The fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem
In the present work symmetrized sequential type decomposition scheme of the fourth degree precision for the solution of inhomogeneous evolution problem is constructed. The fourth degree precision is reached by introducing the complex parameter α = 1/2 ± i(1/2√3) and by the approximation of the semig...
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Інститут прикладної математики і механіки НАН України
2009
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| Цитувати: | The fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem / J. Rogava, M. Tsiklauri // Український математичний вісник. — 2009. — Т. 6, № 3. — С. 385-399. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859918019949494272 |
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| author | Rogava, J. Tsiklauri, M. |
| author_facet | Rogava, J. Tsiklauri, M. |
| citation_txt | The fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem / J. Rogava, M. Tsiklauri // Український математичний вісник. — 2009. — Т. 6, № 3. — С. 385-399. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Український математичний вісник |
| description | In the present work symmetrized sequential type decomposition scheme of the fourth degree precision for the solution of inhomogeneous evolution problem is constructed. The fourth degree precision is reached by introducing the complex parameter α = 1/2 ± i(1/2√3) and by the approximation of the semigroup through the rational approximation. For the considered scheme the explicit a priori estimation is obtained.
|
| first_indexed | 2025-12-07T16:06:24Z |
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Український математичний вiсник
Том 6 (2009), № 3, 385 – 399
The fourth order of accuracy sequential
type rational splitting of inhomogeneous
evolution problem
Jemal Rogava, Mikheil Tsiklauri
(Presented by M. M. Malamud)
Abstract. In the present work symmetrized sequential type decompo-
sition scheme of the fourth degree precision for the solution of inhomo-
geneous evolution problem is constructed. The fourth degree precision
is reached by introducing the complex parameter α = 1
2
± i
1
2
√
3
and by
the approximation of the semigroup through the rational approximation.
For the considered scheme the explicit a priori estimation is obtained.
2000 MSC. 65M12, 65M15, 65M55.
Key words and phrases. Decomposition scheme, rational splitting,
evolution problem, rational approximation.
Introduction
One of the most effective methods to solve multi-dimensional evolu-
tion problems is a decomposition method. Decomposition schemes with
first and second order accuracy were constructed in the sixties of the
XX century (see [8, 11] and references therein). Q. Sheng has proved
that in the real number field there do not exist automatically stable de-
composition schemes with an accuracy order higher than two (see [12]).
Decomposition schemes are called automatically stable if a sum of the ab-
solute values of its split coefficients (coefficients of exponentials’ products)
equals to one, and the real parts of exponential powers are positive. In the
work [1] there is constructed decomposition schemes with the higher order
accuracy, but their corresponding decomposition formulas are not auto-
matically stable. In the works [2–5] introducing the complex parameter,
Received 16.02.2009
The designation project has been fulfilled by Financial support of Georgian National
Science Foundation (Grant GNSF/PRES07/3-149). Any idea in this publication is
possessed by the author and may not represent the opinion of Georgian National Sci-
ence Foundation itself.
ISSN 1810 – 3200. c© Iнститут математики НАН України
386 The fourth order...
we have constructed automatically stable decomposition schemes with
third order accuracy for two- and multi-dimensional evolution problems
and with fourth order accuracy for two-dimensional evolution problem
(evolution problem with the operator A is called m-dimensional, if it can
be represented as a sum of m summands A = A1 + · · ·+ Am ). The new
idea is an introduction of a complex parameter, which allows us to break
the order 2 barrier.
Decomposition formulas constructed in the above mentioned works
represent formulas of exponential splitting. Exponential splitting is called
a splitting which approximates a semigroup by a combination of semi-
groups generated by the summands of the operator generating the given
semigroup. In view of numerical computations, it is important a ratio-
nal splitting of the multi-dimensional problem (We call rational splitting
such a splitting of the evolution problem that is obtained from the expo-
nential splitting by replacing the semigroups generated by the summands
of its main operators with the corresponding rational approximations).
Hence, if we have an exponential splitting with some order precision and
the same order rational approximation of a semigroup, we can construct
a rational splitting of the evolution problem. In the work [7] we have
constructed the rational splitting with the third order precision.
In the present work, we have constructed the fourth order precision
rational splitting for inhomogeneous evolution problem. We say that the
rational approximation of the semigroup used in the work is of Kranc–
Nickolson type, as if we replace the parameter α with 1, we obtain the
classic Kranc–Nickolson approximation. In addition, let us note that
in the scalar case, the considered rational approximation represents a
Pade classic approximation (see [14]). For the rational approximation
constructed in the work, there is obtained the explicit a priori estimate.
1. Statement of the problem and main result
Let us consider the Cauchy abstract problem in the Banach space X :
du(t)
dt
+ Au(t) = f(t), t > 0, u(0) = ϕ, (1.1)
where A is a closed linear operator with the definition domain D [A],
which is everywhere dense in X, ϕ is a given element from X, f(t) ∈
C1 ([0;∞) ; X).
Let the operator (−A) generate the strongly continuous semigroup
{exp(−tA)}t≥0, then the solution of the problem (1.1) is given by the
J. Rogava, M. Tsiklauri 387
following formula ([9, 10]):
u(t) = U(t, A)ϕ +
t∫
0
U(t − s, A)f(s) ds, (1.2)
where U(t, A) = exp (−tA).
Let A = A1 + A2, where Aj (j = 1, 2) are densely defined, closed,
linear operators in X.
As it is well-known, the essence of decomposition method consists
in splitting the semigroup U (t, A) by means of the semigroups U (t, Aj)
(j = 1, 2). In [6] there is constructed the following decomposition formula
with the local precision of Fifth order:
T (τ) = U
(
τ,
α
4
A1
)
U
(
τ,
α
2
A2
)
U
(
τ,
1
4
A1
)
U
(
τ,
α
2
A2
)
U
(
τ,
α
2
A1
)
× U
(
τ,
α
2
A2
)
U
(
τ,
1
4
A1
)
U
(
τ,
α
2
A2
)
U
(
τ,
α
4
A1
)
. (1.3)
where α = 1
2 ± i 1
2
√
3
(
i =
√
−1
)
.
In the above-mentioned work it is shown that:
U (τ, A) − T (τ) = Op
(
τ5
)
,
where Op
(
τ5
)
is the operator, norm of which is of the fifth order with
respect to τ (more precisely, in the case of the unbounded operator∥∥Op
(
τ5
)
ϕ
∥∥ = O
(
τ5
)
for any ϕ from the definition domain of Op
(
τ5
)
).
In the present work (see Section 2) we construct the semigroup approx-
imations with the local precision of the fifth order using the following
rational approximation:
W (τ, A) =
(
I − α
2
τA
) (
I +
α
2
τA
)−1(
I − α
2
τA
) (
I +
α
2
τA
)−1
. (1.4)
The approximation defined by formula (1.4) in the scalar case represent
the Pade approximations for exponential functions [14].
On the basis of formulas (1.3) and (1.4) we can construct the following
decomposition formula:
V (τ) = W
(
τ,
α
4
A1
)
W
(
τ,
α
2
A2
)
W
(
τ,
1
4
A1
)
W
(
τ,
α
2
A2
)
W
(
τ,
α
2
A1
)
× W
(
τ,
α
2
A2
)
W
(
τ,
1
4
A1
)
W
(
τ,
α
2
A2
)
W
(
τ,
α
4
A1
)
. (1.5)
Below we shall show that this formula has the precision of the fifth
order:
U(τ, A) − V (τ) = Op
(
τ5
)
.
388 The fourth order...
In the present work, on the basis of formula (1.5), a decomposition
scheme with the fourth order precision will be constructed for the solution
of problem (1.1).
Let us introduce the following net domain:
ωτ = {tk = kτ, k = 0, 1, . . . , τ > 0}.
According to formula (1.2), we have:
u(tk) = U (tk, A)ϕ +
tk∫
0
U (tk − s, A) f (s) ds
= U (τ, A)U (tk−1, A)ϕ +
tk−1∫
0
U (τ, A)U (tk−1 − s, A) f (s) ds
+
tk∫
tk−1
U (tk − s, A) f (s) ds
= U (τ, A)
[
U (tk−1, A)ϕ +
tk−1∫
0
U (tk−1 − s, A) f (s) ds
]
+
tk∫
tk−1
U (tk − s, A) f (s) .
From here we have
u(tk) = U (τ, A)u (tk−1) +
tk∫
tk−1
U (tk − s, A) f (s) ds.
Let us use Simpson’s formula and rewrite this formula in the following
form:
u (tk) = U (τ, A)u (tk−1) +
τ
6
(
f (tk) + 4U
(τ
2
, A
)
f
(
tk−1/2
)
+ U (τ, A) f (tk−1)
)
+ R5,k (τ) ,
u (t0) = ϕ, k = 1, 2, . . . .
(1.6)
For the sufficiently smooth function f the following estimate is true
(see. Lemma 2.3):
‖Rk,5 (τ)‖ = O
(
τ5
)
. (1.7)
J. Rogava, M. Tsiklauri 389
On the basis of formula (1.6) let us construct the following scheme:
uk = V (τ) uk−1 +
τ
6
(
f (tk) + 4V
(τ
2
)
f
(
tk−1/2
)
+ V (τ) f (tk−1)
)
,
u0 = ϕ, k = 1, 2, . . . .
(1.8)
Let us perform the computation of the scheme (1.8) by the following
algorithm:
uk = u
(0)
k +
2τ
3
u
(1)
k +
τ
6
f (tk) ,
where uk,0 is calculated by the scheme:
u
(0)
k−8/9 = W
(
τ,
α
4
A1
)(
uk−1 +
τ
6
f (tk−1)
)
,
u
(0)
k−7/9 = W
(
τ,
α
2
A2
)
u
(0)
k−8/9, u
(0)
k−6/9 = W
(
τ,
1
4
A1
)
u
(0)
k−7/9,
u
(0)
k−5/9 = W
(
τ,
α
2
A2
)
u
(0)
k−6/9, u
(0)
k−4/9 = W
(
τ,
α
2
A1
)
u
(0)
k−5/9,
u
(0)
k−3/9 = W
(
τ,
α
2
A2
)
u
(0)
k−4/9, u
(0)
k−2/9 = W
(
τ,
1
4
A1
)
u
(0)
k−3/9,
v
(0)
k−1/9 = W
(
τ,
α
2
A2
)
u
(0)
k−2/9,
u
(0)
k = W
(
τ,
α
2
A2
)
u
(0)
k−1/9, u
(0)
0 = ϕ +
τ
6
f (0) ,
(1.9)
and uk,1 by the scheme:
u
(1)
k−8/9 = W
(τ
2
,
α
4
A1
)
f
(
tk−1/2
)
, u
(1)
k−7/9 = W
(τ
2
,
α
2
A2
)
u
(1)
k−8/9,
u
(1)
k−6/9 = W
(τ
2
,
1
4
A1
)
u
(1)
k−7/9, u
(1)
k−5/9 = W
(τ
2
,
α
2
A2
)
u
(1)
k−6/9,
u
(1)
k−4/9 = W
(τ
2
,
α
2
A1
)
u
(1)
k−5/9, u
(1)
k−3/9 = W
(τ
2
,
α
2
A2
)
u
(1)
k−4/9,
u
(1)
k−2/9 = W
(τ
2
,
1
4
A1
)
u
(1)
k−3/9, v
(1)
k−1/9 = W
(τ
2
,
α
2
A2
)
u
(1)
k−2/9,
u
(1)
k = W
(τ
2
,
α
2
A2
)
u
(1)
k−1/9.
(1.10)
To estimate an error of approximate solution we need the natural
powers (As, s = 2, 3, 4, 5) of the operator A = A1 +A2. They are usually
defined as follows:
A2 =
(
A2
1 + A2
2
)
+ (A1A2 + A2A1) ,
A3 =
(
A3
1 + A3
2
)
+
(
A2
1A2 + · · · + A2
2A1
)
+ (A1A2A1 + A2A1A2) ,
Analogously are defined As, s = 4, 5.
390 The fourth order...
It is obvious that the definition domain D (As) of the operator As
represents an intersection of definition domains of its addends.
Let us introduce the following notations:
‖ϕ‖A = ‖A1ϕ‖ + ‖A2ϕ‖ , ϕ ∈ D (A) ;
‖ϕ‖A2 =
∥∥A2
1ϕ
∥∥ +
∥∥A2
2ϕ
∥∥ + ‖A1A2ϕ‖ + ‖A2A1ϕ‖ , ϕ ∈ D
(
A2
)
,
where ‖·‖ is a norm in X. ‖ϕ‖As , (s = 3, 4, 5) is defined analogously.
The following theorem takes place:
Theorem 1.1. Let the following conditions be satisfied:
(a) There exists such τ0 > 0 that for any 0 < τ ≤ τ0 there exist
operators (I + τλγAj)
−1 , j = 1, 2, γ = 1, α, α, λ = α, α and they
are bounded. Besides, the following inequalities are true:
‖W (τ, γAj)‖ ≤ eωτ , ω = const > 0;
(b) The operator (−A) generates the strongly continuous semigroup
U (t, A) = exp (−tA), for which the following inequality is true:
‖U(t, A)‖ ≤ Meωt, M, ω = const > 0;
(c) U (s, A)ϕ ∈ D
(
A5
)
for any s ≥ 0;
(d) f(t) ∈ C4([0,∞); X); f ′(t) ∈ D
(
A3
)
, f ′′(t) ∈ D
(
A2
)
, f ′′′(t) ∈
D (A) and U (s, A) f (t) ∈ D
(
A5
)
for any fixed t and s (t, s ≥ 0) .
Then the following estimate holds:
‖u(tk) − uk‖ ≤ ceω0tktkτ
4
(
sup
s∈[0,tk]
‖U(s, A)ϕ‖A5
+ tk sup
s,t∈[0,tk]
‖U(s, A)f(t)‖A5 + sup
t∈[0,tk]
‖f(t)‖A4
+ sup
t∈[0,tk]
‖f ′(t)‖A3 + sup
t∈[0,tk]
‖f ′′(t)‖A2
+ sup
t∈[0,tk]
‖f ′′′(t)‖A + sup
t∈[0,tk]
‖f (IV )(t)‖
)
,
where c and ω0 are positive constants.
Remark 1.1. In the case when operators A1, A2 are matrices, it is
obvious that conditions (a) and (b) of the Theorem 1.1 are automatically
satisfied. Also these conditions are satisfied, if A1, A2 and A are self-
adjoint, positive definite operators, even more in this case ‖W (τ, γAj)‖ ≤
J. Rogava, M. Tsiklauri 391
1 and ‖U(t, A)‖ ≤ 1. The requirement (a) of the theorem puts the
condition for the spectrum of Aj . Namely, the spectrum of Aj must be
placed within sector with the angle less than 60 degrees, because in case
of turning of spectrum by ±60 degrees (this is caused by multiplying of
Aj on α2 = 1/3
(
cos 600 + i sin 600
)
and α2 parameters) the spectrum
area will stay in the positive (right) half-plane.
2. Auxiliary lemmas
Let us prove the auxiliary lemmas on which the proof of the Theo-
rem 1.1 is based.
Lemma 2.1. If the condition (a) of the Theorem 1.1 is satisfied, then
for the operator W (t, A) the following decomposition is true:
W (t, A) =
k−1∑
i=0
(−1)i ti
i!
Ai + RW,k (t, A) , k = 1, . . . , 5, (2.1)
where, for the residual member, the following estimate holds:
‖RW,k (t, A)ϕ‖ ≤ c0e
ω0ttk‖Akϕ‖, ϕ ∈ D(Ak), (2.2)
where c0 and ω0 are positive constants.
Proof. We obviously have:
(I + γA)−1 = I − I + (I + γA)−1
= I − (I + γA)−1 (I + