IJMMS 2003:63, 3979–3993
PII. S0161171203212242
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
APPROXIMATE SOLUTION FOR EULER EQUATIONS
OF STRATIFIED WATER VIA NUMERICAL
SOLUTION OF COUPLED KdV SYSTEM
A. A. HALIM, S. P. KSHEVETSKII, and S. B. LEBLE
Received 23 December 2002
We consider Euler equations with stratified background state that is valid for internal water waves. The solution of the initial-boundary problem for Boussinesq
approximation in the waveguide mode is presented in terms of the stream function. The orthogonal eigenfunctions describe a vertical shape of the internal wave
modes and satisfy a Sturm-Liouville problem. The horizontal profile is defined by
a coupled KdV system which is numerically solved via a finite-difference scheme
for which we prove the convergence and stability. Together with the solution of
the Sturm-Liouville problem, the stream functions give the internal waves profile.
2000 Mathematics Subject Classification: 76B15, 76B70, 76B55, 37L65, 65M06,
65M12.
1. Introduction. The basic system of Euler equations for internal water
waves in two dimensions (xz), with a stable stratified ambient state and the
buoyancy frequency N(z), is
ux + wz = 0,
→
→
ρo ut = −ρo v , ∇ u − px ,
→
→
ρo wt = −ρo v , ∇ w − pz − ρ g,
(1.1)
→
→
Tt + w T̄z = − v , ∇ T ,
where u, v are velocity components, ρo is the density, p is the pressure, ρ g is
the body force due to stratification, T̄z is the vertical background temperature
gradient, and T is the temperature variable [3]. Combining the equations in
(1.1) and using the state relation for liquid ρ = −ρo αT , α = −ρz /(ρ T̄z ) is the
coefficient of thermal expansion, and T̄z = N 2 /(αg), we obtain
∆wtt + N 2 wxx − N 2
+
→
→
v,∇
→
→
v , ∇ wz dx
t
txz
w dt
0
+
xx
→
→
v , ∇ w txx = 0.
(1.2)
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A. A. HALIM ET AL.
Rescaling the dimensionless variables (primed) using xi = λi xi , t = 2π t /N̄β,
u = λz N̄u /2π , w = βλz N̄w /2π , αT = T , N = N̄N /2π , N̄ is the average
buoyancy frequency and β is a scale parameter. Substitute in (1.2) and omit
primes for simplicity to obtain
λz
wxx wzz
wxx
N̄ 3
+
(βN)2
2 +
2
2π
λx
λx
λz tt
3 t
t
N̄
(βN)2
−
u wx dt + w wz dt
2π
λx
0
0
xx

N̄β 3  λz +
uwx + wwz xxt
2π  λ2x
N̄β
2π
3 (1.3)

t

1
= 0.
+
wz dx
uwz + w
λz
0
z xzt 
Introduce the stream function ψ, w = −σ ψx and u = σ ψz (σ is a scale parameter). Integrate, with respect to x,
ψzztt + N 2 ψxx = −β2 ψxxtt − σ ψz ψxz − ψx ψzz zt
t
t
+ σ N 2 ψz ψxx dt − ψx ψxz dt .
0
0
(1.4)
x
Substitute in (1.4) by the stream function of the form
ψ(z, x, t) =
Z m (z)θ m (x, t),
(1.5)
m
multiply by Z n , integrate with respect to z, and use the separation of variables
that give
n
=−
Zzz
N2 n
Z .
cn2
(1.6)
So (1.4) becomes
n
n
θtt
− cn2 θxx
=
t
m k
cn2 β2 n
n
n
n
2
m k
m
k
θ
+
σ
c
θ
+
b
θ
θ
+
e
θ
θ
dt
a
,
θ
t
n
x t
x
m,k
m,k
m,k x
N 2 xxtt
0
x
m,k
(1.7)
APPROXIMATE SOLUTION FOR EULER EQUATIONS . . .
where
an
m,k
h =N
2
−h
−1
1
−N 2 h m k n
n
zzm zk zn dz,
bm,k
=
zz z z dz,
2 + 2
2
ck
cm
ck −h
h
n
= N2
zm zzk zn dz.
em,k
3981
(1.8)
−h
System (1.7) describes the two oppositely directed propagated modes. The
equations of the separated propagated modes are obtained by substituting
θtn = un , cn θxn = v n , so (1.7) becomes

