Additive schemes for certain operator-differential equations

ISSN 0965�5425, Computational Mathematics and Mathematical Physics, 2010, Vol. 50, No. 12, pp. 2033–2043. © Pleiades Publishing, Ltd., 2010.Original Russian Text © P.N. Vabishchevich, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 12, pp. 2144–2154.

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Additive (splitting) schemes are used for solving various nonstationary problems (see [1–3]); they aredesigned for a more efficient computational implementation of various schemes constructed for solvingthe corresponding grid problem on a new time level. Upon the approximation with respect to space basedon finite difference, finite element, or finite volume methods, splitting with respect to spatial variables,physical processes, or domains (domain decomposition method) is performed.

The main theoretical results concerning the stability and convergence of additive schemes are obtainedfor scalar first�order evolutionary equations and sometimes for second�order equations. For the compu�tational practice, of considerable interest are splitting schemes for systems of evolutionary equations. Forexample, in vector problems, certain components of the vector of unknowns can be related to each other;in this case, the use of splitting schemes can be aimed at obtaining good problems for the individual com�ponents of the solution on the next time level.

For the standard parabolic and hyperbolic systems of equations with a self�adjoint elliptic operator,additive schemes were constructed in [4] on the basis of the regularization principle for finite differenceschemes. For systems of equations, splitting schemes can be constructed using the triangular splitting ofthe operator of the problem into a sum of mutually adjoint operators; this is Samarskii’s alternate�trian�gular method. Such additive schemes are used in [5] for dynamic elasticity problems and in [6] for prob�lems concerning incompressible fluid with variable viscosity.

In [7], the Cauchy problem for a special first�order linear system of equations with self�adjoint opera�tors in a Hilbert space is considered. For example, a similar structure of equations is typical for acousticsproblems (dynamics of compressible fluid) and electrodynamics. In this paper, we consider a somewhatdifferent system of equations that is typical, in particular, for dynamics of incompressible fluid. Standardtwo�level weighted schemes and their computational implementation are considered. The main result ofthis study is the construction of splitting schemes based on the solution of the problems associated withindividual operators.

The results are used to construct splitting schemes with respect to the space variables (locally one�dimensional schemes) for the Stokes (Navier–Stokes) equations of viscous incompressible fluid. A morecomplicated example of splitting associated with the decomposition of the computation domain requiresa special and more detailed investigation. Numerical methods for solving problems of mechanics of con�tinua were in the range of interests of Dorodnicyn (e.g., see [8]). Personally for the author of the presentpaper, his paper [9] was the first one in the field of domain decomposition methods, which are so popularnowadays.

Additive Schemes for Certain Operator�Differential EquationsP. N. Vabishchevich

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russiae�mail: [email protected]

Received May 14, 2010

Abstract—Unconditionally stable finite difference schemes for the time approximation of first�orderoperator�differential systems with self�adjoint operators are constructed. Such systems arise in manyapplied problems, for example, in connection with nonstationary problems for the system of Stokes(Navier–Stokes) equations. Stability conditions in the corresponding Hilbert spaces for two�levelweighted operator�difference schemes are obtained. Additive (splitting) schemes are proposed thatinvolve the solution of simple problems at each time step. The results are used to construct splittingschemes with respect to spatial variables for nonstationary Navier–Stokes equations for incompress�ible fluid. The capabilities of additive schemes are illustrated using a two�dimensional model problemas an example.

DOI: 10.1134/S0965542510120067

Keywords: evolutionary problems, operator�difference schemes, stability, Navier–Stokes equationsfor incompressible fluid.

Dedicated to Academician A.A. Dorodnicynon the Occasion of the Centenary of His Birth

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 50 No. 12 2010

VABISHCHEVICH

1. STATEMENT OF THE PROBLEM

Let Hi (i = 1, 2, …, m) be finite�dimensional Hilbert spaces with the scalar product (·, ·)i and the norm (i = 1, 2, …, m). The components of the solution are denoted by ui(t) (i = 1, 2, …, m) for each t (0 ≤

t ≤ T > 0). We seek a solution to the system of first�order evolutionary equations

(1)

Here, fi(t) ∈ Hi (i = 1, 2, …, m) are given and Ai is a linear (independent of t) operator acting from Hm toHi for i = 1, 2, …, m – 1.

