On the numerical solution of hyperbolic IBVP with high-order stable finite difference schemes

R E S E A R C H

Open Access

On the numerical solution of hyperbolic IBVP

with high-order stable finite difference

schemes

Allaberen Ashyralyev

1

_{and Ozgur Yildirim}

2*

*_{Correspondence:}

[email protected]

2_{Department of Mathematics, Yildiz}

Technical University, Istanbul, 34210, Turkey

Full list of author information is available at the end of the article

Abstract

The abstract Cauchy problem for the hyperbolic equation in a Hilbert spaceHwith self-adjoint positive definite operatorAis considered. The third and fourth orders of accuracy difference schemes for the approximate solution of this problem are presented. The stability estimates for the solutions of these difference schemes are established. A finite difference method and some results of numerical experiments are presented in order to support theoretical statements.

MSC: 65J10; 65M12; 65N12; 35L30

Keywords: abstract hyperbolic equation; stability; initial boundary value problem

1 Introduction

Partial differential equations of the hyperbolic type play an important role in many branches of science and engineering. For example, acoustics, electromagnetics, hydro-dynamics, elasticity, fluid mechanics, and other areas of physics lead to partial differential equations of the hyperbolic type (see,e.g., [–] and the references given therein). The sta-bility has been an important research topic in the development of numerical techniques for solving these equations (see [–]). Particularly, a convenient model for analyzing the stability is provided by a proper unconditionally absolutely stable difference scheme with an unbounded operator.

A large cycle of works on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitude of the grid stepsτ andhwith respect to time and space variables are connected (see,e.g., [–] and the references therein). Of great interest is the study of absolute stable difference schemes of a high order of accuracy for hyperbolic partial differential equations, in which stability was established without any assumptions in respect of the grid stepsτ andh. Such type stability inequalities for the solutions of the first order of accuracy difference scheme for the differential equations of hyperbolic type were established for the first time in [].

It is known (see [, ]) that various initial boundary value problems for a hyperbolic equation can be reduced to the initial value problem

⎧ ⎨ ⎩

du(t)

dt +Au(t) =f(t),  <t<T,

u() =ϕ, u() =ψ, ()

whereAis a self-adjoint positive definite linear operator with the domainD(A) in a Hilbert spaceH.

A functionu(t) is called a solution of problem () if the following conditions are satisfied:

(i) u(t)is twice continuously differentiable on the segment[, ]. The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives. (ii) The elementu(t)belongs toD(A)for allt∈[, ]and the functionAu(t)is

continuous on the segment[, ].

(iii) u(t)satisfies the equations and the initial conditions ().

In recent decades, many scientists have worked in the field of a finite difference method for the numerical solutions of hyperbolic PDEs and have published many scientific papers. For problem (), the first and two types of second order difference schemes were presented and the stability estimates for the solution of these difference schemes and for the first-and second-order difference derivatives were obtained in []. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by the Taylor decomposition on the three points for the numerical solution of the same problem were presented and the stability estimates for approximate solution of these difference schemes were obtained in []. However, the difference methods developed in these references are generated by square roots ofA. This action is very difficult for the realization. Therefore, in spite of theoretical results, the role of their application to a numerical solution for an initial value problem is not great. In this paper, the third order of accuracy difference scheme

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

τ–(uk+– uk+uk–) +_Auk+A(uk++uk–)

+_τAuk+=fk,

fk=_f(tk) +_(f(tk+) +f(tk–))

–_τ(–Af(tk+) +f(tk+)), ≤k≤N– ,

u=ϕ, (I+τ



A+

τ

A)τ–(u–u) = –

τ

Aϕ+ (I–

τ

A)ψ+f,τ,

()

and the fourth order of accuracy difference scheme

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

τ–(uk+– uk+uk–) +_Auk+_A(uk++uk–)

–τ_A_u

k+ τ 

A(uk++uk–) =fk,

fk=_f(tk) +_(f(tk+) +f(tk–)) +τ



(–Af(tk) +f(tk))

–_ τ_(–A(f(t

k+) +f(tk–)) +f(tk+) +f(tk–)),

≤k≤N– ,tk=kτ,Nτ= ,

u=ϕ, (I+τ

_A

 +

τA

 )τ–(u–u) = –

τ

Aϕ+ (I–

τA

)ψ+f,τ

()

for the approximate solution of initial value problem () are constructed using the inte-ger powers of the operatorA, and stability estimates for the solution of these difference schemes are obtained. Here,

f,=

I+τ



A+

τ

A



τ–f_,

=

f() + –f() +τf()  – f

_()τ 

and

f,=

I+τ



A+

τ

A



τ–f_,

=

I–τ

_A



f() +

–

I–τ

_A



f() +τf()

 

+ –Aτf() – f() +τf()τ

+ Af() – f _()τ



. ()

Some results of this paper without proof are accepted and will be published in  (see []).

Note that boundary value problems for parabolic equations, elliptic equations, and equations of mixed type have been studied extensively by many scientists (see,e.g., [–] and the references therein).

2 The stability estimates

In this section, construction of difference schemes (), () and stability estimates for the solutions of these difference schemes are presented.

Let us obtain the third and fourth orders of approximation formulas for the solution of problem (). If the functionf(t) is not only continuous, but also continuously differentiable on [,T],ϕ∈D(A) andψ∈D(A_{), it is easy to show that (see []) the formula}

u(t) =c(t)ϕ+s(t)ψ+ t



s(t–λ)f(λ)dλ ()

gives a solution of problem (). Throughout this paper,{c(t),t≥}is a strongly continuous cosine operator-function defined by the formula

c(t) = e itA/

+e–itA/

 . ()

Then, from the definition of the sine operator-function

s(t)u= t



c(s)u ds, ()

it follows that

s(t) =A–/e

itA/_–e–itA/

i . ()

For the theory of cosine operator-function, we refer to [] and []. A uniform grid is considered on the segment [,T]

[,T]τ={tk=kτ,k= , , . . . ,N,Nτ=T}. ()

In the first step, the approximation of differential equation () is considered. Using Tay-lor’s decomposition on three points, the following formulas for the third order of approx-imation and the fourth order of approxapprox-imation of () are obtained respectively:

u(tk+) – u(tk) +u(tk–) –

 τ

_u(t

k)

– τ

 _u(t

k+) +u(tk–)

+ 

τ

_u()_(t

k+) =o τ

, ()

u(tk+) – u(tk) +u(tk–) –

 τ

_u(t

k)

– 

τ

 _u(t

k+) +u(tk–)

– 

τ

_u()_(t

k) +  τ

 _u()_(t

k+) +u()(tk–)

=o τ. ()

Applying equation (), one can write

u(tk) = –Au(t) +f(t), u()(t) = –Au(t) +f(t) =Au(t) –Af(t) +f(t). () Using () and (), the following formula:

u(tk+) – u(tk) +u(tk–)

τ –



 –Au(tk) +f(tk)

–

 –A u(tk+) +u(tk–)

+f(tk+) +f(tk–)

+ 

τ

 _A_u(t

k+) –Af(tk+) +f(tk+)

=o τ ()

for the third order of approximation of (), and using (), (), the following formula

u(tk+) – u(tk) +u(tk–)

τ –



 –Au(tk) +f(tk)

– 

 –Au(tk+) +u(tk–)

+f(tk+) +f(tk–)

– 

τ

 _A_u(t

k) –Af(tk) +f(tk)

+ 

τ

 _A _u(t

k+) +u(tk–)

–A f(tk+) +f(tk–)

+f(tk+) +f(tk–)

=o τ ()

for the fourth order of approximation of () are obtained. Neglecting the last small term, we get

τ–(uk+– uk+uk–) +

 Auk

+

A(uk++uk–) +  τ

_A_u

k+=fk,

fk=  f(tk) +



 f(tk+) +f(tk–)

– 

τ

 _–Af(t

k+) +f(tk+)

and

τ–(uk+– uk+uk–) +

 Auk+



A(uk++uk–)

– 

τ

_A_u

k+  τ

_A_(u

k++uk–) =fk,

fk=  f(tk) +



 f(tk+) +f(tk–)

()

+ 

τ

 _–Af(t

k) +f(tk)

– 

τ



× –A f(tk+) +f(tk–)

+f(tk+) +f(tk–)

for the third and fourth orders of approximations of (), respectively.

In the second step, the approximation ofu() is considered. Applying (), we can write

u(τ) –u()

τ =

c(τ) –I τ ϕ+

s(τ)

τ ψ+



τ

τ



s(τ–λ)f(λ)dλ. ()

From () it is obvious that for the approximation ofu(), it is necessary to approximate the expressions

s(τ), c(τ), and 

τ

τ



s(τ–λ)f(λ)dλ. ()

Using the definitions of s(τ),c(τ), and Padé fractions for the functione–z (see []), the following approximation formulas are obtained:

c(τ) =R(iτB) +R(–iτB)

 +o τ

_{, s(τ) =B}–R(iτB) –R(–iτB)

i +o τ

 _()

for the third order of approximation of (), where

R(iτB) =R=DE, R(–iτB) =R=DE, ()

and

c(τ) =R(iτB) +R(–iτB)

 +o τ

_,

s(τ) =B–R(iτB) –R(–iτB)

i +oτ



()

for the fourth order of approximation of (), where

R(iτB) =R=CC–_{, R(–iτB) =R=CC}–_{. ()}

HereB=A/_and

D=

I– τ

_A+iτA/

I+  τ

_A

, D=

I– τ

_A–iτA/

I+  τ

_A

,

E=

I+ τ

_A+ 

τ

_A

–

C=

I+iτA

/

 –

τA



, C=

I–iτA

/

 –

τA



.

Using (), (), (), (), we obtain the third order of approximations ofs(τ) andc(τ)

cτ(τ) =

I– τ

_A

I+ τ

_A+ 

τ

_A

–

, ()

sτ(τ) =τ

I+  τ

_A

I+ τ

_A+ 

τ

_A

–

()

and the fourth order of approximations ofs(τ) andc(τ)

cτ(τ) =

I–  τ

_A+ 

τ

_A

I+  τ

_A+ 

τ

_A

–

, ()

sτ(τ) =τ

I–  τ

_A

I+  τ

_A+ 

τ

_A

–

. ()

Let us remark that in constructing the approximation ofu(), it is important to know how to constructf_, andf

,such that



τ

τ



s(τ–λ)f(λ)dλ–f_, =o τ, ()



τ

τ



s(τ–λ)f(λ)dλ–f_, =o τ, ()

and formulas off_, andf_, are sufficiently simple. The choice off_, andf_, is not unique. Using Taylor’s formula and integration, we obtain the following formulas for the third order of approximation:

f_, =

Sτ(τ)f() + –Cτ(τ)f() +Sτ(τ)f()τ

 – Cτ(τ)f _()τ



=

I+τ



A+

τ

A



–

τf() + –f() +τf()τ  – f

_()τ 

()

and for the fourth order of approximation

f_, =

Sτ(τ)f() + –Cτ(τ)f() +Sτ(τ)f()τ 

+ –ASτ(τ)f() – Cτ(τ)f() +Sτ(τ)f()τ





+ ACτ(τ)f() – Cτ(τ)f()τ





=

I+τ



A+

τ

A



–

τ

I–τ

_A



f()

+

–

I–τ

_A



f() +τf()

τ



+ –Aτf() – f() +τf()τ



 + Af() – f _()τ



For simplicity, denote

f,=f() + –f() +τf()

 – f

_()τ

, ()

and

f,=

I–τ

_A



f() +

–

I–τ

_A



f() +τf()

 

+ –Aτf() – f() +τf()τ

+ Af() – f _()τ



, ()

where

f_, =τ

I+τ



A+

τ

A



–

f, ()

and

f_, =τ

I+τ



A+

τ

A



–

f,. ()

Thus, we have the following formula for the approximation ofu():

u–u

τ =

cτ(τ) –I τ ϕ+

sτ(τ)

τ ψ+f

m

,, m= , . ()

Using the approximation formulas above, difference schemes () and () are constructed. Now, we will obtain the stability estimates for the solution of difference scheme (). We consider operatorsR,Rby () and the following operators:

R=

–τ



A

₊τ

A

_–iτA/

I+τ



A+

τ

A



I+  τ

_A

×

–iτA/

I+  τ

_A

I+τ



A+

τ

A



–

, ()

and its conjugateR

R=

I–τ



A

I+τ



A+

τ

A



×

–iA/

I+τ



A+

τ

A



I+  τ

_A

–

, ()

R=

I+τ



A+

τ

A



I+τ



A+

τ

A



–iτA/

I+  τ

_A

–

, ()

R=

I+τ



 A+

τ

A

₊τ

A



×

–iA/

I+  τ

_A

I+τ



A+

τ

A



I+τ



A

–

R=

–τ



A–

τ

A

_+iτA/

I+  τ

_A

I+τ



A+

τ

A



–

, ()

and its conjugateR,

R=

I– τ

_A+iτA/

I+  τ

_A

×

τ

 A+

τ

A

_–iτA/

I+  τ

_A

–

, ()

and its conjugateR.

Let us give one lemma, without proof, that will be needed below.

Lemma  The following estimates hold:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

RH→H≤, RH→H≤,

RH→H≤, RH→H≤,

A/_R

H→H≤, τA/RH→H≤,

A/_R

H→H≤, A–/RH→H≤τ,

A–/_R

H→H≤τ, τA/RH→H≤,

τA/RH→H≤.

()

Now, we will give the first main theorem of the present paper on the stability of difference scheme ().

Theorem  Letϕ ∈D(A), ψ ∈D(A_{), f,}∈_D(A_{). Then, for the solution of difference}

scheme(),the following stability estimates hold:

max

≤k≤NukH≤M

_N–

s=

_A–/_f

s_Hτ+ϕH+A–/ψ_H+τA–/f,_H

, ()

max

≤k≤NA

/_u

k_H+ max

≤k≤N

uk–uk–

τ

H

≤M

_N–

s=

fsHτ+A/ϕ_H+ψH+τf,H

, ()

max

≤k≤NAukH+max≤k≤N

A/uk–uk– τ

H

+ max

≤k≤N–

uk+– uk+uk–

τ

H

≤M

_N–

s=

fs–fs–H+fH+AϕH+A/ψ_H+τA/f,_H

, ()

where M does not depend onτ,ϕ,ψ,f,,and fs, ≤s≤N– .

Proof First, we will obtain the formula for the solution of problem (). It is clear that there exists a unique solution of the initial value problem

and for the solution of (), the following formula is satisfied (see []):

u=ϕ, u=ψ,

uk=RR(R–R)–

Rk––Rk–ϕ+ (R–R)– Rk–Rkψ

+ k–

j=

RR (R–R)a–Rk–j–Rk–jϕj, ()

whereR=q=c+

√ c_–ab

b ,R=q=

c–√c_–ab

b . Hereqandqare roots of equation (), and

a= ,b= . We can rewrite () into the following difference problem:

I+τ



A

uk––

 –τ



 A

uk+

I+τ



A+

τ

A



uk+=τfk, ≤k≤N– ,

u=ϕ, u=ϕ+τ

I+τ



A+

τ

A



–

–τ Aϕ+

I–τ



A

ψ+τf,

.

Replacingawith (I+τ

A),bwith (I+

τ

A+

τ

A),cwith ( – τ

 A), andψwith

ϕ+τ

I+τ



A+

τ

A



–

–τ Aϕ+

I–τ



A

ψ+τf,

,

ϕkwithτfkand applying formula (), we obtain the following formulas foruk, ≤k≤N:

uk=  RR

k_–R

Rk

ϕ+ R

k_–Rk_R

ψ

+ R

k_–Rk_R

τf,+

 R

k–

s=

Rk–s–Rk–sfsτ. ()

Second, we will prove estimates (), (), and (). Using () and the formula foru,

and the following simple estimates:

I– 

τ

_A+ τ

A



I+τ



A+

τ

A



–

_H→H≤, ()

τA/

I–τ



A

I+τ



A+

τ

A



–

_H→H≤, ()

τA/

I+  τ

_A+ 

τ

_A

–

_H→H≤ 

√



 +√, ()

we get

uH≤ I– 

τ

_A+ τ

A



I+τ



A+

τ

A



–

_H→HϕH

+τA/

I–τ



A

I+τ



A+

τ

A



–

_H→HA–/ψ

H

+τA/

I+τ



A+

τ

A



–

H→H

≤ ϕH+ A–/ψ_H+ √  +√τA

–/_f ,_H

≤M

_N–

s=

A–/fs_Hτ+ϕH+A–/ψ_H+τA–/f,_H

.

ApplyingA/_{to the formula foru}

and using estimates (), (), (), and (), we get

A/u_H≤

I–τ

_A

 +

τA



I+τ

_A

 +

τA

 –

H→H

A/ϕ_H

+τA/

I–τ



A

I+τ



A+

τ

A



–

_H→HψH

+τA/

I+τ

_A

 +

τ_A

 –

_H→Hτf,H

≤A/ϕ

H+ ψH+ √

 +√τf,H

≤M

_N–

s=

fsHτ+A/ϕ_H+ψH+τf,H

.

Using the formula

u–u

τ =

I+τ

_A

 +

τ_A

 –

×

–τ

Aϕ+

I–τ



A

ψ+τf,

()

and estimates (), (), and the following simple estimates:

I–τ

_A



I+τ

_A

 +

τ_A

 –

_H→H≤, ()

I+τ

_A

 +

τ_A

 –

_H→H≤, ()

we get

u–u

τ

H

≤

τA

/

I+τ

_A

 +

τA

 –

H→H

A/ϕ_H

+

I–τ



A

I+τ

_A

 +

τA

 –

_H→HψH

+

I+τ

_A

 +

τ_A

 –

_H→Hτf,H

≤ 

√

  +√A

/_ϕ

H+ψH+τf,H

≤M

_N–

s=

fsHτ+A/ϕ_H+ψH+τf,H

ApplyingA/_{to () and using estimates (), (), (), (), we obtain}

A/u–u τ

H

≤τ

A

/

I+τ

_A

 +

τA

 –

H→H

AϕH

+

I–τ



A

I+τ

_A

 +

τA

 –

_H→HA/ψ_H

+

I+τ

_A

 +

τ_A

 –

_H→HτA/f,_H

≤ 

√



 +√AϕH+A

/_ψ

H+τA

/_f ,_H

≤M

_N–

s=

fs–fs–H+fH+AϕH

+A/ψ_H+τA/f,_H

. ()

ApplyingAto the formula foruand using estimates (), (), (), (), we get

AuH≤ I–τ

_A

 +

τ_A



I+τ

_A

 +

τ_A

 –

_H→HAϕH

+τA/

I–τ



A

I+τ



A+

τ

A



–

_H→HA/ψ

H

+τA/

I+  τ

_A+ 

τ

_A

–

H→H

τA/f,_H

≤ AϕH+ A/ψ_H+ √  +√τA

/_f ,_H

≤M

_N–

s=

fs–fs–H+fH+AϕH

+A/ψ

H+τA

/_f ,_H

. ()

Using difference scheme () and formula (), we obtain

u– u+u

τ =

I+τ

_A

 +

τA

 –

×

–

I–τ

_A

 –

τA

 +

τA

,

I+τ

_A

 +

τA

 –

Aϕ

–τ

A+  τ

_A

I–τ



A

I+τ

_A

 +

τ_A

 –

ψ

–

A+τ

_A



I+τ

_A

 +

τ_A

 –

τf,+f

Using formula (), estimates (), (), (), (), and the following simple estimates:

I–τ

_A

 –

τA

 +

τA

,

I+τ

_A

 +

τA

 –

_H→H≤, ()

I+τ

_A



I+τ

_A

 +

τA

 –

H→H≤

, ()

we have

u– u+u

τ

H

≤

I–τ

_A

 –

τA

 +

τA

,

I+τ

_A

 +

τA

 –

_H→HAϕH

+

I+τ

_A



I+τ

_A

 +

τA

 –

H→H

×τA/

I–τ



A

I+τ

_A

 +

τA

 –

_H→HA/_ψ

H

+

I+τ

_A



I+τ

_A

 +

τ_A

 –

_H→H

×τA/

I+τ

_A

 +

τ_A

 –

_H→HA/f,_Hτ

+

I+τ

_A

 +

τA

 –

_H→HfH

≤AϕH+fH+ A/ψ_H+ √  +√τ

A/_f ,_H

≤M

_N–

s=

fs–fs–H+fH+AϕH+A/ψ_H+τA/f,_H

. ()

