Memoirs on Differential Equations and Mathematical Physics
Volume 53, 2011, 29–38
G. Berikelashvili, M. M. Gupta, and M. Mirianashvili
ON THE CHOICE OF INITIAL
CONDITIONS OF DIFFERENCE
SCHEMES FOR PARABOLIC
EQUATIONS
Abstract. We consider the first initial-boundary value problem for linear
heat conductivity equation with constant coefficient in Ω × (0, T ], where Ω
is a unit square. A high order accuracy ADI two level difference scheme
is constructed on a 18-point stencil using Steklov averaging operators. We
prove that the finite difference scheme converges in the discrete L2 -norm
with the convergence rate O(hs + τ s/2 ), when the exact solution belongs to
s,s/2
the anisotropic Sobolev space W2
, s ∈ (2, 4].
2010 Mathematics Subject Classification. 65M06, 65M12, 65M15.
Key words and phrases. Heat equation, ADI difference scheme, high
order convergence rate.
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Ł Œ
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غŒ Œ Ø æ ªŒ
º ºŁ ª
Œ º s,s/2
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, s ∈ (2, 4] ª ø , Ø Œ
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O(hs + τ s/2 ).
On the Choice of Initial Conditions of Difference Schemes
31
1. Introduction
The purpose of this paper is to study the difference schemes approximating the first initial-boundary value problem for linear second order parabolic
equations and to obtain some convergence rate estimates.
The finite difference method is a basic tool for the solution of partial differential equations. When studying the convergence of the finite difference
schemes, Taylor’s expansion was used traditionally. Often, the BrambleHilbert lemma [1], [2] takes the role of Taylor’s formula for the functions
from the Sobolev spaces.
As a model problem, we consider the first initial-boundary value problem for linear second-order parabolic equations with constant coefficients.
We suppose that the generalized solution of this problem belongs to the
s,s/2
anisotropic Sobolev space W2
(Q), s > 2.
In the case of difference schemes constructed for the mentioned problem,
when obtaining convergence rate estimate compatible with smoothness of
the solution, various authors assume that the solution of the problem can
be extended to the exterior of the domain of integration, preserving the
Sobolev class.
Our investigations have shown that if instead of the exact initial condition
its certain approximation is taken, then this restriction can be removed.
A high order alternating direction implicit (ADI) difference scheme is
constructed in the paper for which the convergence rate estimate
ky − ukL2 (Qh,τ ) ≤ c(hs + τ s/2 )kukW s,s/2 (Q) , s ∈ (2, 4],
2
is obtained. Here y is a solution to the difference scheme, Qh,τ is a mesh
in Q, c is a positive constant independent of h, τ and u, and h and τ are
space and time steps, respectively.
2. The Problem and Its Approximation
Let Ω = {x = (x1 , x2 ) : 0 < xα < 1, α = 1, 2} be the unit square in R2
with boundary Γ and let T denote a positive real number. In Q = Ω × (0, T ]
we consider the equation of heat conductivity
∂u
∂2u ∂2u
+
− au + f (x, t), a = const ≥ 0, (x, t) ∈ QT ,
=
∂t
∂x21
∂x22
(1)
under the initial and first kind boundary conditions
u(x, 0) = u0 (x), x ∈ Ω,
u(x, t) = 0, x ∈ Γ, t ∈ [0, T ].
(2)
We mean that the solution to the problem (1), (2) belongs to the anisos,s/2
tropic Sobolev space W2
(Q), s > 2.
Throughout the paper k·kW λ,λ/2 (Q) will denote the norms and |·|W λ,λ/2 (Q)
2
2
the highest semi norms of corresponding Sobolev spaces [6].
We assume that ω̄ is a uniform mesh in Ω with the step h = 1/n. ω =
ω ∩ Ω, γ = ω \ ω. We cover the segment [0, T ] with a uniform mesh ω τ
32
G. Berikelashvili, M. M. Gupta, and M. Mirianashvili
(with the mesh step τ = T /N ). Let ωτ = ω τ ∩ (0, T ), ωτ+ = ω τ ∩ (0, T ],
ωτ− = ω τ ∩ [0, T ), Qh,τ = ω × ω τ . We assume that there exist two positive
constants c1 h2 ≤ τ ≤ c2 h2 . For functions defined on the mesh cylinder
ω × ω τ we use the notation:
y = y(x, t) = y j , x ∈ ω, t = tj ∈ ω τ ,
yb(x, t) = y(x, t + τ ), y̌(x, t) = y(x, t − τ ),
yt =
(I (+α) − I)y
h2
yb − y̌
(I − I (−α) )y
, yxα =
, yxα =
, κ :=
,
τ
h
h
12
where Iy := y, I ±α y := y(x ± hrα , t) and rα represents the unit vector of
the axis xα .
