About Convergence for Finite-difference Equations of Incompressible
Fluid with Boundary Conditions by Woods Formulas
Darkhan Akhmed-Zaki
1
, Nargozy Danaev
1
and Farida Amenova
2
1
Institute of Mathematics and Mechanics, al-Farabi Kazakh National University, Almaty, Kazakhstan
2
Department of Mathematics, D.Serikbaev East Kazakhstan State Technical University, Ust-Kamenogorsk, Kazakhstan
Keywords:
Two-dimensional System of the Navier-stokes Equations for an Incompressible Fluid, Linear Stokes Differ-
ential Problem, Method of a Priori Estimates, Stability, Convergence, Iterative Algorithm.
Abstract:
In this paper, mathematical aspects of stability, convergence and numerical implementation of two-
dimensional differential problem for incompressible fluid equations in “stream function, vorticity” variables
defined on a symmetrical template of finite-difference grid studied by method of a priori estimates are consid-
ered. Approximate boundary conditions for the vorticity are chosen in the form of Woods formula. In case
of a linear Stokes problem, it is shown that the numerical solution of the difference problem converges to the
solution of the differential problem with second order accuracy and two algorithms of numerical implemen-
tation, for which the rates of convergence obtained, are considered. In the case of non-linear Navier-Stokes
equations, estimates of the convergence of a solution of the difference problem to the solution of the differen-
tial problem, as well as estimation of the convergence of a considered iterative algorithm with the assumption
that the condition is equivalent to the condition of uniqueness of nonlinear difference problem are obtained.
1 INTRODUCTION
Sufficient number of scientific publications is de-
voted to the problems of numerical solution of two-
dimensional boundary value problems for incom-
pressible fluid equations in “stream function, vortic-
ity” variables. Descriptions of the most well-known
computing technologies are used during the compu-
tational experiments to study various flows of incom-
pressible fluid can be found in monographs (Chuhg,
2002), (Hirsch, 2002), (Kwak and Kiris, 2013). As
is known, the main difficulties encountered in the nu-
merical solution of the Navier-Stokes equations for an
incompressible fluid, associated with the implementa-
tion of the boundary conditions for the vorticity. Gen-
erally, in practice, to find the values of the vorticity on
the boundary, formulas approximating the conditions
of adhesion and impermeability of the velocity com-
ponents in the physical formulation of the problems
considered are used (Danaev and Smagulov, 1991),
(Vabishchevich, 1983), (Weinan and Liu, 1996). The
most famous among them are Tom and Woods for-
mulas (Tom and Aplt, 1964) having first and second
order accuracy, respectively, for determining the vor-
ticity on the boundary. Sufficient number of papers
devoted to theoretical and practical aspects of using
the Tom’s formula for the calculation of incompress-
ible fluid flow (Li and Wang, 2003). In the paper
(Voevodin, 1993) the absolute stability of the classi-
cal implicit difference schemes for two-dimensional
Stokes equations is proven and stable direct and iter-
ative methods for solving difference boundary value
problems by the method of operator inequalities are
proposed. In the paper (Voevodin and Yushkova,
1999), on the basis of the method of splitting into
physical processes, the numerical method for solving
initial-boundaryvalue problems for the Navier-Stokes
equations written in “stream function, vorticity” vari-
ables is proposed. To solve systems of implicit dif-
ference equations, a modification of “two-field” cal-
culation of the stream function and vorticity values is
used. The investigation of stability is conducted using
the linear approximation of differential schemes. In
the paper (Voevodin, 1998), using the method of a pri-
ori estimates, it is shownthat the solution of the differ-
ence scheme converges to the solution of differential
equations on a symmetrical grid pattern with the order
O(h
3/2
) in the case of choice of the boundary condi-
tion for the vorticity in form of Tom’s formula on the
boundary, where h = max(h
1
,h
2
), h
1
,h
2
are steps
of finite differential grid. Mathematical justification
of implicit iterative methods for their numerical im-
413
Akhmed-Zaki D., Danaev N. and Amenova F..
About Convergence for Finite-difference Equations of Incompressible Fluid with Boundary Conditions by Woods Formulas.
DOI: 10.5220/0005034204130420
In Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2014),
pages 413-420
ISBN: 978-989-758-038-3
Copyright
c
 2014 SCITEPRESS (Science and Technology Publications, Lda.)

plementation is given. Review of existing literature
shows that in the case of selecting Woods formula to
calculate the values of the vorticity at the boundary
theoretical studies are virtually absent. Issues of con-
vergence of difference schemes have not been inves-
tigated. There are no estimates of the rate of conver-
gence of iterative algorithms for numerical implemen-
tation of solutions of corresponding grid equations or
existing studies cover only the case of linear differ-
ence schemes (Danaev and Amenova, 2013).
2 STATEMENT OF THE
PROBLEM AND
FINITE-DIFFERENTIAL
EQUATIONS
In a domain D = {0 ≤ x,y ≤ 1} two-dimensional sys-
tem of stationary Navier-Stokes equations for an in-
compressible fluid of the following form is considered
(Rouch, 1980):

