Revised
Proof
Numer. Math.
DOI 10.1007/s00211-014-0670-9
Numerische Mathematik
Long-time dynamics of 2d double-diffusive convection:
analysis and/of numerics
Florentina Tone · Xiaoming Wang · Djoko Wirosoetisno
Received: 5 March 2014 / Revised: 23 September 2014
© Springer-Verlag Berlin Heidelberg 2014
Abstract We consider a two-dimensional model of double-diffusive convection and
1
its time discretisation using a second-order scheme (based on backward differentiation
2
formula for the time derivative) which treats the non-linear term explicitly. Uniform
3
bounds on the solutions of both the continuous and discrete models are derived (under
4
a timestep restriction for the discrete model), proving the existence of attractors and
5
invariant measures supported on them. As a consequence, the convergence of the
6
attractors and long time statistical properties of the discrete model to those of the
7
continuous one in the limit of vanishing timestep can be obtained following established
8
methods.
9
Mathematics Subject Classification 65M12·35B35·35K45
10
1 Introduction
11
The phenomenon of double-diffusive convection, in which two properties of a fluid
12
are transported by the same velocity field but diffused at different rates, often occurs
13
F. Tone
Department of Mathematics and Statistics,
University of West Florida, Pensacola, FL 32514, USA e-mail: ftone@uwf.edu
X. Wang
Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USA
e-mail: wxm@math.fsu.edu D. Wirosoetisno (
B
)Mathematical Sciences, Durham University, Durham DH1 3LE, UK e-mail: djoko.wirosoetisno@durham.ac.uk
Revised
Proof
in nature [12]. Perhaps the best known example is the transport throughout the world’s
14
oceans of heat and salinity, which has been recognised as an essential part of cli-
15
mate dynamics [15,21]. In contrast to simple convections (cf. [3]), double-diffusive
16
convections support a richer set of physical regimes, e.g., a stably stratified initial
17
state rendered unstable by diffusive effects. Although in this paper we shall be refer-
18
ring to the oceanographic case, the mathematical theory is essentially identical for
19
astrophysical [14,16] and industrial [4] applications.
20
In this paper, we consider a two-dimensional double-diffusive convection model,
21
which by now-standard techniques [18] can be proved to have a global attractor and
22
invariant measures supported on it, and its temporal discretisation. We use a backward
23
differentiation formula for the time derivative and a fully explicit method for the non-
24
linearities, resulting in an accurate and efficient numerical scheme. Of central interest,
25
here and in many practical applications, is the ability of the discretised model to cap-
26
ture long-time behaviours of the underlying PDE. This motivates the main aim of this
27
article: to obtain bounds necessary for the convergence of the attractor and associated
28
invariant measures of the discretised system to those of the continuous system. We do
29
this using the framework laid down in [19,20], with necessary modifications for our
30
more complex model.
31
For motivational concreteness, one could think of our system as a model for the
32
zonally-averaged thermohaline circulation in the world’s oceans. Here the physical
33
axes correspond to latitude and altitude, and the fluid is sea water whose internal
34
motion is largely driven by density differentials generated by the temperature T and
35
salinity S, as well as by direct wind forcing on the surface. Both T and S are also
36
driven from the boundary—by precipitation/evaporation and ice melting/formation
37
for the salinity, and by the associated latent heat release and direct heating/cooling
38
for the temperature. Physically, one expects the boundary forcing for T , S and the
39
momentum to have zonal (latitude-dependent) structure, so we include these in our
40
model. Furthermore, one may also wish to impose a quasi-periodic time dependence
41
on the forcing; although this is eminently possible, we do not do so in this paper to
42
avoid technicalities arising from time-dependent attractors.
43
Taking as our domainD∗ = [0,L∗] × [0,H∗]which is periodic in the horizontal
44
direction, we consider a temperature field T∗and a salinity field S∗, both transported
45
by a velocity fieldv∗=(u∗, w∗)which is incompressible,∇∗·v∗ =0, and diffused
46
at ratesκT andκS, respectively,
47
∂T∗/∂t∗+v∗·∇∗T∗=κT∗T∗
∂S∗/∂t∗+v∗·∇∗S∗=κS∗S∗. (1.1)
48
Here the star∗denotes dimensional variables. Taking the Boussinesq approximation
49
and assuming that the density is a linear function of T∗ and S∗, which is a good
50
approximation for sea water (although not for fresh water near its freezing point), the
51
velocity field evolves according to
52
∂v∗/∂t∗+v∗·∇∗v∗+ ∇∗p∗=κv∗v∗+(αTT∗−αSS∗)ez (1.2)
53
for some positive constantsαT andαS.
