Weierstrafl>=Institut
f?r Angewandte Analysis und Stochastik
im Forschungsverbund Berlin e.V.
Preprint ISSN 0946 >= 8633

The convergence and stability of splitting ønite
diöerence schemes for nonlinear evolutionary type
equations

Mindaugas Radziunas1, Feliksas Ivanauskas 2

submitted: 18th December 1998

1 Weierstrafl Institut

f?r Angewandte Analysis
und Stochastik,

Mohrenstrasse 39

10117 Berlin

Germany

E-Mail: radziunas@wias-berlin.de

2 Faculty of Mathematics at

Vilnius University,
Naugarduko 24,
2600 Vilnius
Lithuania
E-Mail: feliksas.ivanauskas@maf.vu.lt

Preprint No. 464
Berlin 1998

WIAS

1991 Mathematics Subject Classiøcation. 65N06, 65N12.
Key words and phrases. evolutionary equations, ønite diöerence scheme, splitting scheme.

Edited by
Weierstrafl>=Institut f?r Angewandte Analysis und Stochastik (WIAS) Mohrenstrafle 39
D µ 10117 Berlin
Germany

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Abstract

A splitting ønite diöerence scheme for an initial-boundary value problem for a two-dimensional nonlinear evolutionary type equation is considered. The problem is split into nonlinear and linear parts. The linear part is also split into locally one-dimensional equations. The convergence and stability of the scheme in L2 and C norms are proved.

1 Introduction.

We consider the initial-boundary value problem for multidimensional nonlinear evolutionary equation of the type

@u
@t = a?u + f(u):

Here a = a1+ia2 is a complex valued constant and ? is the d-dimensional Laplacian. We consider the following cases:
1. If a1 = 0 and a2 6= 0 we have the Schrfidinger equation.
2. If a1 > 0 and a2 6= 0 we have the Kuramoto>=Tsuzuki equation.
3. If a1 > 0 and a2 = 0 we have a heat equation.

Such linear and nonlinear equations appear in many models of nonlinear optics, quantum mechanics, seismology, plasma physics, in the theory of turbulent Äows and many other øelds of science.

Although evolutionary problems with only one spatial dimension have been investigated during a long period, even recently there appear a lot of papers devoted to the numerical solution of these problems [1] >= [4]. As a rule, the diÖculties in solving such the problems arise due to the diöerent kinds of nonlinearities.

Multi-dimensional nonlinear and linear evolutionary problems are even more complicated. Frequently, solving such problems during small time step, one splits the problem into nonlinear and linear parts [5], [6] or considers multidimensional linear part as locally one-dimensional problems [5], [8], [9].

An object of this paper is the ønite diöerence scheme that approximates the evolutionary equation. There are many papers on ønite diöerence schemes for initial>= boundary linear and nonlinear evolutionary problems. There are two>=layered schemes with weights [2], [5], [7], various three>=layered schemes [2] >= [4] and also splitting schemes [5], [6], [8].

In the paper [6] the splitting of the equation into nonlinear and linear parts

@u(nonl)

@t = f(u(nonl)); u(nonl)

initial = uinitial;

@u(lin)

@t = a?u(lin); u(lin)

initial = u(nonl)
final ; ufinal = u(lin)

final

for a short evolution time ø was used and a proof of the existence and stability of the solution was given.

In a series of papers, for example in [5], locally one-dimensional ønite diöerence schemes for the solution of multidimensional linear evolutionary problems have been constructed. These schemes have allowed to compute a value of the unknown solution on the next time level using diöerential operators in diöerent directions step by step, i.e., splitting the d-dimensional linear part of the equation into the onedimensional problems:

@u(k)

@t = a@2u(k)
@x2k

; k = 1; : : : ; d;

u(1)
initial = uinitial; u(k)
initial = u(k?1)
final ; u(d)
final = ufinal:

The other method for splitting the linear d-dimensional heat equation into local one-dimensional problems was investigated in [8]. When looking for the unknown solution on the next time level, instead of solving step by step the one-dimensional problems, one can solve all these problems simultaneously and get the solution on the new time level from the obtained data afterwards. Such splitting can be described as follows:

ffk

@u(k)

@t = a@2u(k)
@x2k

; k = 1; : : : ; d;

u(k)

initial = uinitial;

k=1

ffk = 1; ufinal =

d
X

k=1

ffku(k)
final:

It seems that such a splitting of the multidimensional problem can be very useful in parallel computations.

A purpose of the present paper is to prove the convergence and stability of the locally one-dimensional diöerence scheme for a broad class of two-dimensional nonlinear evolutionary type equations. The evolutionary problem is split into nonlinear and linear parts, as in [6]. A linear part of this problem is also split into locally onedimensional problems as it had been done in [8].

The nonlinear evolutionary equations which are considered in this paper represent a much broader class of equations than only linear heat equations that were considered in [8]. The diöerence schemes require a more precise investigation due to nonlinearity as well as to some speciøc properties of the Schrfidinger and the Kuramoto-Tsuzuki equations.

Therefore, our paper presents new results proving convergence and stability of the splitting scheme for nonlinear evolutionary type equations. Doing so, we also show convergence of another splitting scheme, where the fully explicit diöerence scheme for the two-dimensional linear part is used.

It should be mentioned that to prove the convergence and stability of diöerence schemes we use a new type of a priori estimates that were introduced and developed

in [6], [7]. This approach allows us to avoid any restrictions on the time and space grid steps.

In Section 2 the diöerential problem and the corresponding ønite diöerence schemes are formulated. In Section 3 some grid embedding and multiplicative inequalities are proved, and some formulae for the ønite diöerence diöerentiation are derived. In Section 4 some estimates for the nonlinear part of the equation are obtained. Section 5 contains various properties of the diöerence schemes. Sections 6 and 7 are devoted to a proof of the convergence of the two ønite diöerence schemes. Section 8 contains an investigation of the stability for both schemes.

2 Formulation of the problem.

Let us consider an initial-boundary value problem with Dirichlet boundary conditions for the nonlinear Schrfidinger, Kuramoto>=Tsuzuki or heat equation:

@t = a ?u + f(u; u?); (x; t) 2 Q;

u(x; 0) = u0(x); x 2 µ?; u(x; t) = 0; (x; t) 2 @? ? [0; T ]: (1)

Here x = (x1; x2); u(x; t) is a complex-valued function; u? is the complex conjugate function; i =

?1; ? = @
x21 + @
x22 is the two-dimensional Laplacian; a = a1 + ia2, a1 >= 0, jaj > 0; µ? = [0; 1] ? [0; 1], µ
Q = µ? ? [0; T ].

Assume that the nonlinear function f(u; u?) satisøes the following conditions:

The partial derivatives of the function f(u; u?) with respect to u and u? are continuous up to the second order and

fififi @jkjf (u; v?)

@uk1@v?k2

fififi <= '(maxfjuj; jvjg); k1 + k2 = jkj 2 f0; 1; 2g; (2)

where ' is a continuous nondecreasing function;

Re(f(u; u?)u?) <= 0: (3)

Condition (2) is necessary for the evaluation of the diöerentiated nonlinear function. Estimate (3) allows to obtain the integral dissipation property ku(t)kL2 <= ku(0)kL2, t 2 [0; T ]. This inequality can be proved multiplying both sides of (1) by 2u?, integrating the obtained equation over ?, taking a real part and using the condition a1 = Re a >= 0.

Note that in the case of the Schrfidinger equation a = ia2 and condition (3) reads as Re(f(u; u?)u?) = 0, the integral conservation law holds: ku(t)kL2 = ku(0)kL2, t 2 [0; T ].

It should be also mentioned that more general a priori estimates hold.

Remark 1 Suppose (2) and (3) are satisøed, u0 2 W 22 (?) and there exists a solution u 2 C( µ
Q) of problem (1). Then the following estimate holds:

ku(t)kW 22 <= c1ku0kW 22 ; t 2 [0; T ]; c1 = c1('(kukC( µQ)); ku0kW 12 ; T ):

Proof. The proof of this remark can be found in [7].

We assume that there exists a solution of (1) which satisøes the following condition:

max

t2[0;T ]

æflflfl @2u(t)

@t2

flflflC ;
flflfl @4u(t)

@x41

flflflC ;
flflfl @4u(t)

@x42

flflflC
ö
<= C < 1: (4)

This smoothness is required to get a good approximation of the diöerence schemes. Note that the estimate maxt2[0;T ] ku(t)kC <= C also follows from (4).

Let us introduce a uniform grid in the domain µ
Q:

!ø = ftj = øj; j = 0; : : : ; M ? 1; Mø = T g; µ!ø = ftj; j = 0; : : : ; Mg;

!x = fx = (x1l1 ; x2l2 ); xjlj = ljhj; lj = 1; : : : ; Nj ? 1; j = 1; 2; Njhj = 1g;

µ!x = fx = (x1l1 ; x2l2 ); lj = 0; : : : ; Nj; j = 1; 2g:

Denote Qhø = !x?!ø , µ
Qhø = µ!x? µ!ø . Let @!x = µ!xn!x be the set of grid boundary points on @?. We require 0 < 1=d <= h1=h2 <= d < 1 for h1; h2 ! 0. For simplicity we assume h = h1 <= h2 = dh.
In the sequel we use the notation

u = u(x; tj) = u(x1; x2; tj); û = u(x; tj+1); _u = (û + u)=2; ut = (û ? u)=ø:

I is the identity operator, T +k and T ?k are shift operators, that is Iu = u and T?k u = u(x ? hkek; tj), where e1 = (1; 0) and e2 = (0; 1) are unit vectors. Let Dhk be a ørst order grid diöerentiation operator and

uµxk = (I ? T ?k )u=hk = Dhku; uxk = (T +k ? I)u=hk = DhkT +k u;
uµxkxk = Dhkuxk = D2hkT +k u; ?hu = uµx1x1 + uµx2x2 :

We associate problem (1) with two diöerent splitting ønite diöerence schemes.

The ønite diöerence equation for the nonlinear part and the initial conditions are the same for both of these schemes:

zt = f( _z; _z?); (x; t) 2 µ!x ? !ø ; z = p; p(x; 0) = u0(x); x 2 µ!x: (5)

In general, the equation is nonlinear and we can solve it using iterations:

z(n+1) ? z

ø = f(z(n) + z
2

?
; z(0) = z; n ! 1:

For the linear part of (1) we have two diöerent ønite diöerence equations.

First, we approximate the two-dimensional linear problem:

gt = a ?h^g; (x; t) 2 Qhø ; g = ^z; ^p = ^g; ^g = 0; (x; t) 2 @!x ? !ø : (6)

Second, we introduce the locally one-dimensional equations:

g(k) ? g

2ø = a

?g(k) + g

?
µxkxk ; (x; t) 2 Qhø ;
g(k) = 0; (x; t) 2 @!x ? !ø ; g = ^z; ^p = ^g = (g(1) + g(2))=2: (7) It appears that we can exclude the functions g(1) and g(2) in (7) and obtain the following equivalent equation:
gt =a ?h^g ? 2a2ø _gµx1x1µx2x2; (x; t) 2 Qhø ; g = ^z; ^p = ^g; ^g = 0; (x; t) 2 @!x ? !ø : (8)

Lemma 1 The ønite diöerence equations (7) and (8) are equivalent.

Proof. Adding the equations (7) for k = 1; 2 and using the expression for ^g via g(k) we obtain

^g ? g

ø = a

?hg + g(1)

µx1x1 + g(2)

µx2x2

= a

?hg + 2?h^g ? g(2)
µx1x1 ? g(1)
µx2x2
?
:

Applying the ønite diöerence operators (?)µx2x2 and (?)µx1x1 respectively to the equations in (7) where k = 1 and k = 2 one can ønd that

gt = a

?hg + 2?h^g) ? a

?
?hg + aø(g(1) + g(2) + 2g)µx1x1µx2x2
?
:

It is exactly the same equation as (8). The lemma is proved.

Due to the equivalence of the schemes, all the results obtained for (8) are also valid for (7).

3 Some properties of the grid functions.

Let us introduce grid analogues of some functional spaces. We shall say that the grid function v belongs to some functional space if there exists h0 such that for all positive h1; h2 <= h0 the corresponding norm of the function v is bounded by a constant which does not depend on grid steps.

Let l be any operator acting on some grid function. Denote by l(µ!x) the subset of the grid µ!x where the operator l is deøned.

We shall deøne the Lp = Lp(!x) norm of the grid functions u or lu, and an inner product in L2 as follows:

kukp =

h1h2

x2µ!x

jujp?1=p
; klukp =

h1h2

x2l(µ!x)

jlujp?1=p

; (u; v) = h1h2

x2!x

uv?:

For simplicity, we deøne k ? k2 = k ? k.

Let us introduce the grid analogues of the Sobolev spaces W lp = W lp(!x) with norms

kukl;p =

? X

0<=jkj<=l

kDkhukpp
?1=p
; 1 <= p < 1; l = 1; 2; : : :

Here Dkh = D(k1;k2)
h = Dk1h1Dk2h2 are ønite diöerence operators.