t
3 2

β
c
vk
n
n
n
n
n
2
m
a
v
+ σ cn
u dt ·
ut − cn vx =
 m,k 0
N 2 xxx
ck t
m,k

t

m t
vk
n
n v
m
k
+ bm,k u dt · u + em,k
dt
,

cm 0 ck
0
x
vtn − cn un
x = 0.
(1.9)
Using projection operators,


c 2 β2 2
1
1
+
∂


x
1
2N 2  ,
P+ = 
2 2


β
c
2
1−
∂x2
1
2
2N


c 2 β2 2
∂
1
−1
−

1
2N 2 x  .
P− = 

c 2 β2 2
2
−1 +
∂x
1
2
2N
(1.10)
So
un
ϕn+
P+
=
,
vn
kϕn+
un
ϕn−
P−
=
vn
−kϕn−
(1.11)
or
un = ϕn+ + ϕn− ,
c 2 β2 2 n+
v n = ϕn+ − ϕn− −
∂ ϕ − ϕn− .
2N 2 x
(1.12)
Operating P+ , P− on (1.9) and using (1.12), we obtain the equations for the
separated modes ϕn+ , ϕn− as
+
k t
σ cn2 n
c 3 β2 n+
ϕ − ϕ−
n+
n+
+
− m
ϕ t − cn ϕ x −
ϕ
−
a
dt ·
ϕ +ϕ
2N 2 xxx
2 m,k m,k 0
ck
t
t
+
m
k
n
ϕ + ϕ− dt · ϕ+ + ϕ−
+ bm,k
0
n
+ em,k
+
m t +
k ϕ − ϕ−
ϕ − ϕ−
dt
= 0,
cm
ck
0
x
3982
A. A. HALIM ET AL.
+
k t
σ cn2 n
c 3 β2 n−
ϕ − ϕ−
+
− m
ϕ
−
a
+
ϕ
dt
·
ϕ
2N 2 xxx
2 m,k m,k 0
ck
t
ϕtn− + cn ϕxn− +
n
+ bm,k
t
0
n
+ em,k
+
m
k
ϕ + ϕ− dt · ϕ+ + ϕ−
+
m t +
k ϕ − ϕ−
ϕ − ϕ−
dt
= 0.
cm
ck
0
x
(1.13)
n−
= ηn−
Let ϕn+ = ηn+
t , ϕ
t , so (1.13) becomes
n+
ηn+
t − c n ηx =
m +
k
σ cn2 n +
c 3 β2 n+
η + η−
η − η− x
ηxxx +
a
2
2N
2 m,k m,k
+
m k
n
+ bm,k
η + η− · η+ + η− x
+
n−
ηn−
t + c n ηx
n
em,k
+
m k
η − η− x η + − η− ,
ck cm
m +
k
σ cn2 n +
c 3 β2 n−
η − η− x
=−
ηxxx +
am,k η + η−
2
2N
2 m,k
(1.14)
+
m k
n
+ bm,k
η + η− · η+ + η− x
+
n
em,k
+
m k
η − η− x η + − η− .
ck cm
In what follows we will consider both directed modes. In this work we evaluate only for short time to describe the phenomena, so we consider only one
direction of the propagating modes. Hence the system that describes one direction of (1.7) has the form (return to θ variable for convenient)
θtn + cn θxn + σ
n
n
gm,k
θ m θxk + β2 dn θxxx
= 0,
(1.15)
m,k
n
=
gm,k
σ N 2 cn2
2
h −h
−1
2
1
k m
m k
n
Z
+
Z
−
Z
Z
z
z Z dz,
2
cm ck
cm
ck2
dn =
cn3
.
2N 2
(1.16)
Hence we can summarize as follows: in the commonly accepted approximations (incompressibility, Boussinesq approximation, and so forth) the solution
of system (1.1) is constructed as the representation for the stream function
(1.5) which is completely defined by solving (1.6) and (1.15).
APPROXIMATE SOLUTION FOR EULER EQUATIONS . . .
3983
2. Physical model. In this model, we simulate the initial stage of McEwan
experiment [4, 5] for a rectangular tank of dimension 50 cm in x by 25 cm in z
directions filled by a linearly stratified water of constant buoyancy frequency
1.23 s −1 . The internal water waves are described by system (1.1). The solution
of this system is constructed as the representation for the stream function
(1.5), where Z n (z) are solutions of the correspondent Sturm-Liouville problem
(1.6), Zzz +(N 2 /cn2 )Z = 0, Z(0) = Z(h) = 0, and describe a vertical shape of the
wave modes. The linear propagation velocities cn play the role of eigenvalues.
The coefficient functions θ n (x, t) are solutions of the coupled KdV system
(1.15) with coefficients from (1.16). We select a localized initial condition along
x-axis described by a smooth enough function that models the paddle motion
as in the experiments of McEwan [5]. The function is also chosen antisymmetric
along x-axis in relation to the paddle axis centered in the middle of the tank.
The time intervals of simulations are taken such that the initial disturbance
decays essentially but does not reach the boundaries.
3. Computation. When the coefficients in the Sturm-Liouville equation (1.6)
are constant, it has very simple general solution
Z n = Bn sin
nπ z
,
h
n = 1, 2, 3, . . . , L,
(3.1)
which tends to zero at boundaries. The eigenvalues
cn =
Nh
,
nπ
n = 1, 2, 3, . . . , L,
(3.2)
have the sense of linear internal gravity waves velocities. Normalization is deh
termined by 0 (Z n )2 N 2 dz = 1 and gives Bn = (2/N 2 h)1/2 . Hence
Z n (z) =
2
N2h
1/2
sin
nπ z
.
h
(3.3)
To solve the coupled KdV system (1.15) an initial condition is required. We
can select the initial perturbation for the stream function (1.5) which has the
general form
ψ(z, x, 0) =
L
Z n (z)θ n (x, 0) = ϕ(x, z) = ϕ1 (x)ϕ2 (z).
n=1
Taking the exponentially localized functions ϕ1 (x), ϕ2 (z):
(3.4)
3984
A. A. HALIM ET AL.
a
,
cosh(x/l)
1/2
2
ϕ2 (z) =
sech b z − z0 tanh b z − z0 ,
N 2h
ϕ1 (x) =
(3.5)
as the initial condition we model the impact of a wave-productor.
The choice of the form and the constants a, b, l qualitatively reflects the
paddle movement (we restrict the movement by some isolated pulse), and the
numerical value for the amplitude is estimated also from the description of
the experiment of McEwan [5]. Then the scalar product gives
L
j n n
j Z , Z θ (x, 0) = Z j , ϕ2 (z) ϕ1 (x),
Z ,ψ =
(3.6)
n=1
j
Z , ϕ2 (z) =
h
0
N 2 Z j ϕ2 (z)dz,
(3.7)
and using the orthogonality,
Zj, Zn =
h
0
N 2 Z j Z n dz = 1, (j = n) , 0, (j ≠ n) .
(3.8)
Hence, (3.6) gives θ n (x, 0), the initial condition of system (1.15) which is solved
by the numerical method introduced below.