54
Revised
Proof
Our system is driven from the boundary by the heat and salinity fluxes (which could
55
be seen to arise from direct contact with air and latent heat release in the case of heat,
56
and from precipitation, evaporation and ice formation/melt in the case of salinity),
57
∂T∗/∂n∗=QT∗ and ∂S∗/∂n∗=QS∗. (1.3)
58
Here n∗denotes the outward normal, n∗ =z∗at the top boundary and n∗= −z∗at
59
the bottom boundary. We also prescribe a wind-stress forcing,
60
∂u∗/∂n∗=Qu∗ (1.4)
61
along with the usual no-flux conditionw∗=0 on z∗=0 and z∗=H∗.
62
Largely following standard practice, we cast our system in non-dimensional form
63
as follows. Using the scalest ,˜ l,˜ T and˜ S, we define the non-dimensional variables˜
64
t =t∗/˜t , x = x∗/l,˜ v = v∗t˜/l, T˜ = T∗/T and S˜ = S∗/S, in terms of which our˜
65
system reads
66
p−1(∂tv+v·∇v)= −∇p+v+(T −S)ez
∂tT+v·∇T =T
∂tS+v·∇S =βS.
(1.5)
67
To arrive at this, we have putl˜=H∗and taken the thermal diffusive timescale for
68
˜
t= ˜l2/κT, (1.6)
69
as well as scaled the dependent variables as
70
T˜ =pl˜/(αTt˜2) and S˜=pl˜/(αSt˜2), (1.7)
71
where the non-dimensional Prandtl number and diffusivity ratio (also known as the
72
Lewis number in the engineering literature) are
73
p=κv/κT and β =κT/κS. (1.8)
74
Another non-dimensional quantity is the domain aspect ratioξ = L∗/l. The sur-˜
75
face fluxes are non-dimensionalised in the natural way: QT = pQT∗/(αTt˜2),
76
QS=pQS∗/(αSt˜2)and Qu=Qu∗t.˜
77
For clarity and convenience, keeping in mind the oceanographic application, we
78
assume that the fluxes vanish on the bottom boundary z=0,
79
Qu(x,0)=QT(x,0)=QS(x,0)=0. (1.9)
80
Revised
Proof
For boundedness of the solution in time, the net fluxes must vanish, so (1.9) then
81
implies that the net fluxes vanish on the top boundary z=1,
82
ξ
0
Qu(x,1)dx= ξ
0
QT(x,1)dx = ξ
0
QS(x,1)dx =0. (1.10)
83
These boundary conditions can be seen to imply that the horizontal velocity flux is
84
constant in time, which we take to be zero, viz.,
85
1
0
u(x,z,t)dz= 1
0
u(x,z,0)dz≡0 for all x∈ [0, ξ]. (1.11)
86
For some applications (e.g., the classical Rayleigh–Bénard problem), the fluxes on the
87
bottom boundary may not vanish, which must then be balanced by the fluxes on the
88
top boundary,
89 ξ
0
[QT(x,1)−QT(x,0)]dx=0 (1.12)
90
and similarly for Qu and QS. With some modifications (by subtracting background
91
profiles from u, T and S), the analysis of this paper also apply to this more general
92
case. This involves minimal conceptual difficulty but adds to the clutter, so we do not
93
treat this explicitly here.
94
Defining the vorticityω := ∂xw−∂zu, the streamfunctionψ byψ = ωwith
95
ψ=0 on∂D[this is consistent with (1.11)], and the Jacobian determinant∂(f,g):=
96
∂xf∂zg−∂xg∂zf = −∂(g, f), our system reads
97
p−1{∂tω+∂(ψ, ω)} =ω+∂xT −∂xS
∂tT +∂(ψ,T)=T
∂tS+∂(ψ,S)=βS.
(1.13)
98
The boundary conditions are,
99
∂zT =QT, ∂zS=QS, ω=Qu and ψ=0 on∂D. (1.14)
100
We note that for the solution to be smooth at t =0, the initial data and the boundary
101
conditions must satisfy a compatibility condition; cf. e.g., [17, Thm. 6.1] in the case
102
of Navier–Stokes equations. In the rest of this paper, we will be working with (1.13)–
103
(1.14) and its discretisation. We assume thatω, T and S all have zero integral overD
104
at t=0. Thanks to the no-net-flux condition (1.10), this persists for all t ≥0.
105
Another dimensionless parameter often considered in studies of (single-species)
106
convection is the Rayleigh number Ra. When the top and bottom temperatures are
107
held at fixed values T1and T0, Ra is proportional to T0−T1. The relevant parameters
108
in our problem would be RaT ∝ |QT|L2(∂D)and RaS∝ |QS|L2(∂D), but we will not
109
consider them explicitly here; see, e.g., (2.11) in [2]. For notational conciseness, we
110
denote the variables U :=(ω,T,S), the boundary forcing Q:=(Qu,QT,QS)and
111
the parametersπ :=(p, β, ξ).
112
Revised
Proof
We do not provide details on the convergence of the global attractors and long time
113
statistical properties. Such kind of convergence can be obtained following established
114
methods once we have the uniform estimates derived here. See [10] for the convergence
115
of the global attractors and [20] for the convergence of long time statistical properties.
116
The rest of this paper is structured as follows. In Sect. 2 we review briefly the
117
properties of the continuous system, setting up the scene and the notation for its
118
discretisation. Next, we describe the time discrete system and derive uniform bounds
119
for the solution. In the appendix, we present an alternate derivation of the boundedness
120
results in [20], without using Wente-type estimates but requiring slightly more regular
121
initial data.
122
2 Properties of the continuous system
123
In this section, we obtain uniform bounds on the solution of our system and use them
124
to prove the existence of a global attractorA. For the single diffusion case (of T only,
125
without S), this problem has been treated in [6] which we follow in spirit, though not
126
in detail in order to be closer to our treatment of the discrete case.
127
We start by noting that the zero-integral conditions onω, T and S imply the Poincaré
128
inequalities
129
|ω|2L2(D)≤c0|∇ω|2L2(D), (2.1)
130
as well as the equivalence of the norms
131
|ω|H1(D)≤c|∇ω|L2(D), (2.2)
132
with analogous inequalities for T and S. The boundary conditionψ=0 implies that
133
(2.1)–(2.2) also hold forψ, while an elliptic regularity estimate [7, Cor. 8.7] implies
134
that
135
|∇ψ|2L2(D)≤c0|ω|2L2(D). (2.3)
136
Following the argument in [8], this also holds for functions, such as our T and S, with
137
zero integrals inD.
138
LetΩbe an H2extension of QutoD¯(further requirements will be imposed below)
139
and letωˆ := ω−Ω; we also defineψˆ := ˆωandΨ :=Ω with homogeneous
140
boundary conditions. Nowωˆ satisfies the homogeneous boundary conditionsωˆ =0
141
on∂D, and thus the Poincaré inequality (2.1)–(2.2). Furthermore, let TQ ∈ ˙H2(D)
142
be such that∂zTQ =QT on∂D(with other constraints to be imposed below) and let
143
Tˆ :=T −TQ; analogously for SQ andSˆ :=S−SQ. We note that since bothT andˆ
144
S have zero integrals overˆ D, they satisfy the Poincaré inequality (2.1)–(2.2).
145
We start with weak solutions of (1.13). For conciseness, unadorned norms and inner
146
products are understood to be L2(D),| · | := | · |L2(D)and(·,·):=(·,·)L2(D). With
147
ˆ
ω,T andˆ S as defined above, we haveˆ
148
Revised
Proof
∂tωˆ+∂(Ψ + ˆψ, Ω+ ˆω)=p
ωˆ+Ω+∂xTQ+∂xTˆ−∂xSQ−∂xSˆ
∂tTˆ+∂(Ψ + ˆψ,TQ+ ˆT)=TQ+Tˆ
∂tSˆ+∂(Ψ + ˆψ,SQ+ ˆS)=β (SQ+Sˆ).
(2.4)
149
On a fixed time interval[0,T∗), a weak solution of (2.4) are
150
ˆ
ω∈C0(0,T∗;L2(D))∩L2(0,T∗;H01(D)) Tˆ ∈C0(0,T∗;L2(D))∩L2(0,T∗;H1(D)) Sˆ∈C0(0,T∗;L2(D))∩L2(0,T∗;H1(D))
(2.5)
151
such that, for allω˜ ∈H01(D),T ,˜ S˜∈ H1(D), the following holds in the distributional
152
sense,
153
d
dt(ω,ˆ ω)˜ +(∂(Ψ + ˆψ, Ω+ ˆω),ω)˜
154
+p
(∇Ω+ ∇ ˆω,∇ ˜ω)−(∂xTQ+∂xTˆ,ω)˜ +(∂xSQ+∂xS,ˆ ω)˜
=0
155
d
dt(Tˆ,T˜)+(∂(Ψ + ˆψ,TQ+ ˆT),T˜)+(∇ ˆT,∇ ˜T)−(TQ,T˜)=0 (2.6)
156
d
dt(S,ˆ S)˜ +(∂(Ψ + ˆψ,SQ+ ˆS),S)˜ +β (∇ ˆS,∇ ˜S)−β (SQ,S)˜ =0.
157 158
The existence of such solutions can be obtained by Galerkin approximation together
159
with Aubin–Lions compactness argument [17, §3.3], which we do not carry out explic-
160
itly here.
161
Next, we derive L2inequalities for T , S andω. Multiplying (2.4a) byωˆ in L2(D)
162
and noting that(∂(ψ,ω),ˆ ω)ˆ =0, we find
163
1 2 d
dt| ˆω|2+p|∇ ˆω|2 = −(∂(Ψ, Ω),ω)ˆ −(∂(ψ, Ω),ˆ ω)ˆ
164
+p
(Ω,ω)ˆ +(∂xT,ω)ˆ −(∂xS,ω)ˆ
. (2.7)
165
We bound the rhs as
166
(Ω,ω)ˆ = |∇Ω| |∇ ˆω| ≤1
8|∇ ˆω|2+2|∇Ω|2
167
(∂xT,ω)ˆ = |∂xω| |Tˆ | ≤ 1
8|∇ ˆω|2+2|T|2≤ 1
8|∇ ˆω|2+4c0|∇ ˆT|2+4|TQ|2
168
(∂xS,ω)ˆ = |∂xω| |S| ≤ˆ 1
8|∇ ˆω|2+2|S|2≤ 1
8|∇ ˆω|2+4c0|∇ ˆS|2+4|SQ|2,
169 170
Revised
Proof
and the “nonlinear” terms as (this defines c1)
171
(∂(ψ,ˆ ω), Ω)ˆ ≤c|∇ ˆψ|L∞|∇ ˆω|L2|Ω|L2 ≤ c1
2 |Ω|L2|∇ ˆω|2
172
(∂(Ψ,ω), Ω)ˆ ≤c|∇Ψ|L∞|∇ ˆω|L2|Ω|L2 ≤ p
8|∇ ˆω|2+c
p|∇Ψ|2L∞|Ω|2.(2.8)
173
This brings us to
174
d
dt| ˆω|2+(p−c1|Ω|)|∇ ˆω|2≤4pc0(|∇ ˆT|2+ |∇ ˆS|2)
175
+c
p|∇Ψ|2L∞|Ω|2+4p(|∇Ω|2+ |TQ|2+ |SQ|2).
176
(2.9)
177178
As usual, in the above and henceforth, c denotes generic constants which may take
179
different values each time it appears. Numbered constants such as c0have fixed values;
180
they are independent of the parameterspandβunless noted explicitly.
181
Now forS, we multiply (1.13c), or equivalently,ˆ
182
∂tSˆ+∂(ψ,Sˆ+SQ)=β (Sˆ+SQ), (2.10)
183
byS in Lˆ 2(D)and use(∂(ψ,S),ˆ S)ˆ =0 to find
184
1 2 d
dt| ˆS|2+β|∇ ˆS|2= −(∂(Ψ,SQ),S)ˆ −(∂(ψ,ˆ SQ),S)ˆ +β (SQ,S).ˆ (2.11)
185
The last term on the rhs requires some care,
186
(SQ,S)ˆ =(QS,S)ˆ L2(∂D)−(∇SQ,∇ ˆS)
≤c|QS|H−1/2(∂D)| ˆS|H1/2(∂D)+ |∇QS| |∇ ˆS|
≤ 1
8|∇ ˆS|2+c(QS2+ |∇SQ|2)
(2.12)
187
where we have used the trace theorem [1, Thm. 4.12] for the last inequality and
188
denoted QS := |QS|H−1/2(∂D). We note that|∇SQ|L2(D) ultimately depends on
189
|QS|H−1/2(∂D)plus the constraint (2.16) below. Bounding the “nonlinear” terms as
190
(∂(Ψ,S),ˆ SQ)≤c|∇Ψ|L∞|∇ ˆS|L2|SQ|L2 ≤ β
8 |∇ ˆS|2+ c
β |∇Ψ|2L∞|SQ|2
191
(∂(ψ,ˆ S),ˆ SQ)≤c|∇ ˆψ|L∞|∇ ˆS|L2|SQ|L2 ≤ β
8 |∇ ˆS|2+ c
β |∇ ˆω|2|SQ|2,
192 193
Revised
Proof
we arrive at (this defines c2)
194
d
dt| ˆS|2+β|∇ ˆS|2 ≤ c2
8c0β |∇ ˆω|2|SQ|2
195
+c
β |∇Ψ|2L∞|SQ|2+cβ (|∇SQ|2+ QS2). (2.13)
196
Analogously, we have forT (withˆ QT := |QT|H−1/2(∂D)),
197
d
dt| ˆT|2+ |∇ ˆT|2 ≤ c2
8c0|∇ ˆω|2|TQ|2
198
+c|∇Ψ|2L∞|TQ|2+c(|∇TQ|2+ QT2). (2.14)
199
Adding 8pc0times (2.14) and 8pc0/βtimes (2.13) to (2.9), we find
200
d dt
| ˆω|2+8pc0| ˆT|2+8pc0 β | ˆS|2
+4pc0(|∇ ˆT|2+ |∇ ˆS|2)
201
+
p−c1|Ω| −c2p|TQ|2−c2p β2|SQ|2
|∇ ˆω|2 (2.15)
202
≤cp|∇Ψ|2L∞
|Ω|2/p2+ |TQ|2+ |SQ|2/β2
203
+cp(|∇Ω|2+ |∇TQ|2+ |∇SQ|2+ QT2+ QS2).
204205
If we now chooseΩ, TQ and SQsuch that
206
|Ω|L2 ≤p/(8c1), |TQ|2L2 ≤1/(8c2) and |SQ|2L2 ≤β2/(8c2), (2.16)
207
(given the BC (1.14), this can always be done at the price of making∇Ω,∇TQ and
208
∇SQ large) we obtain the differential inequality
209
d dt
| ˆω|2+8pc0| ˆT|2+8pc0 β | ˆS|2
+p
2|∇ ˆω|2+4pc0(|∇ ˆT|2+|∇ ˆS|2)≤ F2, (2.17)
210
with F2 denoting the purely “forcing” terms on the rhs of (2.15). Integrat-
211
ing this using the Gronwall lemma, we obtain the uniform bounds, with | ˆU|2 =
212
| ˆω|2+8pc0| ˆT|2+8pc0| ˆS|2/β,
213
| ˆU(t)|2≤e−λt| ˆU(0)|2+ F2/λ
214
c3p t+1
t
|∇ ˆω|2+ |∇ ˆT|2+ |∇ ˆS|2
(t)dt≤e−λt|U(0)|2+(1+1/λ)F2
215
(2.18)
216217
valid for all t ≥ 0, for some λ(π) > 0. It is clear from (2.18a) that we have an
218
absorbing ball, i.e.|U(t)|2≤M0(Q;π)for all t ≥t0(|U(0)|;π).