Ω
∂Ψ
∂y

x
−

Ω
∂Ψ
∂x

y
= ν∆Ω+ f(x,y), (1)
∆Ψ = Ω, (x,y) ∈ D, (2)
with boundary conditions
Ψ =
∂Ψ
∂n
= 0, (x,y) ∈ ∂D, (3)
where~n is the outward normal to the boundary of the
domain, ∆ is the two-dimensional Laplace operator,
Ψ is the stream function, Ω is the vorticity, ν is a vis-
cosity factor, and f (x,y) is a given function.
For approximation of equations (1), (2) in the
computational domain
D
h
=

(kh
1
,mh
2
), k ∈
1,N
1
− 1, m ∈ 1, N
2
− 1

,
where h
1
and h
2
are steps of the finite-differential grid
in the directions of x and y, respectively, the differen-
tial scheme on the symmetrical template of the fol-
lowing form is considered:
L
h
(Ω)Ψ = ν∆
h
Ω+ f, (4)
∆
h
Ψ = Ω, (5)
where the differential operator L
h
corresponds to the
approximation of convective terms of equations (1)
and is given in the form
L
h
(Ω)Ψ =

ΩΨ
0
y

0
x
−

ΩΨ
0
x

0
y
, (6)
here and further Ψ
0
x
, Ψ
0
y
means symmetrical differ-
ence derivatives in the directions of x and y, respec-
tively.
On the border
Ψ
0,m
= Ψ
N
1
,m
= 0, m ∈
1,N
2
− 1,
Ψ
k,0
= Ψ
k,N
2
= 0, k ∈ 1,N
1
− 1 (7)
for the vorticity, boundary conditions are selected in
the form of Woods formula (Rouch, 1980), for exam-
ple:
Ω
0,m
+
1
2
Ω
1,m
=
3
h
1
Ψ
x,0,m
..., m ∈
1,N
2
− 1. (8)
3 LINEAR DIFFERENTIAL
STOKES PROBLEM
Study of the stability and convergence of iterative al-
gorithms of numerical implementation of solving the
Navier-Stokes grid equations for an incompressible
fluid (4)-(8) is essentially based on the results which
can be obtained for the case of a linear Stokes prob-
lem.
∆Ω = f(x,y), (9)
∆Ψ = Ω, (x,y) ∈ D, (10)
with boundary conditions of the form (3). Here, for
simplicity, we assume that ν = 1.
In this case relations (4),(5) can be presented in
the following form:
∆
h
Ω
k,m
=
Ω
k+1,m
− 2Ω
k,m
+ Ω
k−1,m
h
2
1
+
+
Ω
k,m+1
− 2Ω
k,m
+ Ω
k,m−1
h
2
2
= f
k,m
, (11)
∆
h
Ψ
k,m
=
Ψ
k+1,m
− 2Ψ
k,m
+ Ψ
k−1,m
h
2
1
+
+
Ψ
k,m+1
− 2Ψ
k,m
+ Ψ
k,m−1
h
2
2
= Ω
k,m
, (12)
k ∈
1,N
1
− 1, m ∈ 1, N
2
− 1.
Hereinafter, the following well-known inequali-
ties will be used (Samarski, 1989)
δ
0
||u||
2
≤ ||∇
h
u||
2
, δ
0
||u|| ≤ ||∆
h
u||,
||∆
h
u||
2
≤
8
h
2
||∇
h
u||
2
, (13)
which hold for any grid function u ∈
0
Ω
h
, where h =
min(h
1
,h
2
), δ
0
> 0 is minimal eigenvalue of the dif-
ference Laplace operator,
0
Ω
h
(D
h
) is the space of grid
functions with zero boundary values defined at the
grid D
h
.
SIMULTECH2014-4thInternationalConferenceonSimulationandModelingMethodologies,Technologiesand
Applications
414

Let us investigate the stability of a solution of dif-
ference problem (11),(12) with the boundary condi-
tions of the form (7),(8). The relation (11) is mul-
tiplied by Ψ
k,m
h
1
h
2
summed over the internal nodes
of grid D
h
, next, using the formulas of summation by
parts and the boundary conditions (7) we have the en-
ergy identity
N
2
−1
∑
m=1