4. Numerical method. For the coupled KdV system (1.15) we introduce a
numerical (finite-difference) method of solution, a two-step three-time-level
scheme similar to the Lax-Wendroff one [1, 7]. The usual Lax-Wendroff scheme
is modified such that the order of the first derivative becomes of order O(x 4 ).
The approximation of the nonlinear terms is changed such that the integral of
θ 2 is a conserved one. The scheme has the form
n j+1/2 n j !
θ i
− θ i
τ/2
+
k,m
+
m j
n
gmk
θ i
cn
n j
j !
θ i+1 − θ n i−1
2h
j !
k j
θ i+1 − θ k i−1
2h
(4.1)
!
θ n j − 2θ n j + 2θ n j − θ n j
cn h2
i+2
i+1
i−1
i−2
+ dn −
6
2h3
= 0,
where n, m, and k are the modes numbers and i and j are discrete space and
time variables, respectively. The time step is denoted by τ while the spatial
step size is denoted by h. Equation (4.1) is accompanied by a discrete equation
3985
APPROXIMATE SOLUTION FOR EULER EQUATIONS . . .
for the intermediate layer as
n j+1 n j !
θ i − θ i
τ
+
+
cn
n j+1/2 n j+1/2 !
θ i+1 − θ i−1
m j+1/2
n
gmk
θ i
2h
k j+1/2 k j+1/2 !
θ i+1 − θ i−1
k,m
+
en
(4.2)
2h
j+1/2
j+1/2 j+1/2 !
n j+1/2
θ i+2 − 2 θ n i+1 + 2 θ n i−1 − θ n i−2
2h3
= 0.
To support the results, we prove stability and convergence of the scheme in
Appendices A and B [2]. Beside these proofs, we would like to mention that the
order of errors of the difference formulas is improved at the same time they
preserve conservation laws of the KdV-type equations [6].
5. Results. Figure 5.1 is a contour plot presenting the initial perturbation
ψ(z, x, 0) for the upper half of the tank due to antisymmetry.
Figure 5.2 is one-dimensional (x) plots for the second tenth modes.
Figure 5.3 is a three-dimensional plot for the wave profile (1.5) at t = 0.02
second for the same half of the tank.
0.2
25
0.15
0.1
0.05
20
15
0
100
10
200
5
300
400
Figure 5.1. Three-dimensional plot for the initial perturbation
ψ(z, x, 0) for the upper half of the tank. The horizontal numbers
(25, 400) indicate the number of mesh points used in plotting in x
and z directions while the dimensions are 12.5 cm in z and 50 cm in
x, respectively.
3986
A. A. HALIM ET AL.
0.01
−0.25 −0.15
−0.05
−0.01
0.07
0.05
0.15
0.25
x-axis
0.05
−0.03
0.03
−0.05
0.01
−0.25 −0.15
−0.07
0.05
0.15
0.25
x-axis
Fourth mode, series 1 (t = 0),
series 2 (t = 0.02 s)
Second mode, series 1 (t = 0),
series 2 (t = 0.02 s)
Series 1
Series 2
Series 1
Series 2
0.07
0.01
−0.25 −0.15
−0.05
−0.01
−0.05
−0.01
0.05
0.15
0.25
x-axis
0.05
−0.03
0.03
−0.05
0.01
−0.25
−0.07
−0.15
−0.05
−0.01
0.05
0.15
0.25
x-axis
Eighth mode, series 1 (t = 0),
series 2 (t = 0.02 s)
Sixth mode, series 1 (t = 0),
series 2 (t = 0.02 s)
Series 1
Series 2
Series 1
Series 2
0.01
−0.25 −0.15
−0.05
−0.01
0.05
0.15
0.25
x-axis
−0.03
−0.05
−0.07
Tenth mode, series 1 (t = 0),
series 2 (t = 0.02 s)
Series 1
Series 2
Figure 5.2. One-dimensional (x) plots of the wave profile at t = 0
and t = 0.02 second.
3987
APPROXIMATE SOLUTION FOR EULER EQUATIONS . . .
25
0.4
20
0.2
15
0
10
100
200
5
300
400
Figure 5.3. Three-dimensional (x, y, z) plot of the wave profile
ψ(z, x, t) at t = 0.02 second for the upper half of the tank. The horizontal numbers (25, 400) indicate the number of mesh points used
in plotting in x and z directions while the dimensions are 12.5 cm
in z and 50 cm in x, respectively.
All the plots for the modes and for the sum (stream function) show the behavior that is typical for the multisolitonic perturbation. It looks like a decay of
the initial condition to solitons in the single KdV equation theory. The process
of the wave propagation is accompanied by interaction that implies the energy
transfer between modes. This phenomenon may be considered as a possible
reason for the vertical fine-structure generation [3]. The combination of the
fine structures may explain McEwan experiments [4, 5].
6. Summary. In the commonly accepted approximations (incompressibility, Boussinesq approximation, and so forth), the solution of the system of
Euler equations with stratified background state is constructed as the representation for the stream function. The horizontal profile is defined by a coupled KdV system which is numerically solved via a finite-difference scheme for
which stability and convergence are proved. Together with the solution of the
Sturm-Liouville problem that describes the vertical profile, the stream function
gives the complete internal waves profile.
Appendices
A. Stability proof of the scheme. We prove stability with respect to small
perturbations (because we consider nonlinear equations) of initial conditions.
Strictly speaking, it is the boundness of the discrete solution in terms of small
3988
A. A. HALIM ET AL.
perturbation of the initial data. Consider the differential
n,j+1
Ti,r
n,j
dθr
=
n,j
θi−2
"
=
n,j+1 #
∂θi
n,j
∂θr
n,j
θi−1
n,j
θi
,
n,j
θi+1
(A.1)
!
n,j t
θi+2
and define the norm
1/2

n,j 2
$ j$
$dθ $ = 
dθr
h .
r
(A.2)
n
We can write
n,j+1
dθi
n,j+1
= Ti,r
n,j
dθr
n,j+1
= Ti,r
n,j
n,j−1
Ti,r dθr
n r
= Πr Ti,r
dθrn,o ,
n,j+1
(A.3)
n,o
is the perturbation of the discrete solution and dθr is a small
where dθi
n r
) ,
perturbation of the initial data. Stability required the boundness of Πr (Ti,r
r
that is, T is bounded. We calculate T from the difference scheme as follows:
n,j+1
Ti,r
cn τ δi+1,r − δi−1,r
2h
n
&
%
gm,k
k,j
m,j k,j δi+1,r − δi−1,r + δi,r θi+1 − θi−1
−τ
θi
2h
m,k
= δi,r −
−
(A.4)
τen δi+2,r − 2δi+1,r + 2δi−1,r − 2δi−2,r .
3
2h
Rewriting the matrix T (A.4) in terms of identity (E), symmetric (S), and antisymmetric (A), matrices yields T = E + S + A,
m,j
n,j+1 τ n % m,j
m,j m,j g
θi − θi+1 δi+1,r − θi − θi−1 δi−1,r
S
i,r = −
4h m,k m,k
&
k,j
k,j + 2δi,r θi+1 − θi−1 ,
n,j+1 cn τ δi+1,r − δi−1,r
A
i,r = −
2h
&
m,j
τ n % m,j
m,j m,j −
gm,k θi + θi+1 δi+1,r − θi + θi−1 δi−1,r
4h m,k
en τ δi+2,r − 2δi+1,r + 2δi−1,r − δi−2,r ,
3
2h
$ j+1 $
'
'
''
'
'
$S
$ ≤ τ max 'g n ' max 'θ m,j ', 'θ`k,j ' ,
x,i
x,i
m,k
−
n,m,k
i,m,k
APPROXIMATE SOLUTION FOR EULER EQUATIONS . . .
3989
m,j
k,j
m,j k,j − θi
θ
θ − θi−1
m,j
k,j
θx,i = i+1
,
θ`x,i = i+1
,
h
2h
' n '
' '
' '
'
$ j+1 $ τ max n,m,k 'gm,k
' m,j ' τ max n 'cn ' 3τ maxn 'en '
$A $ ≤
'
'
max θi
+
,
+
h
h
h3
m,i
$ $
$ j+1 $2 $
∗
$
$
$T
$ =$
$ T j+1 T j+1 $ = $ E − Aj+1 + S j+1 E + Aj+1 + S j+1 $
$
$ $
$ $
$2
≤ 1 + 2$S j+1 $ + $Aj+1 $ + $S j+1 $
' n '
'
'
' max 'θ m,j '
≤ 1 + 2τ max 'gm,k
x,i
n,m,k
+τ
2
m,i
' n '
'
' n '
' m,j ' max n,m,k 'gm,k
' m,j '
'
'
'
'
max gm,k max θx,i +
max 'θi '
h
n,m,k
m,i
m,i
' '
' ' 2
max n 'cn ' 3 maxn 'en '
+
+
h
h3
≤ eaτ ,
' n '
'
'
' max 'θ m,j '
a = 1 + 2τ max 'gm,k
x,i
n,m,k
+τ
2
m,i
' n '
'
' n '
' m,j ' max n,m,k 'gm,k
' m,j '
'
'
'
'
max gm,k max θx,i +
max 'θi '
h
n,m,k
m,i
m,i
' '
' ' 2
max n 'cn ' 3 maxn 'en '
,
+
+
h
h3
(A.5)
which is a necessary condition of stability. The scheme is stable if a ≤ constant
in spite of τ, h → 0. This is a conditional stability of the scheme. It means that
it is required for stability that τ → 0 more faster than h → 0 or
τ ≤ (constant) · h6 .
(A.6)
B. Convergence proof of the scheme. We would prove that the solution of
the difference equations (4.1) and (4.2) converges to the solution of (1.15) if the
exact solution is a continuously differentiable one. We place here for brevity
a proof that uses only one-step time difference equation and the whole proof
may be developed by similar ideas. Therefore, we now consider the difference
equation
n j
j
j
j
n j+1 n j
θ i+1 − θ n i−1 n m j θ k i+1 − θ k i−1
θ i − θ i
gm,k θ i
+ cn
+
τ
2h
2h
m,k
(B.1)
n j
n j
n j
n j
θ i+2 − 2 θ i+1 + 2 θ i−1 − θ i−2
= 0.
+ en
2h3
3990
A. A. HALIM ET AL.
∞
We know that the KdV-type equation has the conservation law −∞ u2 (x, t)dx =
constant. It may also be shown that if a smooth enough solution un (x, t) of
(L ∞
the set of equations (1.15) exists, then it satisfies the inequality n=1 −∞ (un (x,
t))2 dx < B, for a finite t, where B is a constant dependent on initial conditions
only, L-number of modes taken into account by system (1.15). Therefore, it is
natural to use L2 -norm in the proof, and
1/2