In the sequel we also use the shift operator T k = T (k1;k2) = (T ?1 )k1(T ?2 )k2. It is easy to check that T l(vw) = T lvT lw, T l(v + w) = T lv + T lw. The operators T l and Dkh are commutative. Note that, due to a deønition of the Lp(!x) norm for the grid function u, the estimate kT kDlhukp <= kDlhukp holds.

Denote by

ffi

W 12 a subset of functions of the space W 12 which have zeroes on the

boundary @!x, W =

ffi

W12 \ W 22 .

ffi

C=
ffi
C (µ!x) is the space of the grid functions having zeroes on the boundary and with norm kukC = maxx2µ!xfjujg.
We can prove some relations concerning diöerent norms of the grid functions.

Lemma 2 Suppose that u is the grid function deøned on µ!x with zero boundary conditions and that kDkhuk <= C < 1 for all positive h1; h2 <= h0. Then the following estimates hold:

kDk?ej
h uk <= c2kDkhuk; kj >= 1; 0 <= k1; k2 <= 2: (9)

Proof. Let us denote by wlj(xj) the functions

p
2 sin(ljßxj), j = 1; 2. Here x = (x1; x2) is some point of the grid µ!x.

Since zero boundary conditions hold for the grid function u we can deøne the function u on the grid µ!x as follows:

u(x1; x2) =

N1?1
X

l1=1

N2?1
X

l2=1
al1;l2wl1(x1)wl2(x2):

Due to completeness and orthogonality of the functions wl1(x1)wl2(x2) in the space ffi
W12, the Fourier coeÖcients al1;l2 can be found as

al1;l2 = h1h2

1
X

x1=h1

1
X

x2=h2

u(x1; x2)wl1(x1)wl2(x2):

Due to Parsevall's equality, the L2h norm of the grid function u can be expressed as follows:

kuk2 = h1h2

x2µ!x

ju(x)j2 = h1h2

1
X

x1=h1

x2=h2

ju(x)j2 =

N1?1
X

l1=1

N2?1
X

l2=1

jal1;l2j2:

Let us introduce the eigenvalues of the operator ?D2hjT +j corresponding to the eigenfunctions wl1(x1)wl2(x2):

?(wl1(x1)wl2(x2))µxjxj=>=(j)
lj (wl1(x1)wl2(x2)); >=(j)
lj = 4
h2j
sin2(ljßhj=2); j = 1; 2:

Note that for hj small enough, say, hj <= 2=ß, we have

min

lj f>=(j)
lj g = 4
h2j
sin2(ßhj=2) >= ß2
2 :

Using the formula of summation by parts for the grid functions and the zero boundary conditions, we have

kDh1uk2 = h1h2

1
X

x1=h1

x2=0

jDh1u(x)j2 = ?h1h2

1?h1
X

x1=h1

1
X

x2=0

uµxx(x)u?(x)

= h1h2

1
X

x1=h1

1
X

x2=h2

?N1?1
X

l1=1

N2?1

l2=1

>=(1)

l1 al1;l2wl1(x1)wl2(x2)
?

u?(x) =

N1?1
X

l1=1

N2?1
X

l2=1
>=(1)
l1 jal1;l2j2:

Similarly, we can ønd the expressions for some other diöerentiated function norms via the Fourier coeÖcients and eigenvalues:

kDkhuk2 =

N1?1
X

l1=1

N2?1

l2=1

(>=(1)
l1 )k1(>=(2)
l2 )k2 jal1;l2 j2; 0 <= k1; k2 <= 2:

Note that for other values of k this equality can be not satisøed, since Dkh(µ!x) is just a subset of the grid !x.

Finally, the estimates from below for the eigenvalues >=(j)
lj imply (9) where c2 =

p
2=ß. The lemma is proved.

Lemma 3 Let v1; v2 be the grid functions deøned on µ!x. Suppose that l1; l2 are two operators, ~!x = l1(!x) \ l2(!x) = [l01h1; l001h1] ? [l02h2; l002h2] æ µ!x, (l00j ? l0j + 1)hj >= 0:5, j = 1; 2 and kDh1l1v1k; kDh2l2v2k <= C < 1 for all positive h1; h2 <= h0. Then the following inequality holds:

kl1v1 l2v2k2 <= 4kl1v1kkl2v2k(kl1v1k + kDh1l1v1k)(kl2v2k + kDh2l2v2k) (10)

Proof. For any øxed grid point coordinate x2 we denote by ~x1(x2) the value of x1 such that jl1v1(~x1(x2); x2)j = minx12[l01h1;l001 h1]fjl1v1(x1; x2)jg. Therefore,

jl1v1(~x1(x2); x2)j2 <= 2h1
l001 h1

x1=l01h1

jl1v1(x1; x2)j2; h2
l002 h2
X

x2=l02h2

jl1v1(~x1(x2); x2)j2 <= 2kl1v1k2:

Similarly, we deøne ~x2(x1), jl2v2(x1; ~x2(x1))j = minx22[l02h2;l002h2]fjl2v2(x1; x2)jg, and

have h1

Pl001 h1

x1=l01h1 jl2v2(x1; ~x2(x1))j2 <= 2kl2v2k2.

Now we can have the following estimate:

kl1v1 l2v2k2 <= h2
l002 h2

x2=l02h2

jl1v1(~x1(x2); x2)j2 + h1
l001 h1
X

?=(l01+1)h1

jDh1jl1v1(?; x2)j2j

? h1
l001 h1

x1=l01h1

jl2v2(x1; ~x2(x1))j2 + h2
l002 h2

?=(l02+1)h2

jDh2jl2v2(x1; ?)j2j
?
:

Estimate (10) follows from here. The lemma is proved.

A direct consequence of the two previous lemmas is the following corollary:

Corollary 1 Suppose that v1; v2 are the grid functions deøned on µ!x with zero boundary conditions and that kDk0
h v1k; kDk00
h v2k <= C < 1 for all positive h1; h2 <= h0.

Then the following inequalities hold:

kT r0Dk0?e1

h v1 T r00Dk00?e2

h v2k <= c3
?

kDk0?e1

h v1kkDk00?e2
h v2kkDk0
h v1kkDk00
h v2k
?1=2

<= c2c3kDk0
h v1kkDk00
h v2k; k01; k002 2 f1; 2g; k02; k001 2 f0; 1; 2g: (11)

Proof. Let us denote l1 = T r0Dk0?e1
h , l2 = T r00Dk00?e2
h . The ørst part of (11) follows from (10), kT rvk <= kvk and (9). Here c3 = 2(1 + c2). The second part of this estimate immediately follows from (9). The corollary is proved.

In the two-dimensional case the following multiplicative inequality and grid embedding theorem holds:

kukC <= c4kuk(1=2?fl)kuk(1=2+fl)
2;2 ; 0 < fl < 0:5;
kukp <= c5kuk1;2 p 2 [2; 1); c5 = c5(p): (12)

Both estimates can be found in [7].

