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Abstract

The main purpose of this paper is to investigate the spectral Galerkin method for spatial discretization. We combine it with the method introduced by Jentzen et al. (2011, Efficient simulation of nonlinear parabolic SPDEs with additive noise. Ann. Appl. Probab., 21, 908–950) for temporal discretization of stochastic partial differential equations and study pathwise convergence. We consider the case of coloured noise, instead of the usual space-time white noise that was used before for the spatial discretization. The rate of convergence in uniform topology is estimated for the stochastic Burgers' equation. Numerical examples illustrate the estimated convergence rate.
IMA Journal of Numerical Analysis (2013) Page 1 of 24
doi:10.1093/imanum/drs035
Full discretization of the stochastic Burgers equation with correlated noise
Dirk Blömker
Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany
Corresponding author: dirk.bloemker@math.uni-augsburg.de
Minoo Kamrani and S. Mohammad Hosseini
Department of Applied Mathematics, Tarbiat Modares University, P.O. Box 14115-175, Tehran, Iran
[Received on 8 December 2011; revised on 13 May 2012]
The main purpose of this paper is to investigate the spectral Galerkin method for spatial discretization. We
combine it with the method introduced by Jentzen et al. (2011, Efficient simulation of nonlinear parabolic
SPDEs with additive noise. Ann. Appl. Probab.,21, 908–950) for temporal discretization of stochastic
partial differential equations and study pathwise convergence. We consider the case of coloured noise,
instead of the usual space-time white noise that was used before for the spatial discretization. The rate of
convergence in uniform topology is estimated for the stochastic Burgers’ equation. Numerical examples
illustrate the estimated convergence rate.
Keywords: stochastic partial differential equations; coloured noise; Galerkin approximation; stochastic
Burgers’ equation.
1. Introduction
In this article, the numerical approximation of nonlinear parabolic stochastic partial differential equa-
tions (SPDEs) is considered. Following the ideas of Blömker & Jentzen (2009) for the case of space-
time white noise, a numerical method for simulating nonlinear SPDEs with additive noise for the case of
coloured noise is proposed and analysed. The main novelty in this article is the estimation of the spatial
and temporal discretization error in the Ltopology in the case of coloured noise. This is different from
the usual space-time white noise, which was considered before in Blömker & Jentzen (2009) for spatial
discretization.
We consider as a forcing term an infinite-dimensional stochastic process expanded in the eigenfunc-
tions of the linear operator Apresent in the SPDE. We focus on the case where the Brownian motions
are not independent. This is due to the fact that the spatial covariance operator of the forcing does not
commute with A.
In order to illustrate the main result of this article, we consider the stochastic Burgers’ equation
with Dirichlet boundary conditions on a bounded domain. To be more precise, let T>0, ,F,P)be
a probability space, and let the space-time continuous stochastic process X:[0,T]×ΩC([0, 1], R)
be the unique solution of the SPDE
dXt=2
x2XtXt·
xXtdt+dWt,Xt(0)=Xt(1)=0, X0=0, (1.1)
for t[0, T] and x(0, 1). The noise is given by a cylindrical Wiener process Wt,t[0, T] defined later.
c
The authors 2013. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
IMA Journal of Numerical Analysis Advance Access published January 8, 2013
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2of24 D. BLÖMKER ET AL.
There are numerous publications for coloured or correlated noise of the type presented here, which
is white in time and coloured in space. For Burgers’ equation, see Da Prato et al. (1994), Da Prato &
Gatarek (1995), Goldys & Maslowski (2005) and Blömker & Duan (2007) for example. Here we refrain
from the usual assumption that the covariance of Wand the Laplacian are jointly diagonal.
The existence and uniqueness of solutions of the stochastic Burgers’ equation was studied by Da
Prato & Gatarek (1995) for coloured noise. Da Prato & Zabczyk (1992,1996) studied (1.1) for space-
time white noise and Gyöngy & Nualart (1999) studied the equation on the whole real line.
Alabert and Gyöngy obtained a spatial discretization of this equation in L2topology (Alabert &
Gyöngy,2006). Recently, Blömker & Jentzen (2009) obtained a bound on the spatial discretization
error in uniform topology by the spectral Galerkin method for the case of space-time white noise.
The spectral Galerkin method has been extensively studied for SPDEs with space-time white noise;
see, for example, Jentzen (2009), Kloeden & Shott (2001), Liu (2003), Lord & Rougemont (2004) and
Lord & Shardlow (2007).
Hausenblas investigated the discretization error of semilinear stochastic evolution equations in Lp
spaces and Banach spaces, and quasi-linear evolution equations driven by nuclear or space-time white
noise in Hausenblas (2002,2003). Shardlow (1999) and Gyöngy (1999) apply finite differences in order
to approximate the mild solution of parabolic SPDEs driven by space-time white noise. Yoo investigates
the mild solution of parabolic SPDEs by finite differences in Yoo (1999).
Our aim here is to extend the result of Blömker & Jentzen (2009). First, we discuss the case of
coloured noise not diagonal with respect to the eigenfunctions of the Laplacian. Secondly, using the
time discretization that was introduced in Jentzen et al. (2011), we obtain an error estimate for the full
space-time discretization.
The remainder of this paper is organized as follows. Section 2 gives the setting and the assump-
tions. In Section 3, we investigate the spatial discretization error and in Section 4 the temporal error is
obtained. Finally, in the last section numerical examples are presented.
2. Setting and assumptions
Fix T>0, and let ,F,P)be a probability space and both (V,·
V)and (W,·
W)be R-Banach
spaces. Moreover, let PN:VV,NNbe a sequence of bounded linear operators.
Throughout this article the following assumptions will be used.