% 2 &j
$ j$
n
$Θ $ = 
h  .
θ
i
(B.2)
i
n
Here, h is a grid step, i is a discrete space variable, and j is the discrete time.
t
The symbol Θ denotes a column Θj ≡ (θ 1 )j (θ 2 )j (θ 3 )j · · · and the
components of this column are columns also:
n j
≡ ···
θ
n j
θ i−1
n j
θ i
n j
θ i+1
!t
··· ,
(B.3)
j
where (θ n )i is a solution of the finite-difference equation (B.1).
j
Let the vector (un )i = un (xi , t j ) be an exact solution of (B.1) in the points
j
of grid. Then the error (v n )i is given by
n j n j n j
v i = θ i − u i.
j
(B.4)
j
The difference solution (θ n )i converges to the exact solution (un )i if V j → 0
as τ, h → 0, where V j ≡ Θj − U j . Substitute (B.4) into (B.1) and obtain
n j
j
n j+1 n j
v i+1 − v n i−1
v i − v i
+ cn
τ
2h
k j
j
k j
j
m j uk i+1 − uk i−1
v
j
i+1 − v
i−1
n
n
+
gm,k
um i
v i
+ gm,k
2h
2h
m,k
j j
m j v k i+1 − v k i−1
n
+ gm,k
v i
2h
n j
j
j
n j
v i+2 − 2 v i+1 + 2 v n i−1 − v n i−2
+ en
2h3
j+1 j
j
n j
u n i − un i
u i+1 − un i−1
+ cn
=−
τ
2h
k j
k j
m j u i+1 − u i−1
n
+
gm,k
u i
2h
m,k
n j
j
j n j
u i+2 − 2 u i+1 + 2 un i−1 − un i−2
.
+ en
2h3
(B.5)
APPROXIMATE SOLUTION FOR EULER EQUATIONS . . .
3991
Pick out (in v) a linear part of expression (B.5) and introduce for convenience
an operator T j by the expression
 j
j
j
j
n j
v n i+1 − v n i−1 n m j v k i+1 − v k i−1