In the sequel we also use the results of the following lemma:

Lemma 4 In all points of the grid Dkh(µ!x) the following formula for the function

Dkh

?Qsj=1 vj
?

is valid:

Dkh

? sY

j=1

l(1)+:::+l(s)=k

k1!k2!

Qsj=1 l(j)

1 ! l(j)
2 !

j=1

T l(1)+:::+l(j?1)Dl(j)vj: (13)

Here l(j) = (l(j)
1 ; l(j)
2 ), l(j)
1 2 f0; 1; : : : ; k1g, l(j)
2 2 f0; 1; : : : ; k2g, j = 1; : : : ; s.

Proof. Let us prove (13) for s = 2. We use the method of mathematical induction with respect to k = (k1; k2). We must prove the formula

Dkh(v1v2) =

l(1)+l(2)=k

k1!k2!

l(1)

1 ! l(2)

1 ! l(1)

2 ! l(2)

2 !

Dl(1)v1 T l(1)Dl(2)v2:

Note that here we have l(2)

j = kj ? l(1)

j and, therefore, kj !
l(1)
j ! l(2)
j ! = C l(1)
jkj , j = 1; 2.

This formula is obvious in the case when k = 0 = (0; 0), k = e1 = (1; 0) and k = e2 = (0; 1). Suppose that it is also valid for all r, 0 <= r1 <= k1, 0 <= r2 <= k2. Let us consider this formula for Dk+e1
h = Dk0
h :

Dk+e1

h (v1v2) = De1h (Dkh(v1v2)) =

k1
X

l1=0

l2=0

C l1k1 C l2k2 De1h

Dlv1T lDk?lv2

k1
X

l1=0

k2
X

l2=0

C l1k1 C l2k2 (Dlv1T lDk?l+e1v2 + Dl+e1v1T l+e1Dk?lv2)

l2=0

C l2k2
<= k1

l1=1

(C l1k1 + C l1?1
k1 )Dlv1T lDk0?lv2 + v1Dk0v2 + Dk0v1T k0v2

>=
:

Here the last expression is equivalent to the formula for Dkh(v1v2) with k0 instead of k. We can complete the proof of the formula for Dk+e2
h in the same way and, therefore, complete the proof of the induction step.

Now we can prove a general form of (13) using the mathematical induction with respect to s.

Formula (13) is obvious for s = 1. We have also proved it above for s = 2. Suppose that it is also valid for all r, 2 <= r <= s. Let us introduce w = Qsj=1 vj and consider

this formula for Dkh(Qs+1
j=1 vj) = Dkh(wvs+1):

Dkh

?s+1
Y

j=1

=Dkh(wvs+1) =

r+l(s+1)=k

k1!k2!

r1!l(s+1)
1 !r2!l(s+1)
2 !

DrwT rDl(s+1)vs+1

r+l(s+1)=k

l(1)+:::+l(s)=r

k1!k2!

Qs+1

j=1 l(j)

1 !l(j)

2 !

s+1
Y

j=1
T l(1)+:::+l(j?1)Dl(j)vj:

According to this, (13) is valid for Dkh(Qs+1
j=1 vj). It proves the induction step and, therefore, (13) is valid. The lemma is proved.

4 Estimates of the nonlinear function.

We can also derive some estimates for the nonlinear part of the diöerence schemes. First, we show some properties which can be obtained for the nonlinear functions f(u; u?) due to requirement (2).

Lemma 5 Let (2) be satisøed. Then the following estimate holds:

jf(v; v?) ? f(w; w?)j <= 2'(maxfjvj; jwj)g)jv ? wj:

Proof. We have the following equalities:

f(v; v?)?f(w; w?) = f(v; v?)?f(w; v?)

v ? w (v?w)+ f(w; v?)?f(w; w?)
v? ? w? (v?w)?

= @
@uf(~v1; v?)(v ? w) + @
@u? f(w; ~v?2)(v ? w)?:
Here ~vj = ?jv +(1??j)w, ?j 2 [0; 1], j = 1; 2. The supposition of this lemma follows from (2).

Lemma 6 Assume that v 2 W and (2) is satisøed. Then the following estimates are valid:

kDkhf(v; v?)k <= dkkvkjkj;2; jkj = 1; 2; dk = dk('(kvkC); kvkjkj?1;2): (14)

If, additionally, condition (2) for the function f(u; u?) is valid up to the fourth order derivatives, the following estimates are valid:

kDkhf(v; v?)k <= dk

kD(1;2)
h vk + kD(2;1)
h vk
?

; k 2 f(1; 2); (2; 1)g;

kD(2;2)
h f(v; v?)k <= d(2;2)kD(2;2)
h vk: (15)

Here we suppose that the norms in the right-hand side of the inequalities are bounded kDkhvk <= C < 1 for all positive h1; h2 <= h0 and

dk =

(
dk('(kvkC); kvk2;2) if k 2 f(1; 2); (2; 1)g;

dk('(kvkC); kD(1;2)
h vk; kD(2;1)
h vk) if k = (2; 2).

Proof. If jkj = 1 the statement of this lemma follows from Lemma 5.

Assume jkj >= 2. The ønite diöerence diöerentiation Dkhv can be written as

Dkhv

fififix=µx = 1

hk11 hk22

y2Ak(µx)

e(y)v(y) = 1

hjkjdk2

y2Ak(µx)

e(y)v(y):

Here Ak(µx) = ([µx1 ? k1h1; µx1] ? [µx2 ? k2h2; µx2]) \ µ!x is the set of neighbouring points of µx on which the diöerentiation Dkh is deøned. For y 2 A(µx) we have e(y) = e(µx1 ? l1h1; µx2 ? l2h2) = (?1)jljC l1k1C l2k2 and P
y2A(µx) je(y)j = 2jkj.

Using these notation and an expansion of the function f(v; v?) into the Taylor series in the neighbourhood of µv = v(µx) we get the following formula:

Dkhf(v; v?)

fififix=µx = 1

hjkjdk2

y2Ak(µx)

e(y)f(v(y); v?(y))

= 1

hjkjdk2

y2Ak(µx)

e(y)

jsj<jkj

@sf(µv; µv?)

@s1v @s2v?

(v(y) ? µv)s1(v(y) ? µv)?s2

s1! s2!

+ 1

hjkjdk2

y2Ak(µx)

e(y)

jsj=jkj

@sf(~v1y; ~v?2y)

@s1v @s2v?

(v(y) ? µv)s1(v(y) ? µv)?s2

s1! s2!

1<=jsj<jkj

s1! s2!

@sf(µv; µv?)

