A NOTE ON THE DIFFERENCE SCHEMES
FOR HYPERBOLIC EQUATIONS
A. ASHYRALYEV AND P. E. SOBOLEVSKII
Received 26 March 2001
The initial value problem for hyperbolic equations d 2 u(t)/dt 2 + Au(t) = f (t)
(0 ≤ t ≤ 1), u(0) = ϕ, u (0) = ψ, in a Hilbert space H is considered. The first
and second order accuracy difference schemes generated by the integer power of
A approximately solving this initial value problem are presented. The stability
estimates for the solution of these difference schemes are obtained.
1. Introduction
We consider the initial value problem
d 2 u(t)
+ Au(t) = f (t) (0 ≤ t ≤ 1),
dt 2
u(0) = ϕ,
u (0) = ψ,
(1.1)
for a differential equation in a Hilbert space H with unbounded linear selfadjoint
and positive definite operator A = A∗ ≥ δI (δ > 0) with dense domain D̄(A) =
H . It is known (cf. [3]) that various initial boundary value problems for the
hyperbolic equations can be reduced to problem (1.1). A study of discretization,
over time only, of the initial value problem also permits one to include general
difference schemes in applications, if the differential operator in space variables,
A, is replaced by the difference operators Ah that act in the Hilbert spaces and are
uniformly positive definite and selfadjoint in h for 0 < h ≤ h0 . In the paper [4],
the following first order accuracy difference scheme for approximately solving
problem (1.1)
τ −2 uk+1 − 2uk + uk−1 + Auk+1 = fk , fk = f tk , tk = kτ,
τ
1 ≤ k ≤ N − 1,
N τ = 1,
u1 − u0 + iA1/2 u1 = iA1/2 u0 + ψ,
−1
Copyright © 2001 Hindawi Publishing Corporation
Abstract and Applied Analysis 6:2 (2001) 63–70
2000 Mathematics Subject Classification: 35L, 34G, 65J, 65N
URL: http://aaa.hindawi.com/volume-6/S1085337501000501.html
(1.2)
u0 = ϕ,
64
A note on the difference schemes for hyperbolic equations
was considered. The stability estimates for the solution of the difference scheme
(1.2) were obtained. The proof of these statements is based on the transform of
second order difference equations to equivalent system of first order difference
equations. Application of this approach in [1, 2] with similar results for the
solutions of the second order accuracy of the following difference schemes
τ2
τ −2 uk+1 − 2uk + uk−1 + Auk + A2 uk+1 = fk ,
4
fk = f tk , 1 ≤ k ≤ N − 1,
Nτ = 1,
iτ
τ −1 u1 − u0 + iA1/2 I + A1/2 u1 = z1 ,
2
τ
z1 = I + iτ A1/2 ψ + f0 + iA1/2 − τ A u0 , f0 = f (0), u0 = ϕ,
2
1 τ −2 uk+1 − 2uk + uk−1 + A uk+1 + 2uk + uk−1 = fk ,
4
fk = f tk , 1 ≤ k ≤ N − 1,
Nτ = 1,
(1.3)
i
τ −1 u1 − u0 + A1/2 u1 + u0 = z1 ,
2
iτ
1
τ
z1 = I + A1/2 ψ+ f0 + iA1/2 −τ A u0 , f0 = f (0), u0 = ϕ
2
2
2
for approximately solving the initial value problem (1.1) were obtained. However, for practical realization of these difference schemes it is necessary to first
construct an operator A1/2 . This action is very difficult for a 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 the present paper, first and second order accuracy difference schemes for
approximate solutions of problem (1.1) are constructed using the integer powers
of the operator A, and the stability estimates for the solution of these difference
schemes are obtained.
2. First order difference schemes
We consider the first order accuracy difference scheme for approximately solving the initial value problem (1.1)
τ −2 uk+1 − 2uk + uk−1 + Auk+1 = fk ,
fk = f tk , tk = kτ,
1 ≤ k ≤ N − 1,
N τ = 1,
τ −1 u1 − u0 = ψ, u0 = ϕ.