Ω
0,m
Ψ
x,0,m
− Ω
N
1
,m
Ψ
¯x,N
1
,m

h
2
+
+
N
1
−1
∑
k=1

Ω
k,0
Ψ
y,k,0
− Ω
k,N
2
Ψ
¯y,k,N
2

h
1
+
+k∆
h
Ψk
2
= ( f,Ψ).
Hereinafter, k fk is the norm of the grid function in
the space L
2,h
(D
h
).
Hence, using the boundary conditions of the form
(8), after simple transformations, we have
k∆
h
Ψk
2
+
h
1
h
2
4

N
2
−1
∑
m=1

|Ω
0,m
|
2
+ |Ω
N
1
,m
|
2

+
+
N
1
−1
∑
k=1

|Ω
k,0
|
2
+ |Ω
k,N
2
|
2


−
−
h
1
h
2
12

N
2
−1
∑
m=1
(|Ω
1,m
|
2
+ |Ω
N
1
−1,m
|
2
)+ (14)
+
N
1
−1
∑
k=1
(|Ω
k,1
|
2
+ |Ω
k,N
2
−1
|
2
)

+
+
h
1
h
2
12

N
2
−1
∑
m=1

(Ω
0,m
+ Ω
1,m
)
2
+ (Ω
N
1
,m
+ Ω
N
1
−1,m
)
2

+
+
N
1
−1
∑
k=1

(Ω
k,0
+ Ω
k,1
)
2
+ (Ω
k,N
2
+ Ω
k,N
2
−1
)
2


= ( f,Ψ).
Therefore, we can write
11
12
k∆
h
Ψk
2
≤ |( f,Ψ)|.
Hence, using the Cauchy-Bunyakovsky’s inequal-
ity, we obtain the estimation
k∆
h
Ψk ≤ c
0
k fk.
Here and below, we will designate the bounded posi-
tive constants non-dependent from the grid parameter
h
1
,h
2
by c
0
.
3.1 On the Convergence of Linear
Difference Stokes Problem
Assuming that the solution of differential problem
(9),(10) with the boundary conditions (3) has a suf-
ficient smoothness required for our analysis, we will
study the order of convergenceof the difference prob-
lem (7),(8),(11),(12) to the solution of the differential
problem.
Let us designate discrepancy of differential equa-
tions (11),(12) as R
h
and Q
h
, respectively, i.e.
R
h
= ∆
h
Ω
h
− f (x,y),
Q
h
= ∆
h
Ψ
h
− Ω
h
, (x,y) ∈ D
h
,
where solutions of differential problem (3),(9),(10) in
nodes of finite-differential grid are designated as Ψ
h
,
Ω
h
.
Obviously, since chosen approximation formulas
for derivatives are symmetrical,
R
h
= O

h
2

, Q
h
= O

h
2

, h = max(h
1
,h
2
).
Let us introduce the following designations:
Φ = Ψ
h
− Ψ, Z = Ω
h
− Ω.
Then for solution errors we have the following re-
lations:
∆
h
Z = R
h
, (15)
∆
h
Φ = Z + Q
h
(16)
with boundary conditions
Φ(x,y) = 0, (x,y) ∈ ∂D
h
, (17)
Z
0,m
+
1
2
Z
1,m
=
3
h
1
Φ
x,0,m
+ r
0,m
, ... (18)
m =
1,N
2
− 1,
where r
0,m
= r
N
1
,m
= r
k,0
= r
k,N
2
= O

h
2

.
To obtain an estimate of convergence, let us mul-
tiply the relation (15) by Φh
1
h
2
and sum over the in-
ternal nodes of grid D
h
. In this case, the main energy
identity considering conditions (17) has the form
(Z,∆
h
Φ) +
N
2
−1
∑
m=1
(Z
0,m
Φ
x,0,m
− Z
N
1
,m
Φ
¯x,N
1
,m
)h
2
+
+
N
1
−1
∑
k=1

Z
k,0
Φ
y,k,0
− Z
k,N
2
Φ
¯y,k,N
2

h
1
= (R
h
,Φ).
Considering the relation (16) and the boundary
conditions (18) we will get:
(∆
h
Φ− Q
h
,∆
h
Φ) +
h
1
3
N
2
−1
∑
m=1
h
Z
0,m

Z
0,m
+
1
2
Z
1,m
−
−r
0,m

+ Z
N
1
,m
(Z
N
1
,m
+
1
2
Z
N
1
−1,m
− r
N
1
,m
)
i
h
2
+
AboutConvergenceforFinite-differenceEquationsofIncompressibleFluidwithBoundaryConditionsbyWoodsFormulas
415

+
h
2
3
N
1
−1
∑
k=1
h
Z
k,0

Z
k,0
+
1
2
Z
k,1
− r
k,0

+
+Z
k,N
2

Z
k,N
2
+
1
2
Z
k,N
2
−1
− r
k,N
2

i
h
1
= (R
h
,Φ).
Applying simple transformations, we have:
11
12
k∆
h
Φk
2
≤
h
1
6
N
2
−1
∑
m=1
(|r
0,m
|
2
+ |r
N
1
,m
|
2
)h
2
+
+
h
2
6
N
1
−1
∑
k=1
(|r
k,0
|
2
+ |r
k,N
2
|
2
)h
1
+
+|(Q
h
,∆
h
Φ)| + |(R
h
,Φ)|.
Using the “ε” inequality and relations (13), we
will get the inequality which holds for any positive
ε
1
,ε
2
:

11
12
− ε
1
− ε
2

k∆
h
Φk
2
≤ c
0

1
4ε
1
kQ
h
k
2
+
1
4ε
2
kR
h
k
2

+
h
1
h
2
6

N
2
−1
∑
m=1
(|r
0,m
|
2
+ |r
N
1
,m
|
2
) +
N
1
−1
∑
k=1
(|r
k,0
|
2
+ |r
k,N
2
|
2
)

.
Choosing ε
1
,ε
2
satisfying condition
11
12
− ε
1
− ε
2
≥ δ > 0,
considering the order of smallness of values
R
h
, Q
h
, r
0,m
, r
N
1
,m
, r
k,0
, r
k,N
2
, we finally have
δk∆
h
Φk
2
≤ c
0
h
4
,
that is
k∆
h
Φk ≤ c
0
h
2
,
which means that the solution of the difference
scheme convergesto the solutions of differentialprob-
lem with the second order of accuracy.
3.2 Study of Convergence of Iterative
Algorithm I
For the numerical solution of equations
(7),(8),(11),(12) first we will consider iterative
algorithm of the following form (Algorithm I)
Ω
n+1
k,m
− Ω
n
k,m
τ
= ∆
h
Ω
n
k,m
− f
k,m
, (19)
∆
h
Ψ
n+1
k,m
= Ω
n+1
k,m
, (20)
with boundary conditions
Ψ
n+1
= 0, (x, y) ∈ ∂D
h
, (21)
Ω
n+1
0,m
+
1
2
Ω
n+1
1,m
=
3
h
1
Ψ
n+1
x,0,m
, ... m ∈
1,N
2
− 1. (22)
Hereinafter, for iterative algorithms, we assume that
initial values assignment for stream function is ex-
pected.
Let us introduce designations
Φ
n
k,m
= Ψ
n
k,m
− Ψ
k,m
, Z
n
k,m
= Ω
n
k,m
− Ω
k,m
,
where Ψ
n
k,m
, Ω
n
k,m
are solutions of differential prob-
lem (19)-(22), Ψ
k,m
, Ω
k,m
are solutions of differential
problem (7),(8),(11),(12).
Then for iteration errors we have the following re-
lations:
Z
n+1
k,m
− Z
n
k,m
τ
= ∆
h
Z
n
k,m
, (23)
∆
h
Φ
n+1
k,m
= Z
n+1
k,m
, (24)
with boundary conditions
Φ
n+1
= 0, (x,y) ∈ ∂D
h
, (25)
Z
n+1
0,m
+
1
2
Z
n+1
1,m
=
3
h
1
Φ
n+1
x,0,m
,... m =
1,N
2
− 1. (26)
We multiply the relation (25) by 2τΦ
n+1
k,m
h
1
h
2
and
sum over internal nodes of the grid D
h
. Considering
boundary conditions , we have
k∇
h
Φ
n+1
k
2
− k∇
h
Φ
n
k
2
+ k∇
h
(Φ
n+1
− Φ
n
)k
2
+
2τ

N
2
−1
∑
m=1

Z
n
0,m
Φ
n+1
x,0,m
− Z
n
N
1
,m
Φ
n+1
¯x,N
1
,m

h
2
+
+
N
1
−1
∑
k=1

Z
n
k,0
Φ
n+1
y,k,0
− Z
n
k,N
2
Φ
n+1
¯y,k,N
2

h
1

+
+2τ(∆
h
Φ
n
,∆
h
Φ
n+1
) = 0, (27)
where
k∇
h
Φk
2
=

N
1
∑
k=1
N
2
−1
∑
m=1
|Φ
¯x,k,m
|
2
+
N
1
−1
∑
k=1
N
2
∑
m=1
|Φ
¯y,k,m
|
2

h
1
h
2
,
∀Φ ∈
0
Ω
h
(D
h
).
Let us rewrite (27) in the following form
k∇
h
Φ
n+1
k
2
− k∇
h
Φ
n
k
2
+ k∇
h
(Φ
n+1
− Φ
n
)k
2
+
+2τ