@s1v @s2v? Dkh

(v ? µv)s1(v ? µv)?s2?fififix=µx

y2Ak(µx)

e(y)

jsj=jkj

dk2s1! s2!

@sf(~v1y; ~v?2y)

@s1v @s2v?

?v(y) ? µv

?s1?v(y) ? µv

??s2:

Here we have set ~vjy = ?jyv(µx) + (1 ? ?jy)v(y), ?jy 2 [0; 1], j = 1; 2.

Using (2), Newton's binom and the expression of e(y) we have

fififiDkhf (v; v?)
fififi <=

1<=jsj<jkj

'(kvkC)

s1! s2!

fifififiDkh

? s1
X

r1=0

s2
X

r2=0

Cr1s1 Cr2s2 vr1 v?r2 µvs1?r1 µv?s2?r2?fifififi

+'(kvkC)2jkj

dk2jkj!

k1
X

l1=0

l2=0

C l1k1 C l2k2
fifififi

l1?1

r1=0

Dh1T (r1;l2)v + d

l2?1

r2=0

Dh2T (0;r2)v
fifififi
jkj
:

Taking L2(!x) norms, applying Hfilder's inequality and taking into account that kT kDlhukp <= kDlhukp, we obtain the following estimates:

kDkhf(v; v?)k <= (4jkjd)jkj

jkj jkj! '(kvkC)

k1kDh1vkjkj
2jkj + k2kDh2vkjkj
2jkj
?

1<=jsj<jkj

s1
X

r1=0

s2
X

r2=0

'(kvkC)kvkjs?rj
C
r1!r2!(s1 ? r1)!(s2 ? r2)!kDkhvr1v?r2k = µ1 + µ2:

Assume that jkj = 2. Using (11) we ønd that µ1 <= 32d2'(kvkC)c23kvk1;2kvk2;2 and µ2 = 2'(kvkC)kDkhvk. This completes the proof of (14).

For k 2 f(1; 2); (2; 1); (2; 2)g we estimate µ1 using the second inequality from (12) and estimate (9):

µ1 <= (4jkjd)jkj

jkj jkj! '(kvkC)cjkj

5 kvkjkj?1
2;2
?

k1kDh1vk1;2 + k2kDh2vk1;2

<= (4jkjdc5)jkj

jkj jkj! kvkjkj?1

2;2 '(kvkC)c2

2 + c22

k1kD(2;1)
h vk + k2kD(1;2)
h vk
?

<= (4jkjdc5)jkj

jkj! kvkjkj?1

2;2 '(kvkC)c22

q
2 + c22 kD(2;2)
h vk:

We write v1; v2; v3 instead of T rv or T rv?. We do not specify a value of the vector r since kDlhvjk <= kDlhvk for any l and r. Let l0 + l00 = k and jl0j; jl00j >= 1. For the given k we can suppose that l01 < k1 and l002 < k2. Let us also set kj = maxfk1; k2g = 2. Then (11) and (9) lead to the following estimate:

kDl0

hv1Dl00
h v2k <= c2c3kDl0+e1
h vkkDl00+e2
h vk <= cjkj?2
2 c3kDk?ej
h vkkDkhvk:

For k = (2; 2) we put l + ej + el = k, µej = (1; 1) ? ej. Using (10) we can derive the estimate

kDlhv1Dejh v2Delhv3k<=

kDejh v2Delhv3k+kDejh v2Del+µej
h v3k+kD(1;1)
h v2Delh T µejv3k
i1=2

?kDejh v2Delh v3k1=2p
2c2c3kDl+ej

h vk <=

p
2c42c23kD(1;2)
h vkkD(2;1)
h vkkDkhvk:

Therefore, for k 2 f(1; 2); (2; 1); (2; 2)g we can use (13) and obtain

kDkhv1v2k <= (2kvkC + (2jkj? 2)cjkj?2

2 c3kDk?ej
h vk)kDkhvk; here kj = 2;

kD(2;2)

h v1v2v3k <=
?

3kvk2C +6

7kvkC+6

2c22c3kD(2;1)

h vk
i

c22c3kD(1;2)
h vk
?
kD(2;2)
h vk:

Let us suppose that k 2 f(1; 2); (2; 1)g. Then we get the following estimate:

µ2 = 2'(kvkC)kDkhvk +

jsj=2

s1
X

r1=0

s2
X

r2=0

'(kvkC)kvkjs?rj
C

r1!r2!(s1?r1)!(s2?r2)!kDkhvr1v?r2k

<= 2'(kvkC)

1 + 4kvkC + 6c2c3kD(1;1)
h vk
?
kDkhvk:

Due to the ørst inequality of (12) we obtain the ørst estimate of (15) with a corresponding coeÖcient dk for these values of k.

Finally, let us consider k = (2; 2). Similarly as above we ønd that

µ2 <= '(kvkC)

2+4kvkC+4kvk2C

kD(2;2)
h vk+'(kvkC)(2+4kvkC)kD(2;2)
h v1v2k

+ 4

3'(kvkC)kD(2;2)

h v1v2v3k <= 2'(kvkC)

?
1 + 4kvkC + 8kvk2C

+c22c3

14 + 56kvkC + 24

2c22c3kD(2;1)

h vk
i

kD(1;2)
h vk

?
kD(2;2)
h vk:
Due to the ørst inequality of (12) and (9) this leads to (15). The lemma is proved.

5 General properties of the diöerence schemes.

In order to prove convergence and stability of diöerence schemes (5), (6) and (5), (8) we use some auxiliary statements.

Lemma 7 Let g 2 W . Then there exists a unique solution ^g 2 W of (6) (or (8)) and the following estimates are valid:

kDkh^gk <= kDkhgk; 0 <= jkj = k1 + k2 <= 2

If, additionally, kDkhgk <= C < 1 for all positive h1; h2 <= h0 then these estimates are also valid for k 2 f(1; 2); (2; 1); (2; 2)g.