219
Revised
Proof
On to H1, we multiply (2.4a) by−ωˆ in L2to find
220
1 2 d
dt|∇ ˆω|2+p|ω|ˆ 2 = −(∂(∇ψ,ω),ˆ ∇ ˆω)+(∂(ψ, Ω), ω)ˆ
221
−p(Ω, ω)ˆ −p(∂xT, ω)ˆ +p(∂xS, ω).ˆ (2.19)
222
Bounding the linear terms in the obvious way, and the nonlinear terms as
223
(∂(∇ψ,ω),ˆ ∇ ˆω)≤c|∇ ˆω|2L4|∇2ψ|L2 ≤c|∇ ˆω| |ω| |ψ|ˆ
224
≤ p
8|ω|ˆ 2+c
p|∇ ˆω|2(| ˆω|2+ |Ω|2)
225
(∂(ψ, Ω), ω)ˆ ≤ p
8|ω|ˆ 2+c p|∇Ω|2
|∇ ˆω|2+ |∇Ψ|2L∞ ,
226 227
we find
228
d
dt|∇ ˆω|2+p|ω|ˆ 2≤ c
p|∇ ˆω|2(| ˆω|2+ |Ω|2+ |∇Ω|2)+c
p|∇Ψ|2L∞|∇Ω|2
229
+8p
|∇ ˆT|2+ |∇ ˆS|2+ |∇TQ|2+ |∇SQ|2+ |Ω|2 . (2.20)
230231
Sinceω,ˆ T andˆ S have been bounded uniformly in Lˆ 2t,1Hx1in (2.18b), we can integrate
232
(2.20) using the uniform Gronwall lemma to obtain a uniform bound for|∇ ˆω|2,
233
|∇ ˆω(t)|2≤M1(· · ·) and
t+1
t
|ω(tˆ )|2dt≤ ˜M1(· · ·). (2.21)
234
Similarly, multiplying (2.10) by−S in Lˆ 2, we find
235
1 2 d
dt|∇ ˆS|2+β|Sˆ|2= −β (SQ, Sˆ)
236
−(∂(∇ψ,Sˆ),∇ ˆS)+(∂(ψ,SQ), Sˆ). (2.22)
237
Bounding as we did forω, we arrive atˆ
238
d
dt|∇ ˆS|2+β|Sˆ|2≤8β|SQ|2
239
+ c
β |∇ ˆS|2(| ˆω|2+ |Ω|2)+ c
β |∇SQ|2
|∇ ˆω|2+ |∇Ψ|2L∞ ,
240
(2.23)
241242
which can be integrated using the uniform Gronwall lemma to obtain
243
|∇ ˆS(t)|2≤M1(· · ·) and
t+1
t |Sˆ(t)|2dt≤ ˜M1(· · ·). (2.24)
244
Revised
Proof
Obviously one has the analogous bound forT ,ˆ
245
|∇ ˆT(t)|2≤M1(· · ·) and
t+1 t
|Tˆ(t)|2dt≤ ˜M1(· · ·). (2.25)
246
These bounds allow us to conclude [18] the existence of a global attractorAand
247
of an invariant measureμsupported onA. The convergence of the global attractors
248
can be deduced following an argument similar to that in [11], while the convergence
249
of the the invariant measures can be inferred from an argument similar to that in [20].
250
In particular, any generalised long-time average generates an invariant measure in the
251
sense that for any given bounded continuous functionalΦ(whose domain is the phase
252
space H and rangeR), we have
253
tlim→∞
1 t
t
0
Φ(S(t)U0)dt=
H
Φ(U)dμ(U) (2.26)
254
where U(t)=S(t)U0is the solution of (1.13) with initial data U0. It is known thatAis
255
unique whileμmay depend on the initial data U0and the definition of the generalised
256
limitlim.
257
Due to the boundary conditions, one cannot simply multiply by2ω, etc., to obtainˆ
258
a bound in H2, but following [17, §6.2], one takes time derivative of (1.13a) and uses
259
the resulting bound on|∂tω|to bound|ω|, etc. We shall not do this explicitly here,
260
although similar ideas are used for the discrete case below (Proof of Theorem2).
261
3 Numerical scheme: boundedness
262
Fixing a timestep k > 0, we discretise the system (1.13) in time by the following
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two-step explicit–implicit scheme,
264
3ωn+1−4ωn+ωn−1
2k +∂(2ψn−ψn−1,2ωn−ωn−1)
=p
ωn+1+∂xTn+1−∂xSn+1 3Tn+1−4Tn+Tn−1
2k +∂(2ψn−ψn−1,2Tn−Tn−1)=Tn+1 3Sn+1−4Sn+Sn−1
2k +∂(2ψn−ψn−1,2Sn−Sn−1)=βSn+1,
(3.1)
265
plus the boundary conditions (1.14). Writing Un = (ωn,Tn,Sn), we assume that
266
the second initial data U1has been obtained from U0using some reasonable one-step
267
method, but all we shall need for what follows is that U1∈ H1(D). The time derivative
268
term is that of the backward differentiation formula (BDF2) and the explicit form of
269
the nonlinear term is chosen to preserve the order of the scheme. This results in a
270
method that is essentially explicit yet second order in time, and as we shall see below,
271
preserves the important invariants of the continuous system.
