutta method with Butcher tableau given by,
c1 a1,1 · · · a1,s ... ... ... cs as,1 · · · as,s
b1 · · · bs
(6.14)
We approximate the flow ΦH between times tn and tn+1,i.e. approximateφn+1 ∈L2(R2)
6.3. Numerical scheme for the SPGPE 189 defined by φn+1 = ΦH,hφn for any φn ∈ L2(R2). To this end, we consider the sequence (un)n∈N, approximating the solutionu of Equation (6.13), defined iteratively by,
un+1 =un+
s
X
k=1
bkhS(tn+ckh)−1N(S(tn+ckh)un,k), where (un,k)1≤k≤s is given for all k∈N∗,k≤sby,
un,k =un+
s
X
l=1
ak,lhS(tn+clh)−1N(S(tn+clh)un,l).
This last system of nonlinearly implicit equations can be solved by a fixed-point method under sufficient conditions given in Theorem 6.2 below. Using the Lawson change of un-known given by (6.12), this method can be written as,
φn+1=S(h)φn+
s
X
k=1
bkhS((1−ck)h)N(φn,k), (6.15) where (φn,k)1≤k≤s is given for all k∈N∗,k≤sby,
φn,k=S(ckh)φn+
s
X
l=1
ak,lhS((ck−cl)h)N(φn,l). (6.16) For all φn ∈ K, we denote by ΦH,h the approximation of the flow ΦH(t+h, t) of Equa-tion (6.8), for allt≥0, given by equations (6.15) and (6.16), if they are well-posed. That is to say, for allφn∈K, we set ΦH,hφn=φn+1 given by Equation (6.15).
Since this numerical scheme is nonlinearly implicit, the well-posedness of ΦH,h is not clear. Actually, since the non-linearity is only locally Lipschitz, as stated in Lemma 6.1, we are not able to prove its well-posedness for all time stepsh, and we need them to be small enough with respect toφn. This is sufficient to show local existence and uniqueness inK by a fixed point method, for a time step h small enough, as stated in Theorem6.2.
Then, since theL2(R2)-norm is conserved by the scheme, and that is a finite dimensional space, it is clear that the numerical scheme is actually globally well-posed in this case.
The classical local Lipschitz property is given by the following lemma.
Lemma 6.1. For allφ1, φ2 ∈K, for all k >1(=d/2)there exists C(k)>0 (independent of the cutoff K) such that,
kN(φ1)− N(φ2)kΣk ≤C(k)kφ1−φ2kΣk
1 +
φ1
2 Σk+
φ2
2 Σk
, with for all φ∈K,
kφkΣk =hAkφ, φiL2,
where the operator A is given by Equation (6.6), and h·,·iL2 denotes the L2(Rd)-scalar product.
Theorem 6.2. Let k > d/2. For all M > 0, there exists a maximal time step h0 > 0 (uniform with respect to K) such that for all h ≤ h0 and for all φn ∈ K satisfying kφ0kΣk ≤M, the system composed by Equations (6.15) and (6.16) has a unique solution.
Proof of Theorem 6.2. The local existence follows from a Picard argument in any space L∞([0, T],Σk) for k∈N, and Lemma 6.1.
In the following, we use a special kind of Lawson method, called Gauss-Lawson method.
It is actually a special kind of collocation methods. Let us begin by recalling the definition of a collocation method. Suppose that we want to approximate an ODE of first order, given by,
d
dtu(t) =F(t, u(t)). (6.17)
That is to say, we are looking for an approximation ofu(t0+h), knowingu(t0). A collocation method of degree sconsists in finding a polynomial function y of degree s, taking values inK, and satisfying Equation (6.17) at least insprescribed points.
Definition 6.3. Let (ci)1≤i≤s∈[0,1]s be distinct real numbers. We define the collocation polynomialy(t) as the polynomial of degree at most ssuch that,
y(t0) =y0,
˙
y(t0+cih) =F(t0+cih, y(t0+cih)), for i= 1, . . . , s. (6.18) Then, the numerical solution y1 = y(t0 +h) is called a collocation method for Equa-tion (6.17).
Such a method corresponds to a special kind of Butcher tableau where the coefficients (ai,j)1≤i,j≤s and (bj)A≤j≤s are given by the following theorem.
Theorem 6.4 (Theorem 1.4 [91]). The collocation method given by Definition 6.3, for a given set (ci)1≤i≤s ∈ [0,1]s of distinct real numbers, corresponds to a special kind of Runge-Kutta method, with Butcher tableau (6.14) given by,
ai,j = Z ci
0
lj(τ)dτ, bi = Z 1
0
li(τ)dτ, where li is the Lagrange polynomial given byli(τ) =Q
l6=i(τ−cl)/(ci−cl).
The Gauss collocation method consists in a special choice of the nodes (ci)1≤i≤s. They are chosen to be the zeros of thesth shifted Legendre Polynomial, which is given by,
d
dxs(xs(x−1)s).
6.3. Numerical scheme for the SPGPE 191 This choice of nodes is especially interesting since it ensures that such a method conserves quadratic invariants (such as theL2-norm in our case) [92, Theorem 2.2], converges at order 2s(under sufficient regularity assumptions) and is symplectic [92, Theorem 4.2]. We recall that symplectic schemes ensure good long time conservation of invariants (in particular the energy) by reproducing, at the discrete level, some structure of the continuous equation.
To clearly define this property, we consider the isomorphism C → R2, x 7→ (<x,=x), and replace C by R2 using this isomorphism. The multiplication by i of an element of K can be represented by a product with the square matrix J of size 2·dim(K) given
by J = 0 I
−I 0
!
. A numerical scheme is said to be symplectic if its discrete flow ˜ΦH (taking values inR2·dim(K)) is symplectic, i.e.,
(J ac( ˜ΦH)(φ))TJ J ac( ˜Φ)H(φ) =J, ∀φ∈K, whereJ ac( ˜ΦH) denotes the Jacobian of the flow ˜ΦH.
Theorem 6.5. Let k > d/2. The Gauss-Lawson method given by the discrete flowΦH,h conserves the L2(R2)-norm as long as it is well posed in the sens of Theorem 6.2. As a consequence, the Gauss-Lawson method is globally well-posed, whenever it is locally well-posed. More precisely, for all M > 0, there exists h0 > 0 such that for all h ≤ h0 and for all φ0∈K satisfying
φ0
Σk ≤M, the following equation has a unique solution:
(φn+1)n∈N∈KN.
φ0 =φ0, and φn+1= ΦH,hφn,∀n∈N. (6.19) Proof of Theorem 6.5. Global existence follows from the conservation of theL2(R2)-norm.
It is classical that the Gauss collocation method conserves the quadratic first integrals, and in particular the L2(R2)-norm in this case. See Corollary 12 [22]. Suppose that the ODE defined by Equation (6.17) is such that the quadratic formI defined byI(u) =uTCu withC a symmetric matrix, is a first integral of this dynamics. Then, for allu, and for all t≥0 uTCF(t, u) = 0. Let y(t) be the collocation polynomial of the Gauss method. Then it holds,
I(y(t0+h))−I(y(t0)) = 2 Z t0+h
t0
y(t)TCy(t) dt.˙
Since y(t)TCy(t) is a polynomial of degree 2s˙ −1, it is integrated without error by the Gaussian quadrature formula, which vanishes since for alli= 1, . . . , s,
y(t0+cih)TCy(t˙ 0+cih) =y(t0+cih)TCF(t0+cih, y(t0+cih)) = 0.
The symplecticity follows from the fact that every Runge-Kutta scheme that preserve quadratic first integrals, is a symplectic method (see section VI.4 from [91]).
Proposition 6.6. The Gauss-Lawson method defined by the discrete flowΦH,his symplectic whenever it is well-posed (by Theorem 6.5).
As said previously, the Gauss-Lawson method converges at order 2s. This result is the analogous of Theorem 14 [22].
Theorem 6.7. Let φ0 ∈ K and T > 0. Then there exists h0 > 0 such that for all h≤h0, the s-stage Gauss-Lawson method is well-posed in the sense of Theorem 6.5. Let (φn+1)n∈N ∈ KN be the solution of Equation (6.19) with initial condition φ0. Let also (φ(t))t≥0 be the solution of (6.8) with initial condition φ0. Then, there exists C >0 such that,
∀h≤h0,∀n∈Ns.t.0≤nh≤T, kφ(tn)−φnkΣk ≤Ch2s. (6.20) Remark 6.8. Let k > d/2. In our stochastic setting, Theorem6.2 is quite weak. Since the global scheme to solve Equation (6.4) consists in alternating the discrete flows ΦL,ε,h and ΦH,h, the initial condition for the flowΦH,h becomes random. Thus the maximal time steps h0 that ensures well-posedness (in Theorem6.5) is also random. If it becomes smaller than the initial choice of h, then we will not be able to solve the Hamiltonian dynamics after this time without refining the time step. This maximal time step h0 may become very small if the L2(R2)-norm of the solution of the flowΦL,ε,h becomes large. Because of the additive noise, theL2(R2)-norm of the scheme is not conserved, and worse, it can take arbitrarily large values with positive probability. Then, we cannot conclude to the almost sure well-posedness of the global scheme for any time step small enough. One way to rigorously define a well-posed version of this scheme would be to introduce a stopping time when the solution of the global numerical scheme becomes larger (for the L2(R2) norm) than a prescribed threshold. Then, it is enough to stop the dynamics at this stopping time. We believe that such a procedure would at least enable to prove a convergence in probability. This kind of argument has been used in [20,142]. In practice we do not observe this problem of definition for sensible time steps h. This is related to the fact that the solution of Equation (6.4) takes large values only with small probability thanks to the dissipative term.