Proof. Analogously as in Lemma 2, for both cases of equations (6) and (7) we deøne the following functions on the grid µ!x:

g(x1; x2) =

N1?1
X

l1=1

N2?1
X

l2=1

al1;l2wl1(x1)wl2(x2);

g(j)(x1; x2) =

N1?1
X

l1=1

N2?1
X

l2=1

q(j)
lj al1;l2 wl1(x1)wl2(x2);

^g(x1; x2) =

N1?1
X

l1=1

N2?1
X

l2=1

ql1;l2al1;l2wl1(x1)wl2(x2):

For diöerence scheme (6) the constants ql1;l2 , lj = 1; : : : ; Nj ? 1, j = 1; 2 can be found from the equations

ql1;l2 ? 1

2ø = ?a(>=(1)
l1 + >=(2)
l2 )ql1;l2 :

It follows from here and from the condition Re a = a1 >= 0 that

jql1;l2 j2 =
fifififi

1+ aø(>=(1)

l1 + >=(2)

l2 )

fifififi
2

= 1

(1+ a1ø(>=(1)
l1 +>=(2)
l2 ))2+ (a2ø(>=(1)
l1 +>=(2)
l2 ))2 <= 1:

Similarly, in the case of diöerence scheme (7) we can ønd the constants

ql1;l2 = (q(1)
l1 + q(2)
l2 )=2; lj = 1; 2; : : : ; Nj ? 1; j = 1; 2;

where q(j)
lj can be derived from the equations

(q(j)
lj ? 1)=2ø = ?a>=(j)
lj (q(j)
lj + 1)=2:

It follows that

jq(j)

lj j2 =

fifififi

1 ? aø>=(j)

1 + aø>=(j)

fifififi
2

= 1 ?

4a1ø>=(j)
lj

(1 + a1ø>=(j)
lj )2 + (a2ø >=(j)
lj )2 <= 1

and
jql1;l2j <= (jq(1)
l1 j + jq(2)
l2 j)=2 <= 1:

Thus, similarly as in Lemma 2, for both schemes (6) and (7) and for all 0 <= k1; k2 <= 2 we have

kDkh^gk2 =

N1?1
X

l1=1

N2?1

l2=1

(>=(1)
lj )k1(>=(1)
ll )k2jql1;l2al1;l2j2

N1?1
X

l1=1

N2?1
X

l2=1

(>=(1)
l1 )k1(>=(2)
l2 )k2jal1;l2j2 = kDkhgk2:

Since the coeÖcients ql1;l2 and ql1;l2 exist and can be written in unique way for all lj = f1; 2; : : : ; Nj ? 1g, j = 1; 2, the unique function ^g exists in the case of both schemes. The function ^g belongs to the space W due to the estimates derived above. The lemma is proved.

Lemma 8 Let (2) and (3) be satisøed and z 2

ffi

C (µ!x). Then there is a constant ø0,

such that for all positive ø <= ø0 there is a unique solution of (5) ^z 2
ffi
C (µ!x) and the estimate j^zj <= jzj holds.

Proof. See in [6].

Lemma 9 Let (2) and (3) be satisøed and z 2 W . Then there is a constant ø0, such that for all positive ø <= ø0 a solution of (5) ^z 2 W and the following estimates hold:

k^zkjkj;2 <= (1 + 2djkjø)kzkjkj;2; djkj = djkj('(kzkC); kzkjkj?1;2) jkj = 1; 2:

If, additionally, condition (2) for the function f(u; u?) is valid up to the fourth order

derivatives and kD(2;2)
h zk <= C < 1 for all positive h1; h2 <= h0, then the following

estimates also hold:

kD(1;2)

h ^zk2 + kD(2;1)

h ^zk2?1=2

<= (1 + 2d3ø)

kD(1;2)
h zk2 + kD(2;1)
h zk2?1=2
;

kD(2;2)
h ^zk <= (1 + 2d4ø)kD(2;2)
h zk:

Here d3 = d3('(kzkC); kzk2;2) and d4 = d4('(kzkC); kD(1;2)
h zk; kD(2;1)
h zk).

Proof. Let us denote the coeÖcients

~djkj =
X

1<=jk0j<=jkj
dk0; jkj = 1; 2; ~d3 = maxfd(1;2); d(2;1)g=2; ~d4 = d(2;2):

Here dk0 = dk0( _z) are the coeÖcients from Lemma 6 with the function _z instead of v.

Let us apply the operator Dkh on both sides of equation (5), multiply scalarly both sides of the equation by 2øDkh _z and take real parts. We obtain

kDkh^zk2=kDkhzk2+ 2ø Re(Dkhf( _z; _z?); Dkh _z)<=kDkhzk2+ 2økDkhf( _z; _z?)kkDkh _zk:

Let us summate these equations for all k0, jk0j <= jkj <= 2 and apply the estimate k^zk <= kzk which follows from Lemma 8. Using (14) we obtain

k^zk2jkj;2 <= kzk2jkj;2 + 2ø ~djkjk _zk2jkj;2 ) k^zk2jkj;2 <=
?
1 + 2ø ~djkj
1 ? ø ~djkj

?
kzk2jkj;2:

It follows from Lemmas 6 and 8 that ~d1 = ~d1('(k _zkC)) <= ~d1('(kzkC))= d1('(kzkC)). Taking ø <= ø0 = 1=2d1, we ønd that k^zk1;2 <= (1 + 2d1ø)kzk1;2.

Due to this estimate we can introduce the coeÖcient d2 = d2('(kzkC); kzk1;2) such that ~d2 = ~d2('(k _zkC); k _zk1;2) <= d2. Now for all positive ø <= ø0 = 1=2d2 we have the estimate k^zk2;2 <= (1 + 2d2ø)kzk2;2. This estimate completes the proof of the ørst part of this lemma.

Similarly, using the lower order grid derivative estimates, we can ønd the estimates

for

kD(1;2)

h ^zk2 + kD(2;1)
h ^zk2?1=2
and kD(2;2)
h ^zk where djkj = djkj(z) >= ~djkj( _z). The

lemma is proved.

Corollary 2 Let (2) and (3) hold and a solution of (5), (6), or (5), (8) satisfy the estimate kp(tl)kC <= ff, l = 0; 1; : : : ; j. Then for any u0 2 W there is a constant ø0, such that for all positive ø <= ø0 the following estimate is valid:

kp(tl+1)k2;2 <= kg(tl)k2;2 <= c6ku0k2;2; l = 0; 1; : : : ; j:

If, additionally, condition (2) for the function f(u; u?) is valid up to the fourth order

derivatives, and the norm kD(2;2)
h u0k <= C < 1 is bounded for all h1; h2 <= h0, the

following estimate is also valid:

kD(2;2)
h p(tl+1)k <= kD(2;2)
h g(tl)k <= c06kD(2;2)
h u0k; l = 0; 1; : : : ; j:

Here c6 = c6('(ff); ku0k1;2), c06 = c06('(ff); kD(1;2)
h u0k; kD(2;1)
h u0k). Both these constants do not depend on the grid steps.

Proof. We consider ^p = p(tl+1) = ^g(tl), g = g(tl) = ^z(tl), z(tl) = p(tl) = p. Therefore, the results of Lemmas 7 and 9 lead to the estimates

k^pk1;2 <= kgk1;2 <= (1+ 2d1ø)kpk1;2 <= (1 + 2d1ø)l+1ku0k1;2 <= exp(2d1T )ku0k1;2:

Here d1 = d1('(ff)) and l = 0; : : : ; j. Thus, we have shown that kp(tl+1)k1;2 is bounded with a constant, which depends only on ku0k1;2 and '(ff).