We define also the Steklov averaging operators:
xZ1 +h
¡
1
T1 u(x, t) = 2
h
¢
h − |x1 − ξ| u(ξ, x2 , t) dξ,
x1 −h
1
b
Su(x,
t) =
τ
t+τ
Z
u(x, ζ) dζ.
t
The operator T2 is defined similarly. Note that these operators are commutative and
Tα
∂u
∂2u
= Λα u, Sb
= ut .
∂x2α
∂t
b 1 T2 to the eq. (1), we will get
If we apply the operator ST
b 2 u) + Λ2 (ST
b 1 u) − aST
b 1 T2 u + ST
b 1 T2 f.
(T1 T2 u)t = Λ1 (ST
(3)
It is easy to check that on the set of sufficiently smooth functions the following operators:
Tα ∼ I + κΛα with errors of order O(h4 ),
b
Sb ∼ (I + I)/2
with errors of order O(τ 2 )
are equivalent and, therefore, within the accuracy O(h4 + τ 2 ) we obtain
T1 T2 ∼ (I + κΛ1 )(I + κΛ2 ),
b
b 1 T2 ∼ (I + κΛ1 + κΛ2 ) I + I ,
ST
2
b
b α ∼ (I + κΛα ) I + I .
ST
2
(4)
(5)
(6)
On the Choice of Initial Conditions of Difference Schemes
33
Taking into the account the relations (4)–(6), we denote:
η0 = T1 T2 u − (I + κΛ1 )(I + κΛ2 )u − (τ 2 /4)Λ1 Λ2 u,
b+u
b 3−α u − (I + κΛ3−α ) u
ηα = ST
, α = 1, 2,
2
b+u
b 1 T2 u − (I + κΛ1 + κΛ2 ) u
η = ST
+
2
´
³τκ
τ 2 ´³
aτ 2
+
Λ1 + Λ 2 u t −
ut .
+
4
8
16
(7)
(8)
(9)
In the equalities (7), (9) the additional terms are introduced with the
aim that the resulting difference scheme operator should be factorizable.
Due to (7)–(9), from (3) we get
τ2
Λ1 Λ2 ut + (η0 )t =
4
u
b+u
u
b+u
= Λ1 (I + κΛ2 )
+ Λ2 (I + κΛ1 )
+ Λ1 η1 + Λ2 η2 −
2
2
µ
¶
u
b + u ³τκ
τ2 ´
aτ 2
−a (I + κΛ1 + κΛ2 )
−
+
(Λ1 + Λ2 )ut +
ut + η +
2
4
8
16
b 1 T2 f,
+ST
(I + κΛ1 )(I + κΛ2 )ut +
that is,
³
τ
aτ ´³
τ
aτ ´
I + κΛ1 − Λ1 +
I I + κΛ2 − Λ2 +
I ut =
2
4
2
4
³
´
= Λ1 (I + κΛ2 ) + Λ2 (I + κΛ1 ) − a(I + κΛ1 + κΛ2 ) u+
b 1 T2 f + ψ,
+ST
(10)
ψ = Λ1 η1 + Λ2 η2 − aη − (η0 )t .
(11)
where
Finally, if in the equation (10) we reject the remainder term and change u
by the mesh function y, we will come to the difference scheme
Byt + Ay = ϕ, (x, t) ∈ ω × ωτ− ,
(12)
where
A := A1 (I − κA2 ) + A2 (I − κA1 ) + a(I − κA1 − κA2 ),
³
τ
aτ ´³
τ
aτ ´
B := I − κA1 + A1 +
I I − κA2 + A2 +
I .
2
4
2
4
We define the initial and boundary conditions as follows:
By 0 = T1 T2 u0 +
τ
Au0 , x ∈ ω, y(x, t) = 0, (x, t) ∈ γ × ω τ .
2
(13)
34
G. Berikelashvili, M. M. Gupta, and M. Mirianashvili
3. An a Priori Estimate of the Solution Error
Let H be the space of mesh functions defined on ω and vanishing on γ,
with inner product and norm
X
(y, v) =
h2 y(x)v(x), kyk = kykL2 (ω) = (y, y)1/2 .
x∈ω
Besides, let
kyk0 = kykL2 (Qh,τ ) =
³X
τ ky(·, t)k2L2 (ω)
´1/2
.
t∈ω τ
In the case of self-conjugate positive operators we will use the notation
p
(y, v)D := (Dy, v), kykD := (Dy, y), D = D∗ > 0.
Let
C := B −
τ
A.
2
(14)
It is easy to verify that
³ aτ 2
aτ κ ´
(A1 + A2 )+
8
4
a2 τ 2
τ2
4
τ2
+
I+
A1 A2 ≥ I +
A1 A2 > 0. (15)
16
4
9
4
C = (I − κA1 )(I − κA2 ) +
+
The following lemma plays a significant role in getting the needed a priori
estimate of the solution of the difference scheme.
Lemma 1. Let A = A∗ > 0, B = B ∗ > 0 be arbitrary independent on t
operators and B > (τ /2)A. Then for the solution of the problem
Bvt + Av = ψt , (x, t) ∈ ω × ωτ− ,
0
0
Bv = ψ , x ∈ ω
(16)
(17)
the estimate
kvkL2 (Qh,τ ) ≤ kC −1 ψkL2 (Qh,τ )
is valid with C defined in (14).
Proof. Summing up by t = 0, τ, . . . , (k − 1)τ , from (16) we find
Bv k − Bv 0 +
k−1
X
τ Av j = ψ k − ψ 0 , k = 1, 2, . . . ,
j=0
that is, taking into account the initial condition (17),
Bv k +
k−1
X
j=0
τ Av j = ψ k , k = 1, 2, . . . .
(18)
On the Choice of Initial Conditions of Difference Schemes
35
Since C = C ∗ > 0, the inverse operator C −1 = (C −1 )∗ > 0 exists.
Multiply (18) scalarly by C −1 v k :
(Bv k , C −1 v k ) +
³ k−1
X
´
τ Av j , C −1 v k = (ψ k , C −1 v k ), k = 1, 2, . . . .
(19)
j=0
Denote
χ0 = 0, χk =
k−1
X
τ v j , k = 1, 2, . . . .
j=0
Then (19) yields
³
χk+1 − χk ´
= (ψ k , C −1 v k ),
(Bv k , C −1 v k ) + Aχk , C −1
τ
from which, after some transformations, we obtain
µ³
¶
1
1 k 2
τ ´
2
τ B − A v k , C −1 v k + kχk+1 kAC
kχ kAC −1 =
−1 −
2
2
2
= τ (ψ k , C −1 v k )
or
2τ kv k k2 + kχk+1 k2AC −1 − kχk k2AC −1 = 2τ (C −1 ψ k , v k ), k = 1, 2, . . . . (20)
Using the Cauchy–Bunyakovski inequality, we estimate the right-hand side
of (20)
2τ (C −1 ψ k , v k ) ≤ τ kC −1 ψ k k2 + τ kv k k2
and sum up the obtained result by k = 1, 2, . . . , N . We get
N
X
τ kv k k2 + kχN +1 k2AC −1 − kχ1 k2AC −1 ≤
k=1
N
X
τ kC −1 ψ k k2 .
(21)
k=1
From (14) we have
B 2 = C 2 + τ AC +
τ2 2
A > C 2 + τ AC.
4
Hence
τ AC −1 ≤ B 2 C −2 − I.
Using this inequality and taking into account the relation χ1 = τ v 0 , we get
¡
¢
kχ1 k2AC −1 = (τ AC −1 v 0 , τ v 0 ) ≤ (B 2 CI−2 )v 0 , τ v 0 =
= τ kBC −1 v 0 k2 − τ kv 0 k2 = τ kC −1 ψ 0 k2 − τ kv 0 k2 ,
which together with (21) proves the lemma.
¤
Consider the error z = y−u. From (10)–(13) we get the following problem
for it:
Bzt + Az = A1 η1 + A2 η2 + aη + (η0 )t , (x, t) ∈ ω × ωτ− ,
(22)
Bz 0 = η00 , x ∈ ω, z ∈ H.
36
G. Berikelashvili, M. M. Gupta, and M. Mirianashvili
We define the functions η1 , η2 to be zeros on t = T and substitute z in (22)
by the following expression
z = v + A−1 (A1 η1 + A2 η2 + aη).
(23)
Then for v we obtain the problem (16), (17), where
ψ = η0 − BA−1 (A1 η1 + A2 η2 + aη).
Using Lemma 1 for v, we get the estimate
N
X
τ kv k k2 ≤
k=0
N
X
τ Jk2 ,
k=0
°
°
Jk := °C −1 η0k − C −1 BA−1 (A1 η1k + A2 η2k + aη k )°.