So, fork= , the following estimates are proved:

ukH≤M _N–

s=

A–/fs_Hτ+ϕH+A–/ψ_H+τA–/f,_H

,

A/uk_H≤M _N–

s=

fsHτ+A/ϕ_H+ψH+τf,H

,

uk–uk–

τ

H

≤M

_N–

s=

fsHτ+A/ϕ_H+ψH+τf,H

,

A/uk–uk– τ

H

≤M

_N–

s=

fs–fs–H+fH+AϕH

+A/ψ

H+τA

/_f ,_H

AukH≤M _N–

s=

fs–fs–H+fH+AϕH+A/ψ_H+τA/f,_H

,

uk+– uk+uk–

τ

H

≤M

_N–

s=

fs–fs–H+fH+AϕH

+A/ψ_H+τA/f,_H

.

Now, we will establish these estimates for anyk≥. Using formula (), estimate (), and the triangle inequality, we obtain

ukH≤ 

 RH→HR k

H→H+RH→HRk_H→H

ϕH

+

 A

/_R

_H→HRk_H→H+A/R_H→HRk_H→HA–/ψ_H

+

 τA

/_R

_H→HRk_H→H

+τA/R_H→HRk_H→H

τA–/f,_H

+ τA

/_R _H→H

k–

s=

Rk–s_H→H+Rk–s_H→HA–/fs_Hτ

≤M

_N–

s=

A–/fs_Hτ+ϕH+A–/ψ_H+τA–/f,_H

()

for anyk≥. Combining the estimatesukHfor anyk, we obtain (). ApplyingA/_{to (), using estimate () and the triangle inequality, we get}

A/uk_H≤ 

 RH→HR k

H→H+RH→HRk_H→HA/ϕ_H +

 A

/_R

_H→HRk_H→H+A/R_H→HRk_H→H

ψH

+

 τA

/_R

_H→HRk_H→H

+τA/R_H→HRk_H→H

τf,H

+ τA

/_R _H→H

k–

s=

Rk–s_H→H+Rk–s_H→HfsHτ

≤M

_N–

s=

fsHτ+A/ϕ_H+ψH+τf,H

fork≥. Combining the estimates forA/_u

kHfor anyk, we obtain

max

≤k≤NA

/_u

k_H≤M _N–

s=

fsHτ+A/ϕ_H+ψH+τf,H

Applying formula (), we get

uk–uk–

τ =



τ

 RRR

k–_–R RRk–

ϕ

+ RR

k–_–R Rk–

Rψ+

 RR

k–_–R Rk–

Rτf,

+

[R–R]Rτ

_f

k–+

 R

k–

s=

RRk––s–RRk––s

fsτ

. ()

Using (), estimate (), and the triangle inequality, we obtain

uk–uk–

τ

H

≤  τ



 A

–/_R

_H→HRH→HRk–_H→H

+A–/R_H→HRH→HRk–_H→HA/ϕ_H

+

 A

/_R

_H→HA–/R_H→HRk–_H→H

+A/R_H→HA–/R_H→HRk–_H→H

ψH

+

 τA

/_R

_H→HA–/R_H→HRk–_H→H

+τA/R_H→HA–/R_H→HRk–_H→H

×τf,H+ 

 τA

/_R

_H→HRH→H +τA/R_H→HRH→HA–/fk–_Hτ+

 τA

/_R _H→H

×

k–

s=

A–/R_H→HRk––s_H→H

+A–/R_H→HRk––s_H→H

fsHτ

≤M

_N–

s=

fsHτ+A/ϕ_H+ψH+τf,H

. ()

Combining the estimates forτ–(uk–uk–)Hfor anyk, we obtain

max

≤k≤N–

uk–uk–

τ

H

≤M

_N–

s=

fsHτ+A/ϕ_H+ψH+τf,H

. ()

From estimates (), (), estimate () follows. Now, applying Abel’s formula to (), we obtain

uk–uk–

τ =



τ

 RRR

k–_–R RRk–

ϕ

+ RR

k–_–R Rk–

Rψ+

 RR

k–_–R Rk–

+

[R–R]Rτ

_f

k–+

 Rτ



_k–

s=

RRRk––s–RRRk––s

(fs–fs–)

×[RR–RR]fk––RRRk––RRRk–

f

, ≤k≤N. ()

Next, applyingA/_{to formula () and using estimate () and the triangle inequality, we}

get

A/uk–uk– τ

H

≤  τ



 A

–/_R

_H→HRH→HRk–_H→H

+A–/_R

_H→HRH→HRk–_H→H

AϕH

+

 A

/_R

_H→HA–/R_H→HRk–_H→H

+A/R_H→HA–/R_H→HRk–_H→HA/ψ_H

+

 τA

/_R

_H→HA–/R_H→HRk–_H→H

+τA/R_H→HA–/R_H→HRk–_H→H

τA/f,_H

+

 τA

/_R

_H→HRH→H+τA/R_H→HRH→Hfk–Hτ

+ τA

/_R _H→H

_k–

s=

A–/R_H→HτA/R_H→HRk––s_H→H

+A–/R_H→HτA/R_H→HRk––s_H→H

fs–fs–H + A–/R_H→HτA/R_H→H+A–/R_H→HτA/R_H→H

× fk–H+ A–/R_H→HτA/R_H→HRk–_H→H

+A–/R_H→HτA/R_H→HRk–_H→H

fH

≤M

_N–

s=

fs–fs–H+fH+AϕH

+A/ψ_H+τA/f,_H

.

Combining the estimates forA/_τ–_(u

k–uk–)Hfor anyk, we obtain

max

≤k≤N–

A/uk–uk– τ

H

≤M

_N–

s=

fs–fs–H+fH

+A/ψ

H+AϕH+τA

/_f ,_H

Now, applying Abel’s formula to (), we have

uk=  RR

k_–R

Rk

ϕ+ R

k_–Rk_R

ψ

+ R

k_–Rk_R

τf,+

 τ

_R 

_k–

s=

RRk–s–RRk–s

(fs–fs–)

+ (R–R)fk––RRk––RRk–

f

, ≤k≤N. ()

Next, applyingAto formula () and using () and the triangle inequality, we get

AukH≤ 

 RH→HR k

H→H+RH→HRk_H→HAϕH +

 A

/_R

_H→HRk_H→H+A/R_H→HRk_H→HA/ψ_H

+

 τA

–/_R

_H→HRk_H→H+τA–/R_H→HRk_H→H

×τA/f,_H+

 τA

/_R _H→H

×

_k–

s=

τA/R_H→HRk–s_H→H+τA/R_H→HRk–s_H→H

× fs–fs–H+ τA/R_H→H+τA/R_H→H

fk–H

+τA/R_H→HRk–_H→H+τA/R_H→HRk–_H→H

fH

≤M

_N–

s=

fs–fs–H+fH+AϕH+A/ψ_H+τA/f,_H

fork≥. Combining the estimates forAukHfor anyk, we obtain

max

≤k≤NAukH≤M

_N–

s=

fs–fs–H+fH

+AϕH+A/ψ_H+τA/f,_H

. ()

Now, applying formula (), we get

uk+– uk+uk–

τ =



τ

 RR



Rk––RRRk–

ϕ

+ R



Rk––RRk–

Rψ+

 R



Rk––RRk–

Rτf,

+

[R–R]Rτ

_f

k+ 

 R

 –R

Rτfk–

+ R

k–

s=

R_Rk––s–R_Rk––sfsτ

First, applying Abel’s formula to (), we have

uk+– uk+uk–

τ

= 

τ

 RR



Rk––RRRk–

ϕ

+ R



Rk––RRk–

Rψ+

 R



Rk––RRk–

Rτf,

+

[R–R]Rτ

_f

k+ 

 R

 –R

Rτfk–

+ Rτ



_k–

s=

RRRk––s–RRRk––s

(fs–fs–)

+ R_R–RR

fk––RRRk––RRRk–

f

, ≤k≤N. ()