Assumption 2.1 Let S:(0, T]L(W,V)be a continuous mapping satisfying
sup
0<tT
(tαStL(W,V))<,sup
NN
sup
0tT
(tαNγStPNStL(W,V))<, (2.1)
where α[0, 1)and γ(0, )are given constants.
Assumption 2.2 Let F:VWbe a locally Lipschitz continuous mapping, which satisfies
sup
vV,wVr
F(v)F(w)W
vwV
<(2.2)
for every r>0.
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FULL DISCRETIZATION OF THE STOCHASTIC BURGERS EQUATION 3of24
Assumption 2.3 Let O:[0,T]×ΩVbe a stochastic process with continuous sample paths and
sup
NN
sup
0tT
NγOt(ω) PN(Ot(ω))V<(2.3)
for every ωΩ, where γ(0, )is given in Assumption 2.1.
Assumption 2.4 Let XN:[0,T]×ΩV,NNbe a sequence of stochastic processes with continu-
ous sample paths such that
sup
MN
sup
0sT
XM
s(ω)V<(2.4)
and
XN
t(ω) =t
0
PNStsF(XN
s(ω)) ds+PN(Ot(ω)) (2.5)
for every t[0, T], ωΩand every NN.
Blömker & Jentzen (2009) obtained the following theorem.
Theorem 2.1 Let Assumptions 2.1–2.4 be fulfilled. Then there exists a unique stochastic process X:
[0, T]×ΩVwith continuous sample paths, which fulfils
Xt(ω) =t
0
StsF(Xs(ω)) ds+Ot(ω) (2.6)
for every t[0, T] and every ωΩ. Moreover, there exists an F/B([0, ))-measurable mapping C:
[0, )Ωsuch that
sup
0tT
Xt(ω) XN
t(ω) VC(ω) ·Nγ(2.7)
holds for every NNand every ωΩ, where γ(0, )is given in Assumption 2.1.
3. Spatial discretization for the case of coloured noise
Now we will show that Assumptions 2.1–2.4 are satisfied for Burgers’ equation in the case of coloured
noise. Therefore, from Theorem 2.1 we can conclude the convergence of the Galerkin method for this
equation. Most of the results are already proved in Blömker & Jentzen (2009). We only state the results
needed later in the proofs, and the modifications necessary due to the presence of coloured noise.
In the remainder of the paper define V=C0([0, 1]),W=H1(0, 1). The mapping :VW,
given by
(∂v)(ϕ) =(v)(ϕ) :=−v,ϕL2=−1
0
v(x(x)dx
for every vVand ϕH1(0, 1), is a bounded linear mapping from Vto W.
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4of24 D. BLÖMKER ET AL.
From Blömker & Jentzen (2009, Lemmas 4.6 and 4.8), we have the following lemmas.
Lemma 3.1 The mapping S:(0, T]L(H1(0, 1),C0([0, 1])), given by
(St(w))(x)=
N
n=1
(2·en2π2t·w(sin(nπ(·))) ·sin(nπx))
for every x[0, 1], wH1(0, 1)and every t(0, T], is well defined and satisfies Assumption 2.1.
From Assumption 2.1, we derive
sup
0<tT
(tαStL(V,V))<, (3.1)
where αwas introduced in Assumption 2.1.
Remark 3.1 As we can see from Blömker & Jentzen (2009, Lemma 4.6), Assumption 2.1 is satisfied
for α=3
4and γ[0, 1
2).
Lemma 3.2 The mapping F:C0([0, 1])H1(0, 1), given for every vC0([0, 1])by F(v)=x(v2),
satisfies Assumption 2.2.
In the following, we present details of the Q-Wiener process Wcorresponding to the coloured noise
in order to prove Assumption 2.3 later. We focus on a d-dimensional setting, while the result needed
later is for d=1. Let βi:[0,T]×ΩR,iNdbe a family of Brownian motions that are not neces-
sarily independent. They are correlated as given by
Ek(t l(t)) =Qek,el·t,k=(k1,...,kd)Nd,t>0, l=(l1,...,ld)Nd,
where for every kNd,
ek: [0, 1]dR,ek(x)=2d/2sin(k1πx1)·...·sin(kdπxd),x[0, 1]d
are smooth functions. Furthermore, Qis a symmetric non-negative operator such that
Qek,el=1
01
0
ek(x)el(y)q(xy)dydx(3.2)
for k,lNdand some positive-definite function q.
Moreover, for every kNddefine the real numbers λk=π2(k2
1+···+k2
d)R.
Lemma 3.3 Assume that there exists a ρ>0 such that, in dimension d∈{1, 2, 3},
iNd
jNd
iρ1
2jρ1
2|Qei,ej| <.
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FULL DISCRETIZATION OF THE STOCHASTIC BURGERS EQUATION 5of24
Then there exists a stochastic process O:[0,T]×ΩV, which satisfies
sup
0t1t2T
Ot2(ω) Ot1(ω)V
(t2t1)θ<,
sup
NN
sup
0tT
NγOt(ω) PN(Ot(ω))V<
(3.3)
for every ωΩ, every θ(0, min{1
2,ρ
2}), and every γ(0, ρ). Furthermore, Osatisfies
P
lim
N→∞ sup
0tT
Ot
i∈{1,...,N}dλit
0
eλi(ts)βi
sds+βi
tei
V
=0
=1,
sup
NN
Esup
0tT
OtPNOtp
V1/p
Nγ
+sup
0t1t2T
(E[Ot2Ot1p
V])1/p
(t2t1)θ<
for every p[1, )and every γ(0, ρ).
We need some technical lemmas first in order to prove this lemma.
Lemma 3.4 For every t1,t2[0, T], with t1t2, and every r(0, 1)we have
Et2
0
eλi(t2s)dβi
st1
0
eλi(t1s)dβi
st2
0
eλj(t2s)dβj
st1
0
eλj(t1s)dβj
s
2i+λj)r1(t2t1)rQei,ej(3.4)
for all i,jNd.
Proof. Fix t1,t2[0, T], with t1t2, and i,jNd. Define Δt=t2t1and Λij =λi+λj. We obtain
Et2
0
eλi(t2s)dβi
st1
0
eλi(t1s)dβi
s·t2
0
eλj(t2s)dβj
st1
0
eλj(t1s)dβj
s
=Et2
t1
eλi(t2s)dβi
s+(eλiΔt1)t1
0
eλi(t1s)dβi
s
×t2
t1
eλj(t2s)dβj
s+(eλjΔt1)t1
0
eλj(t1s)dβj
s
=Δt
0
eΛijsQei,ejds+(eΛij ΔteλiΔteλjΔt+1)·Qei,ej· 1eΛijt1
Λij
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6of24 D. BLÖMKER ET AL.
=(1eΛijΔt+(eΛij ΔteλiΔteλjΔt+1)(1eΛijt1)) ·Qei,ej
Λij
(1eΛijΔt+(1eΛij Δt)(1eΛijt1)) ·Qei,ej
Λij
2·1eΛijΔt
Λij
·Qei,ej.
Therefore, for every r(0, 1)we derive
Et2
0
eλi(t2s)dβi
st1
0
eλi(t1s)dβi
s·t2
0
eλj(t2s)dβj
st1
0
eλj(t1s)dβj
s
2·sup
x>0
1
x(1ex)r
·Λr1
ij t)r·|Qei,ej|
=2·Λr1
ij t)r·|Qei,ej|.
Lemma 3.5 For every t1,t2[0, T], with t1t2,NN,p[1, ), and every α,θ(0, 1
2] we have
Esup
x[0,1]d
|ON
t2(x)ON
t1(x)|p1/p
C
i,jIN
i2θ+2α1
2j2θ+2α1
2|Qei,ej|(t2t1)θ,
where C=C(d,p,α,θ) is a constant depending only on d,p,αand θ. The stochastic process ON:
[0, T]×ΩC([0, 1]d)is given by
ON
t=
iINt
0
eλi(ts)dβi
s·ei(3.5)
for every t[0, T] and every NN, where IN={1, ...,N}d.
Proof. Consider first
(ON
t2(x)ON
t1(x)) (ON
t2(y)ON
t1(y))
=
iINt2
0
eλi(t2s)dβi
st1
0
eλi(t1s)dβi
s·(ei(x)ei(y)),
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FULL DISCRETIZATION OF THE STOCHASTIC BURGERS EQUATION 7of24
P-almost surely for every x,y[0, 1]d. Hence, expanding the square of the series as a double sum and
using Lemma 3.4, we obtain (again with Δt=t2t1and Λij =λi+λj)
E|(ON
t2(x)ON
t1(x)) (ON
t2(y)ON
t1(y))|2
i,jIN
Λ2θ1
ij t)2θ|Qei,ej| · |(ei(x)ei(y))(ej(x)ej(y))|
C
i,jIN
Λ2θ1
ij t)2θ|Qei,ej| · (i2
2xy2
2)α(j2
2xy2
2)α
Ct)2θxy4α
2
i,jIN
(i2
2+j2
2)2θ1i2α
2j2α
2|Qei,ej|,
where we used that ekis bounded and Lipschitz. Therefore,
E|(ON
t2(x)ON
t1(x)) (ON
t2(y)ON
t1(y))|2
Ct)2θxy4α
2
i,jIN
i2θ+2α1
2j2θ+2α1
2|Qei,ej|. (3.6)
Again from Lemma 3.4 we derive in a similar way, for every x[0, 1]d,
E[|ON
t2(x)ON
t1(x)|2]C
i,jIN
Λ2θ1
ij t)2θ|Qei,ej|
C
i,jIN
(i2
2+j2
2)2θ1t)2θ|Qei,ej|. (3.7)
The Sobolev embedding of the fractional space Wα,pinto C0([0, 1]d), given in (Runst & Sickel,1996,
Theorem 2.1, Section 2.2.4), yields
E[ON
t2ON
t1p
C0([0,1]d)]C(0,1)d(0,1)d
(E[|(ON
t2(x)ON
t1(x)) (ON
t2(y)ON
t1(y))|2])p/2
xyd+pα
2
dxdy
+C(0,1)d
(E[|ON
t2(x)ON
t1(x)|2])p/2dx,
where we have used Gaussianity for the pth moment. In the following, for shorthand notation, all spatial
integrals are over (0, 1)d.
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8of24 D. BLÖMKER ET AL.
Therefore, by (3.6) and (3.7),
E[ON
t2ON
t1p
C0([0,1]d)]
C ((Δt)2θxy4α
2)p/2
xyd+pα
2
dxdy
i,jIN
(i2j2)2θ+2α1|Qei,ej|
p/2
+Ct)pθ
i,jIN
i2θ1
2j2θ1
2|Qei,ej|
p/2
dx
C1+ xyαpd
2dxdy·t)pθ·
i,jIN
(i2j2)2θ+2α1|Qei,ej|
p/2
.
By the fact that, with arbitrary dN,
(xy2)αdxdy(3d)d
dα
for every α(0, d), we derive
(EON
t2ON
t1p
C0([0,1]d))1/pC
i,jIN
(i2j2)2θ+2α1|Qei,ej|
1/2
t)θ.
Lemma 3.6 For every N,MN,NM,p[1, )and every α(0, 1
2], we have
Esup
0tT
ON
tOM
tp
C0([0,1]d)1/p
C
i,jIN\IM
i4α1
2j4α1
2|Qei,ej|
1/2
,
where IN={1, ...,N}d,IM={1, ...,M}dand C=C(d,p,α,θ) is a constant only depending on d,p,α
and θ.
Proof. Throughout this proof we assume α(0, 1
2)and p>1. Moreover, N>Mis fixed. Define,
for every t[0, T],
YN,M
t=
iIN\IMt
0
(ts)αeλi(ts)dβi
sei.
The celebrated factorization method of Da Prato & Zabczyk (1992) yields
Esup
0tT
ON
tOM
tp
C0([0,1]d)=Esup
0tT
sinα)
πt
0
(ts)α1StsYN,M
sds
p
C0([0,1]d)
Esup
0tT
t
0
(ts)α1StsYN,M
sds
p
C0([0,1]d)
.
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FULL DISCRETIZATION OF THE STOCHASTIC BURGERS EQUATION 9of24
Therefore, using the Hölder inequality and the boundedness of StL(C0([0,1]d)) yields
Esup
0tT
ON
tOM
tp
C0([0,1]d)sup
0tTt
0
(ts)p1)/(p1)dsp1
·ET
0
YN,M
s
p
C0([0,1]d)ds
CT
0
EYN,M
sp
C0([0,1]d)ds.
Hence
Esup
0tT
ON
tOM
tp
C0([0,1]d)1/p
Csup
0tT
(EYN,M
tp
C0([0,1]d))1/p. (3.8)
Again, using the embedding of Wα,pinto C0,
EYN,M
tp
C0([0,1]d)C(0,1)d(0,1)d
(E|YN,M
t(x)YN,M
t(y)|2)p/2
xyd+pα
2
dxdy
+C(0,1)d
(E|YN,M
t(x)|2)p/2dx. (3.9)
For the first term on the right-hand side of (3.9) we proceed completely analogously to Lemma 3.5 in
order to obtain
E|YN,M
t(x)YN,M
t(y)|2
C
i,jIN\IM
0
s2αesds·i+λj)2α1·|Qei,ej| · i2α
2j2α
2xy4α
2.
Therefore,
E|YN,M
t(x)YN,M
t(y)|2C
i,jIN\IM
|Qei,ej|
i14α
2j14α
2
xy4α
2. (3.10)
For the second term on the right-hand side of (3.9) we establish
E|YN,M
t(x)|2
i,jIN\IMt
0
(ts)2αei+λj)(ts)ds|Qei,ejei(x)||ej(x)|
C
i,jIN\IM
i2α1
2j2α1
2|Qei,ej|.(3.11)
Hence, using (3.10) and (3.11), we obtain from (3.9),
sup
0tT
(EYN,M
tp
C0([0,1]d))1/pC
i,jIN\IM
i4α1
2j4α1
2|Qei,ej|
1/2
. (3.12)
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10 of 24 D. BLÖMKER ET AL.
Finally, (3.8) and (3.12) yield
Esup
0tT
ON
t(x)OM
t(x)p
C0([0,1]d)1/p
C
i,jIN\IM
i4α1
2j4α1
2Qei,ej
1/2
.
Now we are ready to present the remaining parts of the proof of Lemma 3.3.
Proof of Lemma 3.3.From Lemma 3.6, we obtain
Esup
0tT
ON
tOM
tp
C0([0,1]d)1/p
C
i,jNd\IM
i4α1
2j4α1
2|Qei,ej|
1/2
CM 4αρ
i,jNd
iρ1
2jρ1
2|Qei,ej|
1/2
for every N,MNwith NM,p[1, ), and α(0, min{1
2,ρ
4}). The processes ONform a Cauchy
sequence in
Vp:=Lp((Ω,F,P),(C0([0, T]×[0, 1]d))).
Hence, there exists a stochastic process ˜
O:[0,T]×ΩC0([0, 1]d)with ˜
OVpand
Esup
0tT
˜
OtON
tp
C0([0,1]d)1/p
CN4αρ
i,jNd
iρ1
2jρ1
2|Qei,ej|
1/2
for every NN,p[1, ), and α(0, min{1
2,ρ
4}).
Therefore
sup
NN
NγEsup
0tT
˜
OtON
tp
C0([0,1]d)1/p
<
for every γ(0, ρ) and every p[1, ). This yields (Jentzen et al.,2009, Lemma 1)
Psup
NN
sup
0tT
{Nγ˜
OtON
tC0([0,1]d)}<=1.
In particular
Plim
N→∞ sup
0tT
˜
OtON
tC0([0,1]d)=0=1
and
Psup
NN
sup
0tT
{Nγ˜
OtPN˜
OtC0([0,1]d)}<=1.
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FULL DISCRETIZATION OF THE STOCHASTIC BURGERS EQUATION 11 of 24
From Lemma 3.5 we derive
(EON
t2ON
t1p
C0([0,1]d))1/pC
i,jIN
(i2j2)2θ+2((ρ/2)θ)1|Qei,ej|
1/2
|t2t1|θ
C
i,jIN
iρ1
2jρ1
2|Qei,ej|
1/2
|t2t1|θ
for every t1,t2[0, T], NNand θ(0, ρ/2). Provided θ1
2this furnishes
(E˜
Ot2˜
Ot1p
C0([0,1]d))1/pC
i,jIN
iρ1
2jρ1
2|Qei,ej|
1/2
|t2t1|θ.
Hence, for every θ(0, min{1
2,ρ
2}),
Psup
0t1,t2T
˜
Ot2˜
Ot1C0([0,1]d)
|t2t1|θ<=1.
Therefore
Pθ0, min 1
2,ρ
2:sup
0t1,t2T
˜
Ot2˜
Ot1C0([0,1]d)
|t2t1|θ<=1.
In conclusion, this shows the existence of a process O:[0,T]×ΩC0([0, 1]d), which satisfies
sup
0t1,t2T
Ot2(ω) Ot1(ω)C0([0,1]d)
|t2t1|θ<,
and
sup
NN
sup
0tT
(NγOt(ω) PNOt(ω)C0([0,1]d))<
for every ωΩ,θ(0, min{1
2,ρ
2})and γ(0, ρ). Moreover, Ois indistinguishable from ˜
O, i.e.,
P[t[0, T]: Ot=˜
Ot]=1.
Summarizing our results, we can state the following lemma.
Lemma 3.7 Assume ρ>0, d∈{1, 2, 3}and
i,jNd
iρ1
2jρ1
2|Qei,ej| <.
Furthermore, suppose that ξ:ΩVis F/V-measurable with
sup
NN
(Nρξ(ω) PN(ξ(ω))V)<
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12 of 24 D. BLÖMKER ET AL.
for every ωΩ. Then there exists a stochastic process O:[0,T]×ΩV, with continuous sample
paths, satisfying
P
lim
N→∞ sup
0<t<T
OtStξ
iINλit
0
eλi(ts)βi
sds+βi
tei
V
=0
=1
and
sup
NN
sup
0tT
{NγOt(ω) PN(Ot(ω))V}<
for every ωΩand γ(0, ρ).
In particular, Osatisfies Assumption 2.3 for every γ(0, ρ).
Note that the process Oin the previous Lemma 3.7 is the solution of the following linear SPDE:
dOt=ΔOtdt+dWt,Ot|∂(0,1)d=0, O0=ξ,
for t[0, T], where Wis a Q-Wiener process.
Lemma 3.8 Let V=C0([0, 1]),W=H1((0, 1)) and S:(0, T]L(W,V), and let F:VWbe given
by Lemmas 3.1 and 3.2. Let O:[0,T]×ΩVbe a stochastic process with continuous sample paths
with
sup
NN
sup
0tT
PN(Ot(ω))V<
for every ωΩ. Then Assumption 2.4 is fulfilled.
Proof. The proof is exactly the same as the one for Blömker & Jentzen (2009, Lemma 4.9).
4. Time discretization
For time discretization of the finite-dimensional stochastic differential equations (SDEs) (2.5)we
study the method introduced by Jentzen et al. (2011). Consider the discretization scheme for Burg-
ers’ equation, i.e., F(u)=xu2in one dimension. This is for simplicity of presentation only, as we need
to bound various terms depending on XNand F(XN).
Through this section assume ρ>0 such that
i,jNd
iρ1
2jρ1
2|Qei,ej| <. (4.1)
Moreover, assume θ(0, min{1
2,ρ
2}). For the time discretization we define the mapping YN,M
m:ΩV
for m∈{1, ...,M}by
YN,M
m+1(ω) =SΔt(YN,M
m(ω) +Δt(PNF)(YN,M
m(ω))) +PN(O(m+1t(ω) SΔtOmΔt(ω)). (4.2)
The purpose of this section is to consider the discretization error in time
XN
mΔt(ω) YN,M
m(ω)V,
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FULL DISCRETIZATION OF THE STOCHASTIC BURGERS EQUATION 13 of 24
where
XN
mΔt(ω) =mΔt
0
PNSmΔtsF(XN
s(ω)) ds+ON
mΔt(ω)
is the solution of the spatial discretization, which is evaluated at the grid points.
Recall that, as we proved in the last section, Assumptions 2.1–2.4 are satisfied for the stochastic
Burgers’ equation in one dimension.
Lemma 4.1 Let XN:[0,T]×ΩVbe the unique adapted stochastic process with continuous sample
paths, defined in Assumption 2.4. Assume that ON:[0,T]×ΩC0([0, 1]d)is the stochastic process
defined in (3.5). Then we obtain
(XN
t2(ω) ON
t2(ω)) (XN
t1(ω) ON
t1(ω))VC(ω)(t2t1)1/4
for every ωΩand all t1,t2[0, T], with t1<t2where Cis a finite random variable C:Ω[0, ).
Proof. For every 0 t1t2Twe have
XN
t2(ω) ON
t2(ω) (XN
t1(ω) ON
t1(ω))V
=
t2
0
PNSt2sF(XN
s(ω)) dst1
0
PNSt1sF(XN
s(ω)) ds
V
=
t2
t1
PNSt2sF(XN
s(ω)) ds+t1
0
(St2sSt1s)PNF(XN
s(ω)) ds
V
t2
t1
PNSt2sL(V,V)(XN
s(ω))2Vds+
t1
0
St1s(St2t1I)PNF(XN
s(ω)) ds
V
.
From (3.1) and using the fact that Stis the semigroup generated by, the Laplacian operator, Δ,we
conclude
XN
t2(ω) ON
t2(ω) (XN
t1(ω) ON
t1(ω))V
C1(ω) t2
t1
(t2s)3/4ds
+t1
0
PNSt1sΔ1/4L(W,V)(St2t1I1/4L(W,V)F(XN
s(ω))Wds
4C1(ω)(t2t1)1/4+t1
0
(t1s)1/4ds(t2t1)1/4F(XN
s(ω))W
4C1(ω)(t2t1)1/4+C2(ω)(t2t1)1/4T3/4
C(ω)(t2t1)1/4,
where C1(ω) =supMNsup0sTXM
s(ω)2
V,C2(ω) =supMNsup0sTF(XM
s(ω))Ware finite due
to Assumptions 2.2 and 2.4, and therefore Cis an almost-surely finite random variable
C:Ω[0, ).
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14 of 24 D. BLÖMKER ET AL.
Before we begin with the first part of the error, we define
R(ω) :=sup
NN
sup
0sT
F(XN
s(ω))W+sup
NN
sup
0sT
XN
s(ω)V
+sup
0t1,t2T
Ot2(ω) Ot1(ω)V|t2t1|θ
+sup
NN
sup
0t1,t2T
XN
t2(ω) ON
t2(ω) (XN
t1(ω) ON
t1(ω))V|t2t1|1/4,
where from Assumption 2.4 and Lemmas 3.3 and 4.1, R:ΩRis a finite random variable.
The main result of this section is stated below.
Theorem 4.1 For m∈{0, 1, ...,M}and every M,NN, there exists a finite random variable C:Ω
[0, )such that
XN
mΔt(ω) YN,M
m(ω)VC(ω)(Δt)min(1/4,θ),
where XN:[0,T]×ΩVis the unique adapted stochastic process with continuous sample paths,
defined in Assumption 2.4, and YN,M
m:ΩVfor m∈{0, 1, ...,M}, and N,MN, are given in (4.2).
Proof. For the proof it is sufficient to prove the result for sufficiently small |t2t1|.Dueto(2.5)
we have
XN
mΔt(ω) =mΔt
0
PNSmΔtsF(XN
s(ω)) ds+ON
mΔt(ω)
=
m1
k=0(k+1t
kΔt
PNSmΔtsF(XN
s(ω)) ds+ON
mΔt(ω) (4.3)
for every m∈{0, 1, ...,M}and every MN.
The mapping YN
m:ΩV,m=1, 2, ...,M, is defined by
YN
m(ω) =
m1
k=0(k+1t
kΔt
PNSmΔtkΔtF(XN
kΔt(ω)) ds+ON
mΔt(ω). (4.4)
Our aim is to bound XN
mΔt(ω) YN,M
m(ω)V. Therefore, we first estimate the difference of the true
solution to YN
m
XN
mΔt(ω) YN
m(ω)V(4.5)
for every m∈{0, 1, ...,M}and then the difference between YN
mand the full discretization in time,
YN
m(ω) YN,M
m(ω)V. (4.6)
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FULL DISCRETIZATION OF THE STOCHASTIC BURGERS EQUATION 15 of 24
For the first error in (4.5) we have
XN
mΔt(ω) YN
m(ω) =
m2
k=0(k+1t
kΔt
PNSmΔtsF(XN
s(ω)) ds
m2
k=0(k+1t
kΔt
PNSmΔtkΔtF(XN
kΔt(ω)) ds
+mΔt
(m1t
PNSmΔtsF(XN
s(ω)) ds
mΔt
(m1t
PNSΔtF(XN
kΔt(ω)) ds. (4.7)
Let us now bound the last two integrals in (4.7). For the first one we derive
mΔt
(m1t
PNSmΔtsF(XN
s(ω)) ds
V
=
mΔt
(m1t
PNSmΔts∂(XN
s(ω))2ds
V
mΔt
(m1t
PNSmΔtsL(V,V)·XN
s(ω)2
Vds
sup
0st
XN
s(ω)2
VmΔt
(m1t
(mΔts)3/4ds
R2(ω)(Δt)1/4.
For the second one we obtain
mΔt
(m1t
PNSΔtF(XN
kΔt(ω)) ds
V
=
mΔt
(m1t
PNSΔt∂(XN
kΔt(ω))2ds
V
mΔt
(m1t
PNSΔtL(V,V)·XN
kΔt(ω)2
Vds
sup
0st
XN
s(ω)2
VmΔt
(m1t
t)3/4ds
R2(ω)(Δt)1/4.
Therefore, we can conclude
XN
mΔt(ω) YN
m(ω)V
m2
k=0(k+1t
kΔt
PNSmΔts(F(XN
s(ω)) F(XN
kΔt(ω))) ds
V
+
m2
k=0(k+1t
kΔt
(PNSmΔtsPNS(mΔtkΔt))F(XN
kΔt(ω)) ds
V
+R2(ω)(Δt)1/4.
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16 of 24 D. BLÖMKER ET AL.
Thus, inserting the nonlinearity with the Ornstein–Uhlenbeck process in the first term yields, for every
m∈{0, 1, ...,M},
XN
mΔt(ω) YN
m(ω)V
m2
k=0(k+1t
kΔt
PNSmΔts(F(XN
s(ω)) F(XN
kΔt(ω) +ON
s(ω) ON
kΔt(ω))) ds
V
+
m2
k=0(k+1t
kΔt
PNSmΔts(F(XN
kΔt(ω) +ON
s(ω) ON
kΔt(ω)) F(XN
kΔt(ω))) ds
V
+
m2
k=0(k+1t
kΔt
(PNSmΔtsPNSmΔtkΔt)F(XN
kΔt(ω)) ds
V
+R2(ω)(Δt)1/4. (4.8)
For the first term in (4.8), by using Lemma 4.1 together with PNStsuVC(ts)3/4uV,we
conclude
m2
k=0(k+1t
kΔt
PNSmΔts(F(XN
s(ω)) F(XN
kΔt(ω) +ON
s(ω) ON
kΔt(ω))) ds
V
m2
k=0(k+1t
kΔt
(mΔts)3/4XN
s(ω) (XN
kΔt(ω) +ON
s(ω) ON
kΔt(ω))V
·XN
s(ω) +(XN
kΔt(ω) +ON
s(ω) ON
kΔt(ω))Vds
R(ω)
m2
k=0(k+1t
kΔt
(mΔts)3/4(skΔt)1/4(2R(ω) +R(ω)(skΔt)θ)ds
2C(R(ω),T)(Δt)1/4, (4.9)
where the constant depends on Rand T.