Applying the same idea for kp(t)k2;2 we obtain analogous result:

kp(tl+1)k2;2 <= kg(tl)k2;2 <= exp(2d2T )ku0k2;2; l = 0; 1; : : : ; j:

Here d2 depends on '(ff) and, due to Lemma 9 and the previous result, on

max
t2µ!ø ;t<=tjfkp(t)k1;2g <= exp(2d1T )ku0k1;2:

A ørst statement of the corollary follows from here with c6 = exp(2d2T ).

Using the lower order grid derivative estimates, similarly we can ønd similar esti-

mates for

kD(1;2)

h p(t)k2 + kD(2;1)
h p(t)k2?1=2
and kD(2;2)
h p(t)k with c06 = exp(2d(2;2)T ). The corollary is proved.

6 Convergence of the scheme (5, 6).

Let us introduce a truncation error ? on the grid µ!ø :

?(tj) = u(tj+1) ? u(tj)

ø ? a?hu(tj+1) ? f

?u(tj+1) + u(tj)

2 ; u?(tj+1) + u?(tj)
2

?
:

Here and below u(x; t) is the solution of (1). This error satisøes the estimate

max
t2µ!ø fk?(t)kg <= c7(ø + h2)

if condition (4) holds.

Now we can show, that ønite diöerence scheme (5), (6) converges to the solution of (1).

Theorem 1 Let (2), (3) and (4) be satisøed and u0 2 W . Then there exist h0 and ø0 such that for all positive h1; h2 <= h0 and ø <= ø0 the solution of diöerence scheme (5), (6) converges to the solution of diöerential problem (1) and the following estimates hold:

max
t2µ!ø fkp(t) ? u(t)kg <= c8(ø + h2);
max
t2µ!ø fkp(t) ? u(t)kCg <= c08(ø (1=2?fl) + h(1?2fl)); fl 2 (0; 0:5) (16)

Here c8; c08 depend only on c7, jaj, ff = 2kukC( µQ), '(ff), T , ku0k2;2. c08 depends also on c4. Both these coeÖcients do not depend on the grid steps.

Proof. Adding equations (5) and (6) we get the scheme

pt = a?h^p + f( _z; _z?):

We have the following diöerence scheme for the error of the solution " = u ? p:

"t = a?h^" + (f( _u; _u?) ? f( _z; _z?)) + ?; (x; t) 2 Qhø ;
"(x; t) = 0; (x; t) 2 @!x ? µ!ø ; "(x; 0) = 0; x 2 !x: (17)

Let as denote the constant ff = 2kukC( µQ). By mathematical induction we can show that there exist constants ø0 and h0 such that for all positive ø <= ø0, h1; h2 <= h0 and for all t 2 µ!ø the estimate kp(t)kC <= ff holds.

It is clear, that kp(t0)kC <= ff. Suppose that for all l = 0; 1; : : : ; j the estimate kp(tl)kC <= ff holds. Then from Corollary 2 and from (12) we know that the estimate kp(tl)k2;2 <= c6ku0k2;2 for all l = 0; 1; : : : ; j +1 is valid. Note also that, due to Lemma 5 and to (6), we have the following estimates:

kf( _u; _u?) ? f ( _z; _z?)k <= 2'(maxfkukC( µQ); kpkCg)(k _"k + 0:5økgtk)
<= 2'(ff)(k _"k + ø jajc6ku0k2;2):

Multiplying scalarly (17) by ø ^", taking real parts, applying Cauchy inequalities, using the property Re a = a1 >= 0, we can obtain the following inequalities:
k^"k2 <= k"k k^"k + ø(k?k + kf( _u; _u?) ? f( _z; _z?)k)k^"k:
From here and from the estimates for k?k and kf( _u; _u?) ? f( _z; _z?)k it follows:

k^"k <= k"k + 2ø'(ff)k _"k + øc7(ø + h2) + 2ø 2jajc6ku0k2;2'(ff):

Taking time step ø <= ø0 = 1=2'(ff), we obtain

k^"k <= (1 + 4'(ff)ø)k"k + ø(2c7 + 4'(ff)jajc6ku0k2;2)(ø + h2):

Adding these estimates for time layers, using the grid Gronwall's inequality and knowing that k"(t0)k = 0, we can obtain

k"(tj+1)k <= c8(ø + h2) ! 0:

Remark 1, condition (4) and the induction's supposition lead to

k"(tj+1)k2;2 <= ku(tj+1)k2;2 + kp(tj+1)k2;2 <= (c1 + c6)ku0k2;2:

Thus, from multiplicative inequality (12) we have

k"(tj+1)kC <= c08(ø + h2)(1=2?fl) ! 0:

Therefore, taking time and spatial grid steps small enough we can achieve that kp(tj+1)kC <= 2ku(tj+1)kC <= ff. Thus, a step of induction is completed. Therefore, the estimates for the L2 and C norms of "(tj+1) are valid for all time layers and (16) holds. The theorem is proved.

7 Convergence of the scheme (5, 7).

Now we can switch to the investigation of the other diöerence scheme. As it was mentioned earlier, it is enough to investigate the scheme (5), (8).
Using similar ideas as earlier we can prove the following theorem:

Theorem 2 Let the assumptions of Theorem 1 be satisøed. Then there exist h0 and ø0 such that for all positive h1; h2 <= h0 and ø <= ø0 the solution of diöerence scheme (5), (8) converges to the solution of diöerential problem (1) and the following estimates are satisøed:

max
t2µ!ø fkp(t) ? u(t)kg <= c9(ø 1=2 + h2);
max
t2µ!ø fkp(t) ? u(t)kCg <= c09(ø (1=4?fl) + h(1?2fl)); fl 2 (0; 0:25): (18)

If, additionally, condition (2) for the function f(u; u?) is valid up to the fourth order

derivatives and kD(2;2)
h u0k <= C < 1 for all positive h1; h2 <= h0, then (16) holds.

CoeÖcients c9; c09; c8; c08 depend on c7, jaj, ff = 2kukC( µQ), '(ff), T , ku0k2;2. c09; c08

additionally depend on c4; c8; c08 depend also on kD(2;2)
h u0k. All these coeÖcients do not depend on the grid steps.

Proof. Let us denote by p0, z0, g0 the solution of (5), (6) and p, z, g the solution of (5), (8). Let us also denote by ", "z, "g the diöerences between these two solutions p0 ? p, z0 ? z and g0 ? g respectively.