(2.1)
A. Ashyralyev and P. E. Sobolevskii
65
Theorem 2.1. Let ϕ ∈ D(A), ψ ∈ D(A). Then for the solution of the difference
scheme (2.1) the following stability inequalities, for 2 ≤ k ≤ N, hold
k−1
−1/2 A
uk ≤ τ
fs H + A−1/2 ψ H + ϕH ,
H
s=1
1/2 A uk ≤ τ
H
k−1
fs + A1/2 ϕ + ψH ,
H
H
s=1
k−1
Auk ≤ 2
fs −fs−1 +f1 +A1/2 ψ +AϕH ,
H
H
H
H
(2.2)
s=2
u1 ≤ ϕH + I + iτ A1/2 A−1/2 ψ ,
H
H
1/2 A u1 ≤ A1/2 ϕ + I + iτ A1/2 ψ ,
H
H
H
Au1 ≤ AϕH + I + iτ A1/2 A1/2 ψ .
H
H
The proof of this theorem uses the method of [4] and is based on the following
formulas:
u1 = ϕ + τ ψ,
uk =
k−1
−1 1 k−1
τ −1/2 k−s
R +R̃ k−1 ϕ+ R−R̃ τ R R k −R̃ k ψ−
A
R −R̃ k−s fs
2
2i
s=1
−1 1
= R k−1 + R̃ k−1 ϕ + R − R̃ τ R R k − R̃ k ψ
2
−1 1
+A
2
k−1
R k−s + R̃ k−s fs−1 − fs + 2fk−1 − R k−1 + R̃ k−1 f1 ,
s=2
(2.3)
for 2 ≤ k ≤ N, where R = (I + iτ A1/2 )−1 , R̃ = (I − iτ A1/2 )−1 and on the
estimates
R̃ RH →H ≤ 1,
≤ 1,
H →H
−1 −1 R R̃ R̃R (2.4)
≤ 1,
≤ 1,
H →H
H →H
1/2 1/2 τ A R τ A R̃ ≤ 1,
≤ 1.
H →H
H →H
Note that formulas (2.3) are generated by the operator A1/2 and are used
to prove stability estimates for the solutions of the difference scheme (2.1).
66
A note on the difference schemes for hyperbolic equations
However, for the practical realization of this difference scheme (2.1) the operator
A1/2 as in [1, 2, 4] is not used. Note also that these stability inequalities
in the case k = 1 are weaker than the respective inequalities in the cases
k = 2, . . . , N . However, obtaining this type of inequalities is important for applications. We denote by a τ = (ak ) the mesh function of approximation. Then
(I +iτ A1/2 )a1 H ∼ a1 H = o(τ ) if we assume that τ Aa1 H tends to 0 as τ
tends to 0 not slower than a1 H . It takes place in applications by supplementary
restriction on the smoothness property of the data in the space variables.
It is clear that the estimate
u1 ≤ ϕH + A−1/2 ψ (2.5)
H
H
is absent. However, estimates for the solution of first order accuracy modification
difference scheme for approximately solving the initial value problem (1.1)
τ −2 uk+1 − 2uk + uk−1 + Auk+1 = fk , fk = f tk , tk = kτ,
1 ≤ k ≤ N − 1,
N τ = 1,
I + τ 2 A τ −1 u1 − u0 = ψ, u0 = ϕ,
(2.6)
are better than the estimates for the solution of the difference scheme (2.1).
Theorem 2.2. Let ϕ ∈ D(A), ψ ∈ D(A1/2 ). Then for the solution of the difference scheme (2.6), the following stability inequalities, for 1 ≤ k ≤ N, hold
k−1
−1/2 uk ≤ τ
A
fs H + A−1/2 ψ H + ϕH ,
H
s=1
k−1
1/2 A uk ≤ τ
fs + A1/2 ϕ + ψH ,
H
H
H
s=1
k−1
Auk ≤ 2
fs −fs−1 +f1 +A1/2 ψ +AϕH .
H
H
H
H
s=2
The proof of this theorem is based on the following formulas:
u1 = ϕ + τ R R̃ψ,
uk =
−1 1 k−1
R
+ R̃ k−1 ϕ + R − R̃ τ R R k − R̃ k R R̃ψ
2
k−1
τ −1/2 k−s
−
A
− R̃ k−s fs
R
2i
s=1
(2.7)
A. Ashyralyev and P. E. Sobolevskii
=
67
−1 1 k−1
R
+ R̃ k−1 ϕ + R − R̃ τ R R k − R̃ k R R̃ψ
2
−1 1
+A
2
k−1
R k−s + R̃ k−s fs−1 − fs
s=2
+ 2fk−1 − R k−1 + R̃ k−1 f1 ,
2 ≤ k ≤ N,
(2.8)
and on the estimates (2.4).
3. Second order difference schemes
We consider the second order accuracy difference schemes for approximate
solutions of the initial value problem (1.1)
τ2
τ −2 uk+1 − 2uk + uk−1 + Auk + A2 uk+1 = fk ,
4
fk = f tk , tk = kτ, 1 ≤ k ≤ N − 1,
Nτ = 1,
(3.1)
τ
I + τ 2 A τ −1 u1 − u0 = f0 − Au0 + ψ, f0 = f (0), u0 = ϕ,
2
1
1 τ −2 uk+1 − 2uk + uk−1 + Auk + A uk+1 + uk−1 = fk ,
2
4
fk = f tk , tk = kτ, 1 ≤ k ≤ N − 1,
Nτ = 1,
(3.2)
τ
I + τ 2 A τ −1 u1 − u0 = f0 − Au0 + ψ,
2
f0 = f (0), u0 = ϕ.
The stability estimates for the solution of these difference schemes are obtained.
Theorem 3.1. Let ϕ ∈ D(A), ψ ∈ D(A1/2 ). Then for the solution of the difference scheme (3.1), the following stability inequalities, for 1 ≤ k ≤ N, hold
k−1
−1/2 A
+ A−1/2 ψ + ϕH ,
uk ≤ τ
f
s
H
H
H
s=0
k−1
1/2 A uk ≤ τ
fs + A1/2 ϕ + ψH ,
H
H
H
s=0
k−1
Auk ≤ 2
fs −fs−1 +f0 +A1/2 ψ +AϕH .
H
H
H
H
s=1
(3.3)
68
A note on the difference schemes for hyperbolic equations
The proof of this theorem is based on the following formulas:
τ2
τ2
I + A ϕ + τ ψ + f0 ,
2
2
u1 = I + τ 2 A−1
−1 k
−1 uk = R k + τ R R − R̃
R − R̃ k I + τ 2 A
− τ A + iA1/2 ϕ
−1
−1 k
R − R̃ k I + τ 2 A ψ
+ τ R R − R̃
τ
−1 k
−1
τ2 R R−R̃
A−1/2 R k−s −R̃ k−s fs
R −R̃ k I +τ 2 A f0 −
2
2i
k−1
+
s=1
−1 k
−1 = R k + τ R R − R̃
R − R̃ k I + τ 2 A
− τ A + iA1/2 ϕ
−1
−1 k
R − R̃ k I + τ 2 A ψ
+ τ R R − R̃
+
−1 k
−1
τ2 R R − R̃
R − R̃ k I + τ 2 A f0
2
+A
k−1 iτ 1/2 −1 k−s iτ 1/2 −1 k−s
R + I− A
I+ A
fs−1 − fs
R̃
2
2
2
−1 1
s=1
iτ
+ 2 I + A1/2
2
−1 iτ
I − A1/2
2
−1
fk−1
iτ 1/2 −1 k−1
iτ 1/2 −1 k−1
R̃
− I+ A
R
+ I− A
f0 ,
2
2
2 ≤ k ≤ N,
(3.4)
where R = (I + iτ A1/2 − (τ 2 /2)A)−1 , R̃ = (I − iτ A1/2 − (τ 2 /2)A)−1 and on
the estimates
R̃ RH →H ≤ 1,
−1 R̃R H →H
I ± iτ A1/2 −1 ≤ 1,
H →H
≤ 1,
H →H
≤ 1,
−1 R R̃ −1 I ± iτ A1/2
2
H →H
H →H
≤ 1,
≤ 1,
1/2 −1 τ A
I ± iτ A1/2 H →H
(3.5)
≤ 1.
A. Ashyralyev and P. E. Sobolevskii
69
Theorem 3.2. Let ϕ ∈ D(A), ψ ∈ D(A1/2 ). Then for the solution of the difference scheme (3.2), the following stability inequalities, for 1 ≤ k ≤ N, hold
k−1
−1/2 uk ≤ τ
A
fs H + A−1/2 ψ H + ϕH ,
H
s=0
k−1
1/2 A uk ≤ τ
fs + A1/2 ϕ + ψH ,
H
H
H
s=0
k−1
A uk + uk−1 ≤2
fs − fs−1 + f0 + A1/2 ψ + AϕH .
H
H
H
2
H
s=1
(3.6)
The proof of this theorem is based on the following formulas:
τ2
τ2
A ϕ + τ ψ + f0 ,
2
2
1/2
k
1
iτ A
R − R̃ k
uk = R k + A−1/2 I −
2i
2
1/2
−1
τ
iτ A
1/2
2
2
A − iτ A
I +τ A I +τ A
ϕ
× I+
2
2
−1
iτ A1/2 k
iτ A1/2
i
I+
R − R̃ k I + τ 2 A
ψ
+ A−1/2 I −
2
2
2
−1
iτ A1/2 k
iτ A1/2 τ
i
R − R̃ k I + τ 2 A
f0
I+
+ A−1/2 I −
2
2
2
2
−1
u1 = I + τ 2 A
I+
k−1
τ −1/2 k−s
A
− R̃ k−s fs
R
2i
s=1
iτ A1/2 k
1 −1/2
k
R − R̃ k
I−
= R + A
2i
2
−1
iτ A1/2 τ
× I+
A − iτ A1/2 I + τ 2 A I + τ 2 A
ϕ
2
2
−1
iτ A1/2 k
iτ A1/2
i −1/2
k
2
R − R̃ I + τ A
ψ
I−
I+
+ A
2
2
2
−1
i −1/2
iτ A1/2 k
iτ A1/2 τ
k
2
R − R̃ I + τ A
f0
+ A
I−
I+
2
2
2
2
−
70
A note on the difference schemes for hyperbolic equations
+A
−1 1
2
iτ A1/2
iτ A1/2
k−s
k−s
R
R̃
+ I+
I−
fs−1 − fs
2
2
k−1 s=1
+ 2fk−1 −
I−
iτ A1/2
iτ A1/2
R k−1 + I +
R̃ k−1 f0 ,
2
2
2 ≤ k ≤ N,
(3.7)
where R = (I−iτ A1/2 /2)(I+iτ A1/2 /2)−1 , R̃ = (I+iτ A1/2 /2)(I−iτ A1/2 /2)−1
and on the estimates
iτ A1/2 −1 R̃ H →H ≤ 1,
≤ 1,
RH →H ≤ 1,
I± 2
H →H
(3.8)
1/2 1/2 −1 I ± iτ A1/2 −1 ≤ 1,
τA
≤ 1.
I ± iτ A
H →H
H →H
References
[1]
[2]
[3]
[4]
A. Ashyralyev and I. Muradov, On one difference scheme of a second order of
accuracy for hyperbolic equations, Trudy Instituta Matematiki i Mechaniki Akad.
Nauk Turkmenistana (Ashgabat), no. 1, 1995, pp. 58–63 (Russian).
, On difference schemes of a second order of accuracy for hyperbolic equations, Modelling Processes of Explotation of Gas Places and Applied Problems of
Theoretical Gasohydrodynamics, Ashgabat, Ilym, 1998, pp. 127–138 (Russian).
S. G. Kreı̆n, Lineikhye Differentsialnye Uravneniya v Banakhovom Prostranstve
[Linear Differential Equations in a Banach Space], Izdat. “Nauka”, Moscow,
1967 (Russian). MR 40#508.
P. E. Sobolevskii and L. M. Chebotaryeva, Approximate solution by method of lines
of the Cauchy problem for abstract hyperbolic equations, Izv. Vyssh. Uchebn.
Zaved. Mat. (1977), no. 5, 103–116 (Russian).
A. Ashyralyev: Department of Mathematics, Fatih University, Istanbul,
Turkey
Current address: International Turkmen-Turkish University, Ashgabat,
Turkmenistan
E-mail address: aashyr@fatih.edu.tr
P. E. Sobolevskii: Institute of Mathematics, Hebrew University, Jerusalem,
Israel
E-mail address: pavels@math.hiji.ac.il