N
2
−1
∑
m=1

Z
n
0,m
Φ
n
x,0,m
− Z
n
N
1
,m
Φ
n
¯x,N
1
,m

h
2
+
+
N
1
−1
∑
k=1

Z
n
k,0
Φ
n
y,k,0
− Z
n
k,N
2
Φ
n
¯y,k,N
2

h
1

+
+2τ

N
2
−1
∑
m=1

Z
n
0,m
(Φ
n+1
− Φ
n
)
x,0,m
−
−Z
n
N
1
,m
(Φ
n+1
− Φ
n
)
¯x,N
1
,m

h
2
+
+2τ

N
1
−1
∑
k=1

Z
n
k,0
(Φ
n+1
− Φ
n
)
y,k,0
−
−Z
n
k,N
2
(Φ
n+1
− Φ
n
)
¯y,k,N
2

h
1
+
+2τ(∆
h
Φ
n
,∆
h
Φ
n+1
) = 0. (28)
SIMULTECH2014-4thInternationalConferenceonSimulationandModelingMethodologies,Technologiesand
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416

Transforming the corresponding terms in the equation
(28) as in the case of the relationship (14) and consid-
ering knowninequalities (13) after simple transforma-
tions we will get
k∇
h
Φ
n+1
k
2
+ (1−
10τ
h
2
)k∇
h
(Φ
n+1
− Φ
n
)k
2
−
−k∇
h
Φ
n
k
2
+
5τ
6

k∆
h
Φ
n+1
k
2
+ k∆
h
Φ
n
k
2

≤
≤
2τ
h
2
1
N
2
−1
∑
m=1

|(Φ
n+1
− Φ
n
)
x,0,m
|
2
+
+|(Φ
n+1
− Φ
n
)
¯x,N
1
,m
|
2

h
1
h
2
+
+
2τ
h
2
2
N
1
−1
∑
k=1

|(Φ
n+1
− Φ
n
)
y,k,0
|+
+|(Φ
n+1
− Φ
n
)
¯y,k,N
2
|
2

h
1
h
2
.
Therefore, under condition
1−
10τ
h
2
≥ 0, (29)
the following inequality holds
k∇
h
Φ
n+1
k
2
− k∇
h
Φ
n
k
2
+
+
5τδ
0
6

k∇
h
Φ
n+1
k
2
+ k∇
h
Φ
n
k
2

≤ 0
i.e.
k∇
h
Φ
n+1
k ≤ qk∇
h
Φ
n
k
where
q =
s
1− τβ
1+ τβ
< 1, β =
5δ
0
6
.
Hence, we can conclude that when condition (29)
holds iterations by Algorithm I converge at a geomet-
ric rate with denominator q < 1. Thus, it is possible
to ensure that value q
n
≤ ε, where ε the number char-
acterizing iteration accuracy if
n ≥ n
0
(ε) ≈ O(
1
h
2
)ln
1
ε
.
3.3 Study of Convergence of Iterative
Algorithm II
Further, let us consider iterative algorithm of the fol-
lowing form (Algorithm II)
Ω
n+1
k,m
− Ω
n
k,m
τ
= ∆
h
Ω
n+1
k,m
− f
k,m
, (30)
∆
h
Ψ
n+1
k,m
= Ω
n+1
k,m
, (31)
with boundary conditions
Ψ
n+1
= 0, (x,y) ∈ ∂D
h
, (32)
Ω
n+1
0,m
+
1
2
Ω
n+1
1,m
=
3
h
1
Ψ
n
x,0,m
,... m ∈
1,N
2
− 1. (33)
For iteration errors we have relationships
Z
n+1
k,m
− Z
n
k,m
τ
= ∆
h
Z
n+1
k,m
, (34)
∆
h
Φ
n+1
k,m
= Z
n+1
k,m
, (35)
with boundary conditions
Φ
n+1
= 0, (x,y) ∈ ∂D
h
, (36)
Z
n+1
0,m
+
1
2
Z
n+1
1,m
=
3
h
1
Φ
n
x,0,m
,... m ∈
1,N
2
− 1. (37)
In this case, considering boundary condition (36)
the main energy identity has the form
k∇
h
Φ
n+1
k
2
− k∇
h
Φ
n
k
2
+ k∇
h
(Φ
n+1
− Φ
n
)k
2
+
+2τ