The specific feature of the system under examination is due to the last equation, which connects theequations into the system. Notice that the case of the last equation in the form

was investigated in [7]. Such a relation between the components is typical for the dynamics of incompress�ible fluid (the equation of continuity). The last equation in (1) is characteristic of the problems concerningincompressible fluid.

System of equations (1) is supplemented with the initial conditions

(2)

We also assume that

(3)

where E is the identity operator in Hm.

Let us derive simple a priori bounds on the solution of problem (1)–(3), which we will later use as land�marks in the investigation of the corresponding operator�difference schemes. To find a bound on um, we

multiply each equation in (1) for ui by and add them to obtain

Taking into account inequality (3), we have

(4)

Taking the scalar product of the equations in (1) with ui in Hi for i = 1, 2, …, m and summing them, weobtain

For the individual terms on the right�hand side, we use the bound

· i

dui

dt�� Aium+ fi, i 1 2 … m 1,–, , ,= =

Ai*ui

i 1=

m 1–

∑– fm, 0 t T.≤<=

dum

dt�� Ai

*ui

i 1=

m 1–

∑– fm, 0 t T≤<=

ui 0( ) vi, i 1 2 … m 1.–, , ,= =

Ai*Ai

i 1=

m 1–

∑ δE, δ 0,>≥

Ai*

Ai*Aium

i 1=

m 1–

∑ Ai*fi

dfm

dt�� .+

i 1=

m 1–

∑=

um m1δ�� Ai

*fi

i 1=

m 1–

∑dfm

dt��+

m

.≤

12��

d ui i2

dt��

i 1=

m 1–

∑ fi ui,( )i fm um,( )m.+i 1=

m 1–

∑=

fi ui,( )i12�� ui i

2 12�� fi i

2, i+≤ 1 2 … m., , ,=


ADDITIVE SCHEMES FOR CERTAIN OPERATOR�DIFFERENTIAL EQUATIONS 2035

Using Gronwall’s lemma and taking into account initial condition (2), we obtain the bound

(5)

With regard to (4), bound (5) reflects the stability of solution of problem (1)–(3) with respect to the initialdata and the right�hand side for the components ui(t) (i = 1, 2, …, m – 1).

2. WEIGHTED SCHEME

Define a uniform grid on the time interval

and an approximate solution yn = y(tn), tn = nτ. For operator�differential problem (1), (2), standard two�level weighted schemes can be naturally used.

We associate with system (1) the finite difference scheme

(6)

where σ is a numerical parameter (weight), which is typically in the range 0 ≤ σ ≤ 1. For simplicity, weconsider the same weight for all the equations in (1). Taking into account conditions (2), we supplement(6) with the initial condition

(7)

For two�level operator�difference schemes, generic stability conditions are known (see [4, 10, 11]).However, the direct use of these conditions for the investigation of operator�difference schemes (6) is noteasy. For that reason, we restrict ourselves to a direct derivation of simple stability bounds for operator�difference scheme (6). In doing so, we use bounds (4) and (5) for operator�differential problem (1)–(3).

First, we obtain an analog of a priori bound (4). We introduce the following notation for the weightedgrid quantities:

Multiplying Eqs. (6) by for i = 1, 2, …, m– 1 and adding them, we obtain

With regard to (3), this yields

(8)

To obtain an a priori bound on the other components of the solution, we write system (6) in the form

(9)

ui t( ) i2

i 1=

m 1–

∑ t( ) vi i2

t θ–( ) fi θ( ) i2

um θ( ) m2

+i 1=

m

∑⎝ ⎠⎜ ⎟⎛ ⎞

exp θ.d

0

t

∫+i 1=

m 1–

∑exp≤

ωτ ωτ T{ }∪ tn nτ n, 0 1 … N τN, , , , T= = ={ }= =

yin 1+

yin

–τ

�� Ai σymn 1+

1 σ–( )ymn

+( )+ σfi n 1+

1 σ–( )fi n

, i+ 1 2 … m 1,–, , ,= =

Ai*yi

n 1+

i 1=

m 1–

∑– fm n 1+

, n 0 1 … N 1,–, , ,= =

yi0

vi, i 1 2 … m 1.–, , ,= =

yiσ n( ) σyi

n 1+1 σ–( )yi

n, i+ 1 2 … m., , ,= =

Ai*

Ai*Aiyi

σ n( )

i 1=

m 1–

∑ Ai*fi

σ n( ) fm n 1+

fm n

–τ

��, n+i 1=

m 1–

∑ 0 1 … N 1.–, , ,= =

yiσ n( )

m1δ�� Ai

*fi σ n( ) fm

n 1+fm

n–

τ��+

i 1=

m 1–

∑m

.≤

yin 1+

yin

–τ

�� Aiymσ n( )

+ fi σ n( )

, i 1 2 … m 1,–, , ,= =

Ai*yi

σ n( )

i 1=

m 1–

∑– fm σ n( )

, n 0 1 … N 1.–, , ,= =

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VABISHCHEVICH

In the subsequent manipulations, we use the equality

Take the scalar product of each equation in (9) with 2τ in Hi and add them. This yields

(10)

For the first terms on the right�hand side in (10) (i = 1, 2, …, m – 1), we have

We consider only the schemes with σ ≥ 1/2 and use the bounds

Substitution into (10) yields

(11)

Without loss of generality, we assume that 2τ ≤ T; therefore,

Taking this inequality into account, we obtain from (11) the following levelwise stability bound:

(12)

Bounds (8) and (12) ensure the stability of weighted scheme (6) for σ ≥ 1/2; in fact, they are differenceanalogs of bounds (4) and (5) for problem (1)–(3). Considering the corresponding problem for the error,we can verify in the standard way (see [4]) that the solution of operator�difference problem (6), (7) con�verges to the solution of differential�difference problem (1), (2) for σ ≥ 1/2 at a rate O((2σ – 1)τ + τ2).

2τyiσ n( ) τ yi

n 1+yi

n+( ) 2σ 1–( )τ2yi

n 1+yi

n–

τ��.+=

yiσ n( )

yin 1+

i

2

yin

i

2

2σ 1–( )τ2 yin 1+

yin

–τ

��i

2

i 1=

m 1–

∑+i 1=

m 1–

∑–i 1=

m 1–

∑ 2τ fi σ n( )

yiσ n( ),( )i 2τ fm

σ n( )ymσ n( ),( )i.+

i 1=

m 1–

∑=

2τ fi σ n( )

yiσ n( ),( )i τ fi

σ n( )yi

n 1+yi

n+,( )i 2σ 1–( )τ2

fi σ n( ) yi

n 1+yi

n–

τ��,⎝ ⎠

⎛ ⎞i.+=

2σ 1–( )τ2fi

σ n( ) yin 1+

yin

–τ

��,⎝ ⎠⎛ ⎞

i2σ 1–( )τ2 yi

n 1+yi

n–

τ��

i

22σ 1–( )

4��τ2

fi σ n( )

i

2

,+i 1=

m

∑≤

τ fi σ n( )

yin 1+

yin

+,( )iτ

2T�� yi

n 1+yi

n+ i

2 τT2

�� fi σ n( )

i

2

,+≤

yin 1+

yin

+ i

2

2 yin 1+

i

2

yin

i

2

+( ).≤

1 τT��–⎝ ⎠

⎛ ⎞ yin 1+

i

2

i 1=

m 1–

∑ 1 τT��+⎝ ⎠

⎛ ⎞ yin

i

2

i 1=

m 1–

∑≤

+ τT2

�� 1 2σ 1–2T

��τ+⎝ ⎠⎛ ⎞ fi

n 1/2+i

2

i 1=

m 1–

∑ 2τ fi σ n( )

m yiσ n( )

m.+

1 τT��+⎝ ⎠

⎛ ⎞ 1 τT��–⎝ ⎠

⎛ ⎞1– 4τ

T��⎝ ⎠

⎛ ⎞ .exp≤

yin 1+

i

2

i 1=

m 1–

∑4τT��⎝ ⎠

⎛ ⎞exp yin

i

2

i 1=

m 1–

∑≤

+ τTexp 2σ 1–T

��τ⎝ ⎠⎛ ⎞ fi

n 1/2+i

2

i 1=

m 1–

∑ 2τ fi σ n( )

m yiσ n( )

m.+



3. ADDITIVE SCHEMES

The computational implementation of scheme (6), (7) requires the following grid problem to be solved

on the next (n + 1)th time level for the given (i = 1, 2, …, m):

(13)

Substitute found from the first m – 1 equations of system (13) into the last equation to obtain

(14)

The other components of the approximate solution are found by solving grid problem (14) using explicitformulas from the first m – 1 equations of system (13).

Unconditionally stable operator–difference schemes (6) and (7) can be poorly suited for computa�tions. For solving problem (1), (2), it is often preferable to use additive (splitting) schemes, in which thetransition to the next time level involves solving simpler problems—inversion of the individual operators

(i = 1, 2, …, m – 1) rather than the inversion of their sum as in (13).

It is convenient to construct additive schemes interpreting system of equation (1) as a first order evo�lutionary equation for the vector u = {u1, u2, …, um}

(15)

in which f = { f1, f2, …, fm}. For the elements of the operator matrices D and A, we have the representations

(16)

(17)

In the direct sum of the spaces H = H1 ⊕ H2 ⊕ … ⊕ Hm, we define

Then, the operator A in H is skew�symmetric (A = –A*) and the operator D is self�adjoint and nonnega�tive definite (D = D* ≥ 0) in H.

Bound (5) for the first m – 1 components of the solution is associated with the seminorm induced by

the operator D ( = (Du, u)). Taking the scalar product of Eq. (15) with u in H, we obtain

χmn

yin 1+ στAiym

n 1++ χi

n, i 1 2 … m 1,–, , ,= =

Ai*yi

n 1+

i 1=

m 1–

∑– χmn

.=

yin 1+

στ Ai*Aiym

n 1+

i 1=

m 1–

∑ χmn

Ai*χi

n.

i 1=

m 1–

∑+=

Ai*Ai

Ddudt�� Au+ f t( ), 0 t T,≤<=

D Dij{ }, Dij

0, i j, i≠ 1 2 … m, j, , , 1 2 … m, , ,= =

E, i j, j 1 2 … m 1–, , ,= =

0, i m, j m,= =⎩⎪⎨⎪⎧

= =

A Aij{ }, Aij

0, i 1 2 … m 1– , j, , , 1 2 … m 1–, , ,= =

Aj, i m, j 1 2 … m 1–, , ,= =

Ai*, i– 1 2 … m 1, j–, , , m= =

0, i m, j m.= =⎩⎪⎪⎨⎪⎪⎧

= =

u v,( ) ui vi,( )i, u2

i 1=

m

∑ ui i2.

i 1=

m

∑= =

u D2

12��

d u D2

dt�� f u,( )D fm um,( )m.+=

2038


VABISHCHEVICH

Therefore,

(18)

where v = {v1, v2, …, vm – 1, 0}, and um satisfies bound (4).We will use vector notation (15)–(17) for system of equations (1). Taking into account (17), we can use

the additive representation

(19)

for the operator A, in which

The main property of the operators A(α) (α = 1, 2, …, m – 1) is due to their skew�symmetry. This enablesus to use various classes of unconditionally stable additive operator�difference schemes (see [2, 3]) toapproximately solve the Cauchy problem for Eqs. (15), (16), (19).