272
Revised
Proof
Subject to some restrictions on the timestep k, we can obtain uniform bounds
273
and absorbing balls for the solution of the discrete system analogous to those of the
274
continuous system. Our first result is the following:
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Theorem 1 With Q ∈ H3/2(∂D), the scheme (3.1) defines a discrete dynamical
276
system in H1(D)×H1(D). Assuming U0, U1∈ H1(D)and the timestep restriction
277
given in (3.20) below,
278
k≤k1
|U0|H1,|U1|H1; |Q|H1/2(∂D), π , (3.2)
279
the following bounds hold
280
|Un|2L2 ≤40 e−νnk/4
|U0|2L2 + |U1|2L2 +M0
|Q|H1/2(∂D);π
281
+c
|Q|H−1/2(∂D);π k e−νnk/4
|U0|2H1+ |U1|2H1 , (3.3)
282
|Un|2H1 ≤N1
nk; |U0|H1,|U1|H1,|Q|H1/2(∂D), π +M1(|Q|H3/2(∂D);π), (3.4)
283 284
whereν(π) >0 and N1(t; · · ·)=0 for t≥t1
|U0|H1,|U1|H1;Q, π .
285
We note that the last term in (3.3) has no analogue in the continuous case; we believe
286
this is an artefact of our proof, but have not been able to circumvent it. Here one can
287
choose the bounds M0 and M1 to hold for both the continuous and discrete cases,
288
although the optimal bounds (likely very laborious to compute) may be different.
289
Unlike in [20], H2bounds do not follow as readily due to the boundary conditions,
290
so we proceed by first deriving bounds for|Un+1−Un|, using an approach inspired
291
by [17, §6.2]. We state our result without the transient terms:
292
Theorem 2 Assume the hypotheses of Theorem 1. Then for sufficiently large time,
293
nk≥t2(U0,U1;Q, π), one has
294
|ωn+1−ωn|2+ |Tn+1−Tn|2+ |Sn+1−Sn|2≤k2Mδ(|Q|H3/2(∂D);π). (3.5)
295
Furthermore, for large time nk≥t2the solution is bounded in H2as
296
|ωn|2+ |Tn|2+ |Sn|2≤ M2(|Q|H3/2(∂D);π). (3.6)
297
We remark that these difference and H2bounds require no additional hypotheses
298
on Q, suggesting that Theorem 1may be sub-optimal. We also note that using the
299
same method (and one more derivative on Q) one could bound|Un+1−Un|H1 and
300
|Un|H3, although we will not need these results here.
301
Following the approach of [20], these uniform bounds (along with the uniform
302
convergence results that follow from them) then give us the convergence of long-time
303
statistical properties of the discrete dynamical system (3.1) to those of the continuous
304
system (1.13).
305
Revised
Proof
Proof of Theorem1Central to our approach is the idea of G-stability for multistep
306
methods [9, §V.6]. First, for f , g∈L2(D)andνk∈ [0,1], we define the norm
307
|[f,g]|2νk= |f|2L2
2 +5+νk
2 |g|2L2−2(f,g)L2. (3.7)
308
Note that our notation is slightly different from that in [11,20]. Since both eigenvalues
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of the quadratic form are finite and positive for allνk∈ [0,1], this norm is equivalent to
310
the L2norm, i.e. there exist positive constants c+and c−, independent ofνk∈ [0,1],
311
such that
312
c−
|f|2L2 + |g|2L2 ≤ |[f,g]|2νk≤c+
|f|2L2+ |g|2L2 (3.8)
313
for all f , g∈ L2(D); computing explicitly, we find
314
c−= 6−√ 32
4 and c+=7+√
41
4 . (3.9)
315
As in [20], an important tool for our estimates is an identity first introduced in [9]
316
forνk=0; the following form can be found in [11, proof of Lemma 6.1]: for f , g,
317
h ∈L2(D)andνk∈ [0,1],
318
(3h−4g+ f,h)L2+νk|h|2L2
= |[g,h]|2νk− 1
1+νk|[f,g]|2νk+|f −2g+(1+νk)h|2L2
2(1+νk) . (3.10)
319
The fact that (3.1) forms a discrete dynamical system in H1×H1can be seen by
320
writing
321
(3−2k)Tn+1=4Tn−Tn−1−2k∂(2ψn−ψn−1,2Tn−Tn−1) (3.11)
322
and inverting: given Un−1and Un ∈ H1(D), the Jacobian is in H−1, which, with
323
the Neumann BC∂zTn+1=QT ∈ H1/2(∂D), gives Tn+1∈ H1. Similarly for Sn+1
324
and, since now Tn+1, Sn+1∈H1andωn+1=Qu∈ H1/2(∂D), forωn+1. Therefore
325
(Un−1,Un)∈H1×H1maps to(Un,Un+1)∈H1×H1.
326
Let ωˆn := ωn −Ω, Tˆn := Tn −TQ and Sˆn := Sn −SQ be defined as in
327
the continuous case, i.e. Ω, TQ, SQ ∈ H2(D)satisfying the boundary conditions
328
Ω =Qu,∂zTQ =QT and∂zSQ =QS, and the constraint (3.29), which is essentially
329
(2.16). The scheme (3.1) then implies
330
Revised
Proof
3ωˆn+1−4ωˆn+ ˆωn−1
2k +∂(2ψn−ψn−1,2ωˆn− ˆωn−1+Ω)
=p
ωˆn+1+Ω+∂xTn+1−∂xSn+1 3Tˆn+1−4Tˆn+ ˆTn−1
2k +∂(2ψn−ψn−1,2Tˆn− ˆTn−1+TQ)=Tˆn+1+TQ
3Sˆn+1−4Sˆn+ ˆSn−1
2k +∂(2ψn−ψn−1,2Sˆn− ˆSn−1+SQ)=β(Sˆn+1+SQ) (3.12)
331
where we have kept some ψn, Tn and Sn for now. We start by deriving difference
332
inequalities for ωˆn,Tˆn and Sˆn. In order to bound terms of the form |∇ ˆψn|2L∞ ≤
333
c| ˆωn|2H1/2, we assume for now the uniform bound
334
| ˆωn|2H1/2 ≤k−1/2Mω(· · ·) for all n=0,1,2, . . . (3.13)
335
where Mωwill be fixed in (3.31) below. We also assume for clarity that k≤1.
336
Multiplying (3.12a) by 2kωˆn+1in L2(D)and using (3.10), we find
337
|[ ˆωn,ωˆn+1]|2νk−νk| ˆωn+1|2+2pk|∇ ˆωn+1|2+|(1+νk)ωˆn+1−2ωˆn+ ˆωn−1|2 2(1+νk)
338
=|[ ˆωn−1,ωˆn]|2νk
1+νk −2k(∂(2ψn−ψn−1,ωˆn+1), (1+νk)ωˆn+1−2ωˆn+ ˆωn−1)
339
+2k(∂(2ψˆn− ˆψn−1,ωˆn+1), Ω)+2k(∂(Ψ,ωˆn+1), Ω)
340
+2pk
(Ω,ωˆn+1)+(ωˆn+1, ∂xTn+1)−(ωˆn+1, ∂xSn+1)
. (3.14)
341
whereν >0 will be set below. We bound the last terms as in the continuous case,
342
2|(Ω,ωˆn+1)≤ 1
8|∇ ˆωn+1|2+8|∇Ω|2
343
2|(∂xTn+1,ωˆn+1)| ≤ 1
8|∇ ˆωn+1|2+16c0|∇ ˆTn+1|2+16|TQ|2
344
2|(∂xSn+1,ωˆn+1)| ≤ 1
8|∇ ˆωn+1|2+16c0|∇ ˆSn+1|2+16|SQ|2
345
2(∂(Ψ,ωˆn+1), Ω)≤ p
8|∇ ˆωn+1|2+c
p|∇Ψ|2L∞|Ω|2,
346 347
and the previous one as
348
2|(∂(2ψˆn− ˆψn−1,ωˆn+1), Ω)| ≤c|2∇ ˆψn− ∇ ˆψn−1|L∞|∇ ˆωn+1|L2|Ω|L2 349
≤ p
8|∇ ˆωn+1|2+c
p(|∇ ˆωn−1|2+ |∇ ˆωn|2)|Ω|2. (3.15)
350
351