gm,k u i
v i − τ cn
2h
2h
m,k
j
j
m j uk i+1 − uk i−1
n
+ gm,k
v i
2h
j
j
j 
n j
v i+2 − 2 v n i+1 + 2 v n i−1 − v n i−2 
+ en
2h3
j
=
T j+1 ir v n i , n = 1, 2, 3, . . . , L.
(B.6)
r
j
Using the above expression for T and utilizing that (un )i is an exact solution
of differential equations, due to the approximation, we use the fact that the
right-hand-side term of (B.5) is a small one of order O(τ + h2 ). Therefore, we
can rewrite (B.5) as follows:
n j+1 j+1 n j
T
v i −
ir v
i
r
+τ
m,k
n
gm,k
j
j
m j v k i+1 − v k i−1
= O τ + h2
v i
2h
(B.7)
or
n j+1 j+1 n j
j
)ir v i + τ f n,m,k i ,
v i = (T
(B.8)
r
where
j
j
nmk j
m j v k i+1 − v k i−1
n
=
−τ
g
f
v
+ O τ + h2 ,
i
i
m,k
2h
m,k
(B.9)
and the following estimation for the norm is valid:
$ $
$ j+1 $ $ j+1 $$ j $
$ ≤ $T
$$V $ + τ $f j $.
$V
(B.10)
If we will consequently substitute (B.10) into itself, we will get
$ j+1 $ $ j+1 $$ j $
$ $
$ ≤ $T
$$v $ + τ $f j $
$v
$$ $$
$
$$
$ $ $
$
$
≤ $T j+1 $$T j $$v j−1 $ + τ $T j+1 $$f j−1 $ + $f j $
$$ $$
$$
$
$
≤ $T j+1 $$T j $$T j−1 $$v j−2 $
$$ $$
$ $
$$
$ $ $
$
+ τ $T j+1 $$T j $$f j−2 $ + $T j+1 $$f j−1 $ + $f j $ .
(B.11)
3992
A. A. HALIM ET AL.
Using the estimate of norm of T in inequality (A.5), (B.11) becomes
$ $
$ $
$ $
$ j+1 $
$ $
$ ≤ eaτj $v 0 $ + τ eaτ(j−1) $f 0 $ + eaτ(j−2) $f 1 $ + · · · $f j $
$v
$ $
$ $
≤ eaτj $v 0 $ + M max $f k $ ,
k≤j
M =τ
(B.12)
eatmax − 1
;
eaτ − 1
M is the sum of a geometric series and tmax is the time of simulation, 0 ≤ t ≤
j
tmax . The norm of (f n,m,k )i is given by simple estimations
1/2