For the second term in (4.8) we derive
m2
k=0(k+1t
kΔt
PNSmΔts(F(XN
kΔt(ω) +ON
s(ω) ON
kΔt(ω)) F(XN
kΔt(ω))) ds
V
2
m2
k=0(k+1t
kΔt
PNSmΔts∂(XN
kΔt(ω) ·(ON
s(ω) ON
kΔt(ω)))Vds
+
m2
k=0(k+1t
kΔt
PNSmΔts((ON
s(ω) ON
kΔt(ω))2)Vds
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FULL DISCRETIZATION OF THE STOCHASTIC BURGERS EQUATION 17 of 24
2
m2
k=0(k+1t
kΔt
PNSmΔtsL(V,V)XN
kΔt(ω)V(ON
s(ω) ON
kΔt(ω))Vds
+
m2
k=0(k+1t
kΔt
PNSmΔtsL(V,V)·(ON
s(ω) ON
kΔt(ω))2Vds
2R2(ω)
m2
k=0(k+1t
kΔt
(mΔt(k+1t)3/4·(skΔt)θds
+R2(ω)
m2
k=0(k+1t
kΔt
(mΔt(k+1t)3/4(skΔt)2θds
C(R(ω),θ )(Δt)θ,
where the constant depends on Rand θ.
Finally, for the third term in (4.8), again by using the fact that Stis the semigroup generated by the
Laplacian, we have
m2
k=0(k+1t
kΔt
(PNSmΔtsPNSmΔtkΔt)F(XN
kΔt(ω)) ds
V
m2
k=0(k+1t
kΔt
PNSmΔtkΔt(SkΔtsI)F(XN
kΔt(ω))Vds
m2
k=0(k+1t
kΔt
(mΔtkΔt)1(kΔts)F(XN
kΔt(ω))Wds
C(R(ω),Tt, (4.10)
where we have used PNΔStL(W,V)Ct1, together with Δ1(StI)L(W,V)t. Hence, from (4.9)
and (4.10) we derive
XN
mΔt(ω) YN
m(ω)VC(R(ω),R2(ω),θ,T)(Δt)min{1/4,θ}.(4.11)
Let us now turn to the second error term in (4.6). Note that YN,M
m:ΩVsatisfies
YN,M
m(ω) =
m1
k=0(k+1t
kΔt
PNSmΔtkΔtF(YN,M
k(ω)) ds+PNOmΔt(ω). (4.12)
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18 of 24 D. BLÖMKER ET AL.
Thus, by using PNStL(V,V)Ct3/4, we can estimate
YN
mYN,M
mV=
m1
k=0(k+1t
kΔt
PNSmΔtkΔt(F(XN
kΔt)F(YN,M
k))
V
m1
k=0(k+1t
kΔt
(mΔtkΔt)3/4(XN
kΔtYN,M
k)2+2XN
kΔt(XN
kΔtYN,M
k)Vds
m1
k=0
Δt(mΔtkΔt)3/4(XN
kΔtYN,M
k2
V+2R(ω)XN
kΔtYN,M
kV). (4.13)
Combining (4.11) with (4.13), we have
XN
mΔt(ω) YN,M
m(ω)VC(R(ω),θ,T)(Δt)min{1/4,θ}
+
m1
k=0
XN
kΔt(ω) YN,M
k(ω)2
V
+2R(ω)
m1
k=0
XN
kΔt(ω) YN,M
k(ω)V. (4.14)
If we assume that, for some δ>0 fixed later,
sup
0kM
XN
kΔt(ω) YN,M
k(ω)Vδ, (4.15)
then
XN
mΔt(ω) YN,M
m(ω)VC(R(ω),θ,T)(Δt)min{1/4,θ}
++2R(ω))
m1
k=0
XN
kΔt(ω) YN,M
k(ω)V. (4.16)
Then, by the discrete Gronwall lemma, we can conclude
XN
mΔt(ω) YN,M
m(ω)Ve(m1)(δ+2R(ω)) C(R(ω),θ,T)(Δt)min{1/4,θ}.
In order to verify (4.15), we need
e(m1)(δ+2R)C(R(ω),θ,T)(Δt)min{1/4,θ}δ,
which is true for any δ>0, provided Δtis sufficiently small. This completes the proof of the time
discretization.
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FULL DISCRETIZATION OF THE STOCHASTIC BURGERS EQUATION 19 of 24
From Theorem 2.1, for the spatial discretization error we verified in Section 3,
XmΔt(ω) XN
mΔt(ω)VC(ω) ·Nγ, (4.17)
and from Theorem 4.1, for the temporal discretization error we just established
XN
mΔt(ω) YN,M
m(ω)VC(R(ω),θ,T)(Δt)min{1/4,θ}.
Therefore we have proved the following theorem for the stochastic Burgers’ equation.
Theorem 4.2 Assume ρ>0 such that
i,jN
iρ1
2jρ1
2|Qei,ej| <.
Let X:[0,T]×ΩVbe the solution of SPDE (2.6) and YN,M
m:ΩV,m∈{0, 1, ...,M},M,NN
be the numerical solution given by (4.2). Fix θ(0, min{1
2,ρ
2})and γ[0, 1
2).
Then there exists a finite random variable C:Ω[0, )such that
XmΔt(ω) YN,M
m(ω)VC(ω)(Nγ+t)min{1/4,θ})(4.18)
for all m∈{0, 1, ...,M}and every M,NN.
5. Numerical results
In this section, we consider the numerical solution of the stochastic Burgers’ equation by the method
given in (4.2).
Consider the stochastic evolution equation (2.6) with S:(0, T]L(W,V),F:VWgiven by
Lemma 3.1, Lemma 3.2 for T=1, d=1, and some initial condition fixed to be (ξ(ω))(x)=6
5sin(x)for
all x[0, π].
We assume that O:[0,T]×ΩVis given by Lemma 3.3 where the Brownian motion βi:[0,T]×
ΩR,iNd, are related by
Ekβl)=Qek,el,k,lN, (5.1)
where the covariance operator Qis explicitly given as a convolution operator
Qek,el=π
0π
0
ek(x)el(y)q(xy)dydx, (5.2)
with kernel
q(xy)=max 0, h−|xy|
h2, (5.3)
where we define the orthonormal basis
ek(x)=2
πsin(kx)for kN. (5.4)
The possibly small quantity h>0 measures the correlation length of the noise. In this case the covari-
ance matrix, i.e., Qek,elk,l, is not diagonal. But, for small h>0, it is close to diagonal. In Fig. 1,
the covariance matrix is plotted for k,l∈{1, 2, ..., 100}for h=0.1, 0.01. Then, by some numerical
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20 of 24 D. BLÖMKER ET AL.
0
20
40
60
80
100
0
50
100
−0.5
0
0.5
1
k
Covariance Matrix
l
Q(k,l )
Q(k,l )
0
20
40
60
80
100
0
50
100
−0.5
0
0.5
1
k
Covariance Matrix
l
(a) (b)
Fig. 1. Covariance matrix Qek,elk,lfor k,l∈{1, 2, ...,c100},for(a)h=0.1 and (b) h=0.01.
calculations we can show that the condition on Qfrom (4.1) is satisfied for any ρ(0, 1
2).
The stochastic evolution equation (2.6) reduces to
dXt=2
x2XtXt·
xXtdt+dWt,X0(x)=6
5sin(x), (5.5)
with Xt(0)=Xt) =0fort[0, 1] and x[0, π].
The finite-dimensional SDE (2.5) reduces to
dXN
t=2
x2XN
tPNXN
t·
xXN
tdt+PNdWt,XN
0(x)=6
5sin(x), (5.6)
with XN
t(0)=XN
t) =0fort[0, 1] and x[0, π], and all NN.
In Fig. 2the solution O:[0,T]×ΩC0([0, π]), the solution of the linear SPDE
dOt=ΔOtdt+dWt,Ot|∂(0,π) =0, O0=6
5sin(x),
is plotted for T=1.
Theorem 4.2 yields the existence of a unique solution X:[0,π]×ΩC0([0, π])of the SPDE
(5.5) such that
sup
0xπ
|XmΔt,x)YN,M
m,x)|C(ω)(Nγ+t)min{1/4,θ})(5.7)
for m=1, ...,M,M=1tsuch that γ(0, 1
2),θ(0, 1
4).
By using Δt=T/N2, the solutions XN
t,x)of the finite-dimensional SDEs (5.6) converge uni-
formly in t[0, 1] and x[0, π] to the solution Xt,x)of the stochastic Burgers’ equation (5.5) with
the rate 1
2,asNgoes to infinity for all ωΩ.InFig.3, the pathwise approximation error
sup
0xπ
sup
0mM
|XmΔt,x)YN,M
m,x)|(5.8)
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FULL DISCRETIZATION OF THE STOCHASTIC BURGERS EQUATION 21 of 24
Fig.2.Ot,x),x[0, π], t[0, 1] and one random ωΩ,for(a)h=0.1 and (b) h=0.01.
100101102103
10−4
10−3
10−2
10−1
N
Pathwise approximation error
Stochastic Burgers equation
Pathwise approximation error
Orderlines 0.25, 0.5, 1
100101102103
10−4
10−3
10−2
10−1
N
Pathwise approximation error
Stochastic Burgers equation
Pathwise approximation error
Orderlines 0.25, 0.5, 1
(a) (b)
Fig. 3. Pathwise approximation error (5.8) against Nfor N∈{16, 32, ..., 256}for two random ωΩ, with h=0.1. These are
only two examples, but all other calculated trajectories behave similarly.
is plotted against Nfor N∈{16, 32, ..., 256}. As a replacement for the unknown solution, we use a
numerical approximation for Nsufficiently large.
Figure 3confirms that, as expected from Theorem 4.2, the order of convergence is 1
2. Obviously,
these are only two examples, but all of a few hundred calculated examples behave similarly. Even their
means seem to behave with the same order of the error. Nevertheless, we did not prove this here and
also did not calculate the mean with a sufficiently good standard deviation.
Finally, as an example, in Fig. 4,Xt(ω),x[0, π] is plotted for t∈{0, 3
200 ,0.2,1},forh=0.01, 0.1.
Acknowledgements
The authors of Tarbiat Modares University would like to thank the Department of Mathematics of the
University of Augsburg for its support during the second-named author’s visit and also for providing an
opportunity for joint research collaboration.
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22 of 24 D. BLÖMKER ET AL.
00.5 11.5 22.5 33.5
0
0.2
0.4
0.6
0.8
1
1.2
x
T=0.2
00.5 11.5 22.5 33.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x
T=0. 2
00.5 11.5 22.5 33.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x
T=0
00.5 11.5 22.5 33.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x
T=0
X(T,x )
X(T,x )
X(T,x )
X(T,x )
X(T,x )X(T,x )X(T,x )
X(T,x )
00.5 11.5 22.5 33.5
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
00.5 11.5 22.5 33.5
0
0.2
0.4
0.6
0.8
1
1.2
x
T=1 T=1
00.5 11.5 22.5 33.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x
T=3/200
00.5 11.5 22.5 33.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x
T=3/200
(a) (b)
Fig. 4. The stochastic Burgers’ equation Xt,x),x[0, π], t∈{0, 3
200 ,0.2,1}, given by (5.5)for(a)h=0.1 and (b) h=0.01, for
one random ωΩ.
at University of Aberdeen on March 9, 2013http://imajna.oxfordjournals.org/Downloaded from
FULL DISCRETIZATION OF THE STOCHASTIC BURGERS EQUATION 23 of 24
Funding
Funded by the University of Augsburg and Tarbiat Modares University.
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