To derive bound (4) on the component um of the solution, property (3) is used. Under such conditions,the construction of additive schemes is possible for the individual terms A(α) (α = 1, 2, …, m – 1) in rep�resentation (19) if

(20)

Let us illustrate the capabilities of the standard componentwise splitting additive schemes. In this case,the computations are organized according to the scheme A(1) A(2) … A(m – 1). The simplestcompletely implicit componentwise splitting additive scheme is

(21)

here,

The investigation of stability of additive scheme (19)–(21) follows the same line as that used for obtainingbounds (8) and (12), but (20) is used instead of (3). For the completeness of the investigation, we mustsupplement our considerations with the investigation of convergence of such total approximation schemestaking into account the features of the problems under examination.

The computational implementation of these additive schemes is much simpler than the implementa�tion of schemes (6) and (7). To explain this point, consider scheme (21) for a single α = 1, 2, …, m – 1. Itis convenient to use the following right�hand sides of scheme (21):

In this case, we obtain from (21) the following formula for yn + α/(m – 1):

u D2

t( ) v D2

exp t θ–( ) f2

um θ( ) m2

+( )exp θ,d

0

t

∫+≤

A Aα( )

, Aα( )

i 1=

m 1–

∑ Aα( )( )*, α– 1 2 … m 1,–, , ,= = =

Aα( )

Aijα( ){ }, Aij

α( )

0, i 1 2 … m 1, j–, , , 1 2 … m 1–, , ,= =

Ai, i i, j m= =

Ai*, i– m, j i= =

0, i m, j m.= =⎩⎪⎪⎨⎪⎪⎧

= =

Ai*Ai δiE, δi 0, i>≤ 1 2 … m 1.–, , ,=

Dyn α/ m 1–( )+

yn α 1–( )/ m 1–( )+

–τ

�� Aα( )

yn α/ m 1–( )+

+ fαn 1+

, α 1 2 … m 1;–, , ,= =

fn 1+

fαn 1+

.

α 1=

m 1–

∑=

fαn 1+

fα n 1+( )i{ }, fα

n 1+( )i

0, i α, i≠ 1 2 … m 1–, , ,=

fα n 1+

, i α, i 1 2 … m 1,–, , ,= =⎩⎨⎧

α 1 2 … m 1.–, , ,= = =

yin α/ m 1–( )+

yin α 1–( )/ m 1–( )+

, i α, i≠ 1 2 … m 1,–, , ,= =

yin α/ m 1–( )+

yin α 1–( )/ m 1–( )+

–τ

�� Aiymn α/ m 1–( )+

+ fα n 1+( )i, i α, i 1 2 … m 1,–, , ,= = =



Therefore, we only need to calculate two components of yn + α/(m – 1) for every α = 1, 2, …, m – 1.

To find , we have the equation

After this problem has been solved, are found using explicit formulas.

4. MORE GENERAL PROBLEMS

Let us discuss the most interesting generalizations of system of equations (1) that have the most impor�tant features of problem (1), (2). We consider the case

(22)

Assume that the operators Bi (i = 1, 2, …, m – 1) are nonnegative in the respective spaces (Bi ≥ 0 in Hi fori = 1, 2, …, m – 1).

System of equations (22) can be written in vector form, which is similar to (15)–(17):

(23)

here,

(24)

The operator thus defined is nonnegative in H: B ≥ 0. As in the case of (15)–(17), we obtain bound (18)for Eqs. (16), (17), (23), (24).

The problem for the component um of the solution must be considered separately. Instead of the generalsystem (22), we consider the particular case in which

In this simple case, we obtain a bound of type (4) for um.The simple two�component splitting of the problem for Eq. (23) (the operators A and B) makes it pos�

sible to construct operator�difference schemes on the basis of the schemes for Eq. (15) considered above.If the completely implicit componentwise scheme is used, we have

(25)

The first step in scheme (25) requires the individual problems

for the components of the solution to be solved. The second step is implemented in the same way as inweighted scheme (6).

Ai*yi

n α/ m 1–( )+– fα

n 1+( )m.=

ymn α/ m 1–( )+

Ai*Aiym

n α/ m 1–( )+Ai

* fα n 1+( )i

fα n 1+( )m f α 1–( )

n 1+( )m–τ

��.+=

yin α/ m 1–( )+

dui

dt�� Aium Biui+ + fi, i 1 2 … m 1,–, , ,= =

Ai*ui

i 1=

m 1–

∑– fm, 0 t T.≤<=

Ddudt�� Au Bu+ + f t( ), 0 t T;≤<=

B Bij{ }, Bij

0, i j, i≠ 1 2 … m, j, , , 1 2 … m, , ,= =

Bi, i j, j 1 2 … m 1–, , ,= =

0, i m, j m.= =⎩⎪⎨⎪⎧

= =

Bi B, BAi* Ai

*B, i 1 2 … m 1.–, , ,= = =

Dyn 1/2+

yn

–τ

�� Byn 1/2+

+ 0,=

Dyn 1+

yn 1/2+

–τ

�� Ayn 1+

+ fn 1+

.=

yin 1/2+

yin

–τ

�� Biyyn 1/2+

+ 0, i 1 2 … m 1–, , ,= =

2040


VABISHCHEVICH

When the operator A is additively split (19), we can be guided by the similar splitting of the operator B:

(26)

In the simplest case, the individual summands of the additive representation can be associated with theindividual operators Bi (i = 1, 2, …, m – 1).

Under conditions (19), (26), the scheme of the componentwise splitting for Eq. (23) is

(27)

This additive scheme is standard for operator terms (26); for the operator terms (19), it was consideredabove.

5. PROBLEMS OF HYDRODYNAMICS

Let us apply the general consideration of additive schemes for the systems of equations under exami�nation for the case of problems of the dynamics of incompressible fluid. For simplicity, we consider onlytwo�dimensional model problems in the rectangle

and simple rectangular grids. This allows us to focus on the approximation with respect to time and notenlarge on the approximation with respect to space.

Consider the problem for the Navier–Stokes equation. In the domain Ω with solid boundary, we con�sider the motion of an incompressible fluid caused by distributed bulk forces. In the natural variables pres�sure, velocity, the equations of motion and continuity have the form

(28)

(29)

Here, u is the velocity, p is the pressure, Re is the Reynolds number, and � is the convective transport oper�ator, which we write in symmetric form (see [12])

Equations (28), (29) are supplemented with the condition ensuring the unambiguous value of the pressure

On the solid boundaries, the impermeability and no�slip conditions yield

(30)

The initial condition

(31)

is also specified.In some cases, one can use the linear approximation to describe nonstationary hydrodynamic pro�

cesses. In the Stoles approximation, the equation of motion is used in the form

(32)

B Bα( )

, Bα( )

i 1=

m 1–

∑ 0, α≥ 1 2 … m 1.–, , ,= =

Dyn α/ 2m 2–( )+

yn α 1–( )/ 2m 2–( )+

–τ

�� Bα( )

yn α/ 2m 2–( )+

+ 0, α 1 2 … m 1,–, , ,= =

Dyn α/ 2m 2–( )+

yn α 1–( )/ 2m 2–( )+

–τ

�� Aα( )

yn α/ 2m 2–( )+

+ fαn 1+

,=

α m m 1 … 2m 2.–, ,+,=

Ω x x x1 x2,( ) 0 xα lα α,< <, 1 2,= ={ }=

∂u∂t�� u( )u gradp 1

Re��divgradu–+ + f x t,( ),=

divu 0, x Ω, 0 t T.≤<∈=

� w( )u 12�� w grad⋅( )u div wv( )+[ ].=

p x t,( ) xd

Ω

∫ 0, 0 t T.≤<=

u x t,( ) 0, x ∂Ω, 0 t T.≤<∈=

u x 0,( ) v x( ), x Ω∈=

∂u∂t�� gradp 1

Re��divgradu–+ f x t,( )=



instead of (28). The further simplification can be achieved by neglecting the viscosity (perfect fluid). Inthis case, we have

(33)

Then, the boundary conditions on the solid boundary (30) are replaced with

(34)

where n is the outward normal to the domain boundary; these conditions represent the impermeabilitycondition.

For system of equations (29), (33), we use the coordinate notation. In the two�dimensional case, wehave

(35)

Differential problem (31), (34), (35) can be considered as a problem of type (1), (2).In the case under examination, we have m = 3 and u3 = p. Define the operators

On the set of functions ui (i = 1, 2) in �i = L2(Ω) (i = 1, 2) satisfying conditions (34) in the rectangle Ω,we have

Therefore, we can write the system of equations in operator form

(36)

The problem for the Stokes equation (30)–(32) is related to the system by an equation of type (22).Define

For the functions satisfying conditions (30) on the domain boundary, we have

in L2(Ω). Thus, we arrive at system of equations

(37)

which is more general than (36).

∂u∂t�� gradp+ f x t,( ).=

un 0, x ∂Ω, 0 t T,≤<∈=

∂u1

∂t�� ∂p

∂x1

��+ f1 x t,( ),=

∂u2

∂t�� ∂p

∂x2

��+ f2 x t,( ),=

∂u1

∂x1

��∂u2

∂x2

��+ 0, x Ω, 0 t T.≤<∈=

�iu3∂u3

∂xi

��, i 1 2.,= =

�i*ui

∂ui

∂xi

��, i– 1 2.,= =

dui

dt�� ium+ fi, i 1 2,,= =

�i*ui

i 1=

2

∑– 0, 0 t T.≤<=

�i �, �ui1

Re��div grad ui, i– 1 2.,= = =

� �* 0>=

dui

dt�� ium �ui+ + fi, i 1 2,,= =

�i*ui

i 1=

2

∑– 0, 0 t T,≤<=

2042


VABISHCHEVICH

For Navier–Stokes equations (28), the basic property of the convective transport operator is due to itsskew�symmetry; namely, due to the fact that the term with the convective transport does not contribute tothe kinetic energy. This property also holds for individual components:

in L2(Ω).We arrive at system of equations (37) in which

is a nonlinear operator.Let us write out the differential–difference analogs of systems (36) and (37) corresponding to the use

of simple approximations on a uniform rectangular grid in Ω. We focus on the consistency of the approx�imations of the gradient operator (its components) and the divergence operator (its summands). Forsimplicity, we consider only the Stokes equation; we do not enlarge on the features of the problem forsystem (29), (33), which are of technical nature, and do not discuss the solution of nonlinear problems,i.e., Eqs. (28), (29).

In the rectangle Ω, we use the uniform rectangular grid

On the set of grid functions that vanish on ∂ω ( = ω ∪ ∂ω, ω = ω1 × ω2), we define the Hilbert spaceH = L2(ω)) with the scalar product and the norm

For the gradient and divergence operators when the unified grid is used, simple approximations byfirst�order directed differences must be used (see [13, 14]). Define the difference operators

which are associated with the approximation of the pressure gradient. It can be immediately verified that

on the set of grid functions satisfying the boundary conditions (see (30))

For the viscous terms, we obtain

It is well known (see [4]) that

in L2(Ω). Thus, with the selected approximations, the basic properties of the operators of differentialproblem (37) are inherited, and we arrive at system of equations (22).

Two�component splitting (25) as applied to hydrodynamic problems for incompressible fluid is inter�preted as a possibility to obtain an acceptable problem for pressure at the next time level (see [14, 15]). Thecomputational implementation is based on the two�dimensional grid problem for the Laplace operatorsubject to the Neumann conditions—assumption (3) is not fulfilled in this case. When additive schemes

� w( ) � w( )* � w( )ui ui,( ) 0 i, 1 2,= =( )–=

� � u1 u2,( ) 1Re��div grad � u1 u2,( ) 0>+–= =

ωα xα xα iαhα iα, 0 1 … Nα Nαhα, , , , lα= = ={ },=

ωα xα xα iαhα iα, 1 2 … Nα 1– Nαhα, , , , lα= = ={ }, α 1 2.,= =

ω

y w,( ) y x( )w x( )h1h2, yx ω∈

∑ y y,( )1/2.≡ ≡

A1u3u3 x1 x2 t, ,( ) u3 x1 h1– x2 t, ,( )–

h1

��, A2u3u3 x1 x2 t, ,( ) u3 x1 x2 h2– t, ,( )–

h2

��, x ω,∈= =

A1*u1

u1 x1 h1+ x2 t, ,( ) u1 x1 x2 t, ,( )–h1

��– , A2*u2

u2 x1 x2 h2+ t, ,( ) u2 x1 x2 t, ,( )–h2

��, x– ω,∈= =

ui x( ) 0, x ∂ω, i∈ 1 2.,= =

Bi B, Bui1

Re��

ui x1 h1+ x2 t, ,( ) 2ui x1 x2 t, ,( )– ui x1 h1– x2 t, ,( )+

h12

��–= =

– 1Re��

ui x1 x2 h2+ t, ,( ) 2ui x1 x2 t, ,( )– ui x1 x2 h2– t, ,( )+

h22

��, i 1 2.,=

B B* 0>=



of type (27) are used, the calculation of pressure is based on the solution of similar one�dimensional prob�lems.

REFERENCES1. N. N. Yanenko, The Method of Fractional Steps (Springer, Berlin, 1971).2. G. I. Marchuk, Splitting Methods (Nauka, Moscow, 1989) [in Russian].3. A. A. Samarskii and P. N. Vabishchevich, Additive Finite Difference Schemes for Problems of Mathematical Physics

(Nauka, Moscow, 1999) [in Russian].4. A. A. Samarskii, Theory of Finite Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).5. F. L. Lisbona and P. N. Vabishchevich, “Operator�Splitting Schemes for Solving Elasticity Problems,” Comput.

Meth. Appl. Math. 1 (1), 188–198 (2001).6. A. A. Samarskii and P. N. Vabishchevich, “Problems in Compressible Fluid Dynamics with a Variable Viscosity,”

Zh. Vychisl. Mat. Mat. Fiz. 40, 1813–1822 (2000) [Comput. Math. Math. Phys. 40, 1741–1750 (2000)].7. P. N. Vabishchevich, “Additive (Splitting) Finite Difference Schemes for Sysytems of Partial Differential Equa�

tions,” Vychisl. Metody Programmirovanie 11 (1), 1–6 (2010).8. A. A. Dorodnicyn, “A Method for Solving Three�Dimensional Problems of an Incompressible Viscous Flow

Past Bodies,” in Modern Problems of Computational Aerohydrodynamics (CRC Press, Moscow, 1992), pp. 93–102.

9. A. A. Dorodnitsyn and N. A. Meller, “Certain Approaches to the Solution of Steady�State Navier–StokesEquations,” Zh. Vychisl. Mat. Mat. Fiz. 8, 393–402 (1968).

10. A. A. Samarskii and A. V. Gulin, Stability of Finite Difference Schemes (URSS, Moscow, 2009) [in Russian].11. A. A. Samarskii, P. P. Matus, and P. N. Vabishchevich, Difference Schemes with Operator Factors (Kluwer, Dor�

drecht, 2002).12. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Convection–Diffusion Problems (URSS, Mos�

cow, 2009) [in Russian].13. O. A. Ladyzhenskaya, Mathematical Methods of the Dynamics of Viscous Incompressible Fluid (Nauka, Moscow,

1970) [in Russian].14. A. G. Churbanov, A. N. Pavlov, and P. N. Vabishchevich, “Operator�Splitting Methods for the Incompressible

Navier–Stokes Equations on Non�Staggered Grids. Part 1: First�Order Schemes,” Int. J. Numer. Meth. Fluids21, 617–640 (1995).

15. J. L. Guermond, P. Minev, and J. Shen, “An Overview of Projection Methods for Incompressible Flows,” Com�put. Meth. Appl. Mech. Eng. 195, 6011–6045 (2006).

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Additive schemes for certain operator-differential equations