%
$ j$
&2
mk j
$f $ = 
h
f
i
i
1/2
 
2
' n ' m j v k ji+1 − v k ji−1
 h
= max 'gm,k ' 
v i

2h
n,m,k
i
m,k
1/2

' n ' % m j &2 % k j &2
'
≤ max 'gm,k
v i h
v i h
n,m,k
i
m
i
k
' n '
' $ $
max n,m,k 'gm,k
$V j $ 2 + O τ + h2 .
≤
3/2
h
(B.13)
1
h3/2
Using this estimate in inequality (B.12) yields
' n '
' $
$ j+1 $
$ $
$
max n,m,k 'gm,k
$V
$V j+1 $ 2 + O τ + h2 .
$ ≤ eaτj $V 0 $ + M
h3/2
(B.14)
Taking into consideration V 0 = 0, the above inequality gives the solution
' n '
$ j+1 $ 1 − 1 − 4M max n,m,k 'gm,k '/h3/2 MO τ + h2
$V
$≤
' n '
2M max n,m,k 'gm,k '/h3/2
MO τ + h2 ,
(B.15)
that is, V j+1 → 0 as τ, h → 0. Hence the convergence is proved.
Acknowledgment. This work was partially supported by the KBN Grant no
5 PO3B 040 20.
References
[1]
[2]
[3]
[4]
[5]
S. P. Kshevetskii, Analytical and numerical investigation of nonlinear internal gravity waves, Nonlinear Processes in Geophysics 8 (2001), 37–53.
P. Lancaster, Theory of Matrices, Academic Press, New York, 1969.
S. B. Leble, Nonlinear Waves in Waveguides: with Stratification, Springer-Verlag,
Berlin, 1991.
A. D. McEwan, Internal mixing in stratified fluids, J. Fluid Mech. 128 (1983), 59–80.
, The kinematics of stratified mixing through internal wavebreaking, J. Fluid
Mech. 128 (1983), 47–57.
APPROXIMATE SOLUTION FOR EULER EQUATIONS . . .
[6]
[7]
3993
Z. Shaohong, A difference scheme for the coupled KdV equation, Commun. Nonlinear Sci. Numer. Simul. 4 (1999), no. 1, 60–63.
J. C. Tannehill, D. A. Anderson, and R. H. Pletcher, Computational Fluid Mechanics
and Heat Transfer, Taylor & Francis, Washington, 1997.
A. A. Halim: Theoretical and Mathematical Physics Department, Gdansk University
of Technology, 80-952 Gdansk, Poland
E-mail address: ahmed@mif.pg.gda.pl
S. P. Kshevetskii: Theoretical Physics Department, Kaliningrad State University, Kaliningrad 236041, Russia
E-mail address: verd@tphys.albertina.ru
S. B. Leble: Theoretical and Mathematical Physics Department, Gdansk University of
Technology, 80-952 Gdansk, Poland
E-mail address: leble@mif.pg.gda.pl