Second, using formulas (), (), and the triangle inequality, we get

uk+– uk+uk–

τ

H

≤  τ

  A

–/_R 



H→HRH→HRk–_H→H

+ A–/R



H→HRH→HRk–_H→H

AϕH

+  A

–/_R 



H→HA/R_H→HRk–_H→H

+ A–/R



H→HA/R_H→HRk–_H→HA/ψ_H

+  A

–/_R 



H→HτA/R_H→HRk–_H→H

+ A–/R



H→HτA/R_H→HRk–_H→H

τA/f,_H

+

 τA

/_R

_H→HRH→H+τA/R_H→HRH→H

τfkH

+  A

–/_R 



H→H+ A–/R



H→H

×A/R_H→HτA/fk–_H+

 τA

/_R _H→H

×

_k–

s=

A–/R



H→HτA/R_H→HRk––s_H→H

+ A–/R



H→HτA/R_H→HRk––s_H→H

fs–fs–H + A–/R



H→HτA/R_H→H+ A–/R



H→HτA/R_H→H

fk–H + A–/R



H→HτA/R_H→HRk–_H→H

+ A–/R



H→HτA/R_H→HRk–_H→H

≤M

_N–

s=

fs–fs–H+fH+AϕH

+A/ψ

H+τA

/_f ,_H

. ()

Combining the estimates forτ–_(u

k+– uk+uk–)Hfor anyk, we obtain

max

≤k≤N–

τ–(uk+– uk+uk–)_H

≤M

_N–

s=

fs–fs–H+fH+A/ψ_H+AϕH+τAf,H

. ()

From estimates (), (), and (), we obtain (). Theorem  is proved.

Now, we will obtain the stability estimates for the solution of difference scheme (). We consider operatorsR,Rby () and the following operators:

J=

I–τ

_A



I–iτA

/

 –

τA

 –

,

J=

I–τ

_A



I+iτA

/

 –

τ_A

 –

,

J=

–iτA/

I–iτA

/

 –

τ_A



–

,

J=

–iτA/

I+iτA

/

 –

τA



–

,

J=iτA/

I–τ

_A



I–iτA

/

 –

τA

 –

,

J= –iτA/

I–τ

_A



I+iτA

/

 –

τA

 –

,

J=

I–τ

_A



iτA/

I+τ

_A

 +

τ_A



–

,

J=

I–τ

_A



–iτA/

I+τ

_A

 +

τ_A



–

,

which will be used in the sequel. It is obvious thatJ,J, andJ,J, andJ,J, andJ,Jare

conjugates.

Let us give one lemma, without proof, that will be needed below.

Lemma  The following estimates hold: ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

RH→H≤, RH→H≤,

JH→H≤, JH→H≤,

τA/JH→H≤, τA/JH→H≤,

A–/_J

H→H≤τ, A–/_J

H→H≤τ,

τA/JH→H≤, τA/JH→H≤.

Now, we will give the first main theorem of the present paper on the stability of difference scheme ().

Theorem  Letϕ∈D(A),ψ ∈D(A_{), f,}∈_D(A_{).Then, for the solution of difference}

scheme(),the following stability estimates hold:

max

≤k≤N

uk+uk–

 H

≤M

_N–

s=

A–/fs_Hτ+ϕH

+A–/ψ_H+τA–/f,_H

, ()

max

≤k≤N–

uk+–uk–

τ

H

+ max

≤k≤N

A/uk+uk–

 H

≤M

_N–

s=

fsHτ+A/ϕ_H+ψH+τf,H

, ()

max

≤k≤N–

A/uk+–uk–

τ

H

+ max

≤k≤N

Auk+uk–

 H

≤M

_N–

s=

fs–fs–H+fH+AϕH+A/ψ_H+τA/f,_H

, ()

where M does not depend onτ,ϕ,ψ,f,,and fs, ≤s≤N– .

Proof First, we will obtain the formula for the solution of problem (). In exactly the same manner as in Theorem , one establishes formula () for the solution of initial value prob-lem ().

We can rewrite () into the following difference problem:

I+τ

_A

 +

τA



uk––

 –τ

_A

 +

τA



uk+

I+τ

_A

 +

τA



uk+

=τfk, ≤k≤N– ,

u=ϕ,

u=ϕ+τ

I+τ

_A

 +

τ_A

 –

–τ Aϕ+

I–τ

_A



ψ+τf,

.

()

Replacingawith (I+τA

 +

τA

 ),bwith (I+

τA

 +

τA

 ),cwith

 –τ

_A

 +

τA



,

andψwith

ϕ+τ

I+τ

_A

 +

τ_A

 –

–τ Aϕ+

I–τ

_A



ψ+τf,