N
2
−1
∑
m=1

Z
n+1
0,m
Φ
n+1
x,0,m
− Z
n+1
N
1
,m
Φ
n+1
¯x,N
1
,m

h
2
+
+
N
1
−1
∑
k=1

Z
n+1
k,0
Φ
n+1
y,k,0
− Z
n+1
k,N
2
Φ
n+1
¯y,k,N
2

h
1

+
+2τk∆
h
Φ
n+1
k
2
= 0.
Considering boundary conditions (37) and trans-
forming it, we will get the inequality
k∇
h
Φ
n+1
k
2
− k∇
h
Φ
n
k
2
+ (1−
2τ
h
2
)k∇
h
(Φ
n+1
− Φ
n
)k
2
+
+
11τ
6
k∆
h
Φ
n+1
k
2
≤ 0.
Consequently, under the condition
1−
2τ
h
2
≥ 0, (38)
we have that
k∇
h
Φ
n+1
k ≤ k∇
h
Φ
n
k,
where
q =
s
1
1+
11τδ
0
6
< 1.
i.e. we can conclude that when condition (38) holds,
iterations by Algorithm II also converge at a geomet-
ric rate with ratio q < 1 and for n
0
(ε) the following
relation holds
n ≥ n
0
(ε) ≈ O(
1
h
2
)ln
1
ε
.
AboutConvergenceforFinite-differenceEquationsofIncompressibleFluidwithBoundaryConditionsbyWoodsFormulas
417

4 STUDY OF BOUNDARY VALUE
PROBLEM FOR NON-LINEAR
NAVIER-STOKES EQUATIONS
Note that for the differential operator L
h
the fol-
lowing relations are valid (Danaev and Amenova,
2013)
|(L
h
(ω)u,v)| ≤ c
0
kωkk∆
h
ukk∆
h
vk, (39)
(L
h
(ω)u,u) = 0, ∀u,v ∈
0
Ω
h
(D
h
), (40)
where c
0
is a uniformly bounded constant.
Due to the fact that the equation (40) holds, pro-
ceeding as in the case of the linear problem, we will
have a priori estimate of the solution for the solution
of the differential problem (4)-(8):
νk∆
h
Ψk ≤ c
0
k fk.
4.1 Uniqueness Condition of Solutions
Non-linear Differential
Navier-stokes Equations
Let us show that under the condition c
0
k fk
ν
2
< 1, the
solution of the problem (4)-(8) is unique.
Assume that there are two solutions (Ψ
1
,Ω
1
) and
(Ψ
2
,Ω
2
). Then for differences Φ = Ψ
1
− Ψ
2
Z =
Ω
1
− Ω
2
, we have the differential problem:
L
h
(Ω
1
)Φ+ L
h
(Z)Ψ
2
= ν∆
h
Z,
∆
h
Φ = Z,
with the following boundary conditions:
Φ = 0, Z
0,m
+
1
2
Z
1,m
=
3
h
1
Φ
x,0,m
,... m ∈
1,N
2
− 1.
We have
νk∆
h
Φk
2
≤ |(L
h
(Z)Ψ
2
,Φ)| ≤ c
0
k∆
h
Φk
2
k∆
h
Ψ
2
k,

ν− c
0
k∆
h
Ψ
2
k

k∆
h
Φk
2
≤ 0.
Hence, if
ν− c
0
k∆
h
Ψ
2
k > 0,
c
0
k fk
ν
2
< 1, (41)
then it should be
k∆
h
Φk = 0,
i.e. the solution is unique.
4.2 Study of Convergence of Non-linear
Difference Equations
Assuming sufficient smoothness of solutions of the
differential problem (1)-(3), we will study the con-
vergence of the solution of grid equations (4)-(8).
Let us designate discrepancy of the differential
scheme (4) for the equation of motion as R
h
, and dis-
crepancy of the differential relation (5) as Q
h
:
R
h
= L
h
(Ω
h
)Ψ
h
− ν∆
h
Ω
h
+ f (x,y) , (x,y) ∈ D
h
,
Q
h
= ∆
h
Ψ
h
− Ω
h
.
where solutions of the differential problem (1)-(3)
in nodes of finite-differential grid are designated as
Ψ
h
,Ω
h
.
Obviously, because chosen approximation tem-
plates of derivatives are symmetrical
R
h
= O

h
2

, Q
h
= O

h
2

,
where h = max(h
1
,h
2
).
Let us introduce
Φ = Ψ
h
− Ψ, Z = Ω
h
− Ω.
Then for solution error we have following relation-
ships:
L
h
(Ω
h
)Φ+ L
h
(Z)Ψ = ν∆
h
Z + R
h
, (42)
∆
h
Φ = Z + Q
h
, (43)
with boundary conditions
Φ(x,y) = 0, (x,y) ∈ ∂D
h
, (44)
Z
0,m
+
1
2
Z
1,m
=
3
h
1
Φ
x,0,m
+ r
0,m
,... m ∈
1,N
2
− 1,
(45)
where r
0,m
= r
N
1
,m
= r
k,0
= r
k,N
2
= O

h
2

.
In order to obtain estimation of convergence, we
multiply the relation (42) by Φ and sum by internal
nodes of grid D
h
. Then we apply Green’s difference
formula, take into account the given condition (44),
and as a result, we have the following main energy
identity:
ν

(Z,∆
h
Φ) +
N
2
−1
∑
m=1
(Z
0,m
Φ
x,0,m
− Z
N
1
,m
Φ
¯x,N
1
,m
)h
2
+
+
N
1
−1
∑
k=1