(24)
Because of (14), (15) we have
C −1 BA−1 = A−1 +
τ −1
9τ
C ≤ A−1 +
I, C −1 ≤ (9/4)I.
2
8
Therefore
°
°
° 9τ °
9
°A1 η1k + A2 η2k + aη k °.
Jk ≤ kη0k k + °A−1 (A1 η1k + A2 η2k + aη k )° +
4
8
Taking into account the operator inequalities
A≥
2
32
3
(A1 + A2 ), A ≥
I, A−1 Aα ≤ I,
3
3
2
we get
´ 9τ °
°
9 k
3³ k
a
°A1 η1k + A2 η2k + aη k °.
kη0 k +
kη1 k + kη2k k +
kη k k +
4
2
16
8
On the basis of this and the following algebraic inequalities
½ X ³ X ´ ¾1/2 X ³ X ´
2
1/2
aik
≤
a2ik
, aik ≥ 0,
Jk ≤
k
i
i
k
we get from (24)
kvk0 ≤
´
9
3³
a
kη0 k0 +
kη1 k0 + kη2 k0 +
kηk0 +
4
2
16
¢
9τ ¡
+
kA1 η1 k0 + kA2 η2 k0 + akηk0 . (25)
8
(23), (25) enable us to assert the validity of the following
Theorem 1. For the solution of the difference problem (22) the following
a priori estimate is true
kzk0 ≤
¢
9τ ¡
9
kη0 k0 + 3(kη1 k0 + kη2 k0 ) +
kA1 η1 k0 + kA2 η2 k0 .
4
8
(26)
On the Choice of Initial Conditions of Difference Schemes
37
4. Convergence of the Finite-Difference Scheme
Let E denote a bounded open set in R2 with Lipschitz continuous boundary, and let G = E × (0, 1). We introduce the set of multi-indices
n
o
Bk = (α1 , α2 , β) : αi , β = 0, 1, 2, . . . ; α1 + α2 + 2β ≤ k .
Further, let [s]− denote the largest integer less than s. The convergence
analysis of our finite difference scheme is based on the following lemma.
s,s/2
Lemma 2. If ϕ is a bounded linear functional on W2
1 α2 β
ϕ(xα
1 x2 t )
(G) such that
= 0, ∀ (α1 , α2 , β) ∈ B[s]− ,
then there exists a positive constant c = c(G, s) such that
s,s/2
|ϕ(v)| ≤ c|v|W s,s/2 (G) , ∀ v ∈ W2
2
(G).
Lemma 2 is an easy consequence of the Dupont–Scott approximation
theorem [4] (see also [5]).
If we use Lemma 2 and the well-known techniques (see, e.g., [1]–[3], [5])
for estimation of the terms in the right-hand side of the equation (26), we
will get convinced in the validity of the following
Theorem 2. Assume that the solution u to the problem (1), (2) belongs
s,s/2
(Qh,τ ), 2 < s ≤ 4. Then the rate of convergence of the
to the space W2
difference scheme (12), (13) in the L2 grid norm is described by the estimate
ky − ukL2 (Qh,τ ) ≤ chs kukW s,s/2 (Q) , s ∈ (2, 4],
2
where the constant c does not depend on h and u.
Remark. A more detailed analysis enables us to obtain the estimate
ky − ukL2 (Qh,τ ) ≤ c(hs + τ s/2 )kukW s,s/2 (Q) , s ∈ (2, 4],
2
2
as well without restriction τ ∼ h .
The results of the paper were announced on Sixth International Congress
on Industrial Applied Mathematics (ICIAM07), Zürich, 2007 [7].
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38
G. Berikelashvili, M. M. Gupta, and M. Mirianashvili
5. B. S. Jovanović, On the convergence of finite-difference schemes for parabolic equations with variable coefficients. Numer. Math. 54 (1989), no. 4, 395–404.
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(Received 29.09.2010)
Authors’ addresses:
G. Berikelashvili
A. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, University Str., Tbilisi 0186
Georgia
Department of Mathematics, Georgian Technical University
77, M. Kostava Str., Tbilisi 0175
Georgia
E-mail: bergi@rmi.ge
M. M. Gupta
Department of Mathematics
The George Washington University
Washington, DC 20052
USA
E-mail: mmg@gwu.edu
M. Mirianashvili
N. Muskhelishvili Institute of Computational Mathematics
8, Akuri Str., Tbilisi 0193
Georgia