We have the following diöerence scheme for ", "g and "z:

("z)t = f( _z0; _z0?) ? f ( _z; _z?); ("g)t = a?h ^"g + 2ø a2( _g)µx1x1µx2x2 ; (x; t) 2 Qhø ;
"z = "; "g = ^"z; ^" = ^"g; ^" = 0; (x; t) 2 @!x ? !ø ; "(x; 0) = 0; x 2 µ!x: (19)

Denote again ff = 2kukC( µQ). Due to Theorem 1 we have the estimates kp0(t)kC <= ff and kp0(t)k2;2 <= c6ku0k2;2 for all t 2 µ!ø .

Following Theorem 1 we suppose that kp(tl)kC <= ff holds for all l = 0; 1; : : : ; j. Analogously as earlier, the estimate kp(tl)k2;2 <= c6ku0k2;2 holds for all l = 0; 1; : : : ; j + 1. Due to Lemmas 5 and 8, the ørst equation of (19) leads to the estimate

k^"zk <= k"zk + 2ø'(maxfkpkC; kp0kCg)k _"zk <= k"zk + 2ø'(ff)k _"zk:

For all positive ø <= ø0 = 1=2'(ff) we can ønd that

k^"zk <= (1 + 4'(ff)ø)k"k ) k^"zk2 <= (1 + 16'(ff)ø)k"k2:

Note that the following estimates for grid functions v; w on µ!x with zero boundary conditions are valid:

j(vµx1x1µx2x2; w)j = j(vµx1x1 ; wµx2x2)j <= kvµx1x1k kwµx2x2k <= kvk2;2kwk2;2:

Thus, multiplying scalarly the second equation of (19) by ø ^"g, taking real parts, applying Cauchy inequalities we can obtain the estimates

k^"gk2 <= k"gk k^"gk + 2ø 2jaj2k _gk2;2k^"gk2;2 <= k^"zk2 + 2ø 2jaj2k^p + ^zk2;2k^p0 ? ^pk2;2:

Using the results of Lemma 9, Corollary 2 and the previous estimate for k^"zk, we have
k^"k2 <= (1 + 16'(ff)ø)k"k2 + 8c26jaj2ku0k22;2ø 2:
Similarly as in Theorem 1 this leads to the estimate

k"(tj+1)k <= ~c9ø 1=2 ! 0:

This estimate, (12) and boundedness of k"(tj+1)k2;2 lead to

k"(tj+1)kC <= ~c09ø (1=4?fl) ! 0; fl 2 (0; 0:25):

It follows from here, that for all ø small enough kp(tj+1)kC <= ff and, therefore, the induction step is proved.

Since ju ? pj <= ju ? p0j + jp0 ? pj ! 0, (18) is valid. The convergence rates for the C or L2 norms of the ørst term at the right-side of the estimate were found in Theorem (1), and the convergence rates for the second term were obtained above in this theorem.

Suppose that condition (2) is satisøed for the functions f(u; u?) up to the fourth

order derivatives and kD(2;2)
h u0k <= C < 1 for all h1; h2 <= h0. Similarly we can

prove the second part of the theorem and obtain (16).

Assume that kp(tl)kC <= ff, l = 0; 1; : : : ; j. Then kp(tl)k2;2 <= c6ku0k2;2 and, due to

Corollary 2, kD(2;2)
h p(tl)k <= c06kD(2;2)
h u0k for all l = 0; 1; : : : ; j + 1.

From the ørst and the second equations of (19) we have the estimates

k^"zk <= (1 + 4'(ff)ø)k"k; k^"k <= k^"zk + 2c06jaj2kD(2;2)
h u0kø 2:

From these two estimates, the grid Gronwall's inequality and multiplicative inequalities (12) we ønd

k"(tj+1)k <= ~c8ø ! 0; k"(tj+1)kC <= ~c08ø (1=2?fl) ! 0; fl 2 (0; 0:5):

Analogously as above, these estimates lead to (16), where coeÖcients c8 and c08 satisfy the supposition of our theorem. The theorem is proved.

8 Stability of the diöerence schemes.

Suppose that p1 and p2 are the two solutions of the same diöerence scheme with diöerent initial conditions u0;1 and u0;2 respectively. We shall say that the scheme is stable in the norm of the space B, if the inequality

max
t2µ!ø fkp1(t) ? p2(t)kBg <= cku0;1 ? u0;2k?B; ? 2 (0; 1]:

holds for all positive ø <= ø0, h1; h2 <= h0. Here we suppose that constant c is not dependent on the grid steps.

Theorem 3 Let the conditions of Theorem 1 be satisøed. Then diöerence schemes (5), (6) and (5), (7) (or (8)) are stable in the norm of the spaces L2 and C and the following estimates hold:

max
t2µ!ø fkp1(t) ? p2(t)kg <= c10ku0;1 ? u0;2k;

max
t2µ!ø fkp1(t) ? p2(t)kCg <= c010ku0;1 ? u0;2k(1=2?fl)
C ; fl 2 (0; 0:5): (20)

here c10; c010 depend on T , ff = 2 maxfku1kC( µQ); ku2kC( µQ)g, '(ff). c010 additionally depends on c4, ku0;1k2;2 + ku0;2k2;2. Both these coeÖcients do not depend on the grid steps.

Proof. We shall investigate both schemes together. Analogously as in Theorem 2 we denote " = p1 ? p2, "z = z1 ? z2, "g = g1 ? g2. The diöerence schemes for ", "z and "g can be written as follows:

("z)t = f( _z1; _z?1) ? f ( _z2; _z?2); ("g)t = a?h ^"g ? 2<=ø a2( _"g)µx1x1µx2x2 ; (x; t) 2 Qhø ;
"z = "; "g = ^"z; ^" = ^"g; ^" = 0; x 2 @!x ? !ø ; "(x; 0) = u0;1? u0;2; x 2 µ!x:

Here <= = 0 in a case of diöerence scheme (5), (6) and <= = 1 in a case of (5), (8).

Analogously as in Theorem 2, from the ørst equation we can obtain the estimate k^"zk <= (1 + 4ø'(ff))k"k for ø <= ø0 = 1=2'(ff).

Due to Lemma 7, the second equation for both diöerence schemes leads to the estimate k^"k <= k"gk.

Combining both these estimates, adding them for all time layers and applying Gronwall's inequality we came to the ørst inequality of (20) indicating stability in L2 norm with the parameter ? = 1 and c10 = exp(4'(ff)T ).

Due to multiplicative inequality (12) and the boundedness of kp1(t)k2;2 + kp2(t)k2;2, the stability in L2 norm leads also to the stability in C norm with the coeÖcient

c010 = c1=2?fl

10
?

c6(ku0;1k2;2 + ku0;2k2;2)

?1=2+fl
and the parameter ? = 1=2 ? fl. The theorem is proved.

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