Z
k,0
Φ
y,k,0
− Z
k,N
2
Φ
¯y,k,N
2

h
1
+
+(R
h
,Φ) = (L
h
(Z)Ψ,Φ).
Considering the relation (43) and boundary condi-
tions (45), we will get:
ν(∆
h
Φ− Q
h
,∆
h
Φ) +
νh
1
3
N
2
−1
∑
m=1
h
Z
0,m

Z
0,m
+
1
2
Z
1,m
−
SIMULTECH2014-4thInternationalConferenceonSimulationandModelingMethodologies,Technologiesand
Applications
418

−r
0,m

+ Z
N
1
,m
(Z
N
1
,m
+
1
2
Z
N
1
−1,m
− r
N
1
,m
)
i
h
2
+
+
νh
2
3
N
1
−1
∑
k=1
h
Z
k,0

Z
k,0
+
1
2
Z
k,1
− r
k,0

+ Z
k,N
2

Z
k,N
2
+
1
2
Z
k,N
2
−1
− r
k,N
2

i
h
1
+ (R
h
,Φ) = (L
h
(Z)Ψ,Φ).
Hence, we will get inequality:
11
12
νk∆
h
Φk
2
≤
νh
1
h
2
6

N
2
−1
∑
m=1
(|r
0,m
|
2
+ |r
N
1
,m
|
2
)+
+
N
1
−1
∑
k=1
(|r
k,0
|
2
+ |r
k,N
2
|
2
)

+
+ν|(Q
h
,∆
h
Φ)| + |(R
h
,Φ)| + |(L
h
(Z)Ψ,Φ)|.
Applying the “ε” -inequality and the inequality
(39) for the operator L
h
, we have

11
12
ν− νε
1
− ε
2
− c
0
k∆
h
Ψk(1+ ε
3
)

k∆
h
Φk
2
≤
≤
ν
4ε
1
kQ
h
k
2
+
1
4ε
2
kR
h
k
2
+ c
0
k∆
h
Ψk
4ε
3
kQ
h
k
2
+
+
h
1
h
2
6

N
2
−1
∑
m=1
(|r
0,m
|
2
+ |r
N
1
,m
|
2
) +
N
1
−1
∑
k=1
(|r
k,0
|
2
+ |r
k,N
2
|
2
)

.
Choosing positive parameters ε
1
, ε
2
, ε
3
satisfying the
inequality
11
12
ν− νε
1
− ε
2
− c
0
k∆
h
Ψk(1+ ε
3
) ≥ δ > 0, (46)
we have
δk∆
h
Φk
2
≤ Mh
4
,
where 0 < M < ∞ uniformly bounded constant non-
dependent from grid steps.
Therefore,
k∆
h
Φk ≤ c
0
h
2
.
i.e. under the condition (46) which is equivalent to
the uniqueness condition (41), the solutions of differ-
ential equation (4)-(8) converge to the solution of the
differential problem as in the case of the linear Stokes
problem with the second order of accuracy.
4.3 Study of Convergence of Iterative
Algorithm for Non-linear Problem
To find a numerical solution of the differential prob-
lem (4)-(8), let us consider the iterative algorithm of
the following form:
Ω
n+1
− Ω
n
τ
+ L
h
(Ω
n
)Ψ
n
= ν∆
h
Ω
n
+ f, (47)
∆
h
Ψ
n+1
= Ω
n+1
, (48)
with boundary conditions
Ψ
0
(x,y) = Ψ
0
(x,y) = 0, (x, y) ∈ D
h
Ψ
n+1
= 0, (x,y) ∈ ∂D
h
, (49)
Ω
n+1
0,m
+
1
2
Ω
n+1
1,m
=
3
h
1
Ψ
n+1
x,0,m
,... m ∈
1,N
2
− 1. (50)
Let us show that the solution of differential
scheme (47)-(51) converges to the solution of the dif-
ferential problem (4)-(8) and we obtain estimation of
the convergence rate.
For iteration errors, we have the relations:
Z
n+1
− Z
n
τ
+ L
h
(Ω)Φ
n
+ L
h
(Z
n
)Φ = ν∆
h
Z
n
, (51)
∆
h
Φ
n+1
= Z
n+1
, (52)
Ψ
n+1
= 0, (x,y) ∈ ∂D
h
, (53)
Z
n+1
0,m
+
1
2
Z
n+1
1,m
=
3
h
1
Φ
n+1
x,0,m
,... m ∈
1,N
2
− 1. (54)
We multiply (51) by 2τΦ
n+1
and sum by nodes
of the grid. Considering conditions (53) we have the
following identity
k∇
h
Φ
n+1
k
2
− k∇
h
Φ
n
k
2
+ k∇
h
(Φ
n+1
− Φ
n
)k
2
+
+τν(∆
h
Φ
n
,∆
h
Φ
n+1
)+
+2τν

N
2
−1
∑
m=1

Z
n
0,m
Φ
n+1
x,0,m
− Z
n
N
1
,m
Φ
n+1
¯x,N
1
,m

h
2
+
+
N
1
−1
∑
k=1

Z
n
k,0
Φ
n+1
y,k,0
− Z
n
k,N
2
Φ
n+1
¯y,k,N
2

h
1

=
+2τ(L
h
(Z
n
)Ψ,Φ
n+1
).
Considering boundary conditions (54), applying
simple transformations, and applying known inequal-
ities (13) we will get
k∇
h
Φ
n+1
k
2
− k∇
h
Φ
n
k
2
+
+τνk∆
h
Φ
n+1
k
2
+
5τν
6
k∆
h
Φ
n
k
2
−
−(1−
10τν
h
2
)

N
2
−1
∑
m=1



(Φ
n+1
− Φ
n
)
x,0,m


2
+
+


(Φ
n+1
−Φ
n
)
¯x,N
1
,m


2

+
N
1
−1
∑
k=1



(Φ
n+1
−Φ
n
)
y,k,0


2
+
+


(Φ
n+1
− Φ
n
)
¯y,k,N
2


2

h
1
h
2
+
+(1−
8τν
h
2
)

N
1
−2
∑
k=2
N
2
−1
∑
m=1

|Φ
n+1
¯x,k,m
− Φ
n
¯x,k,m
|
2

+
+
N
1
−1
∑
k=2
N
2
−2
∑
m=1

|Φ
n+1
¯y,k,m
− Φ
n
¯y,k,m
|
2


h
1
h
2
≤
AboutConvergenceforFinite-differenceEquationsofIncompressibleFluidwithBoundaryConditionsbyWoodsFormulas
419

≤ 2τc
0
k∆
h
Ψkk∆
h
Φ
n
kk∆
h
Φ
n+1
k.
Therefore, under the condition
1−
10τν
h
2
≥ 0, (55)
we have the following inequalities
k∇
h
Φ
n+1
k
2
− k∇
h
Φ
n
k
2
+
+τνk∆
h
Φ
n+1
k
2
+
5τν
6
k∆
h
Φ
n
k
2
≤
≤ τc
0
k∆
h
Ψk

k∆
h
Φ
n+1
k
2
+ k∆
h
Φ
n
k
2

,
k∇
h
Φ
n+1
k
2
− k∇
h
Φ
n
k
2
+
+τ

ν− c
0
k∆
h
Ψk

k∆
h
Φ
n+1
k
2
+
+τ

5
6
ν− c
0
k∆
h
Ψk

k∆
h
Φ
n
k ≤ 0,

1+ τδ
0

ν− c
0
k∆
h
Ψk

k∇
h
Φ
n+1
k
2
≤
≤

1− τδ
0

5
6
ν− c
0
k∆
h
Ψk


k∇
h
Φ
n
k
2
.
Assume that
5ν
6
ν− c
0
k∆
h
Ψk ≥ δ > 0, (56)
then
k∇
h
Φ
n+1
k ≤ qk∇
h
Φ
n
k
where
q =
s
1− τδ
0
δ
1+ τδ
0
δ
< 1.
Therefore, when condition (55) at chosen param-
eters τ,h and the inequality (56) both hold iterations
converge at a geometric rate with denominator less
than one and for n
0
(ε) as in the case of the linear prob-
lem following relationship is valid
n ≥ n
0
(ε) ≈ O(
1
h
2
)ln
1
ε
.
5 CONCLUSION
In the paper, the study of the differential scheme writ-
ten on a symmetrical grid pattern and methods of their
numerical implementation for an incompressible fluid
for equations in case of the choice of boundary con-
ditions for grid values of the vorticity at the boundary
by Woods formula are conducted. It is shown that the
order of accuracy of the differential scheme in case of
the choice of the Woods formula is better in compari-
son with the case of using the Tom’s formula. For the
Stokes difference problem, two algorithms for the nu-
merical implementation of the solution of the differ-
ence problem are considered and the influence of the
boundary conditions on the iteration layers is studied.
In the case of the non-linear Navier-Stokes problem
for the considered iterative algorithm, it is shown that
the assumption of certain conditions which are equiv-
alent to the uniqueness condition of the differential
problem, the convergencerate coincides with the con-
vergence rate in the case of the linear Stokes problem.
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Chuhg, T. (2002). Computational Fluid Dynamics. Cam-
bridge University Press, New York.
Danaev, N. and Amenova, F. (2013). Vorticity bound-
ary condition and related issues for finite difference
sabout one method to solve navier-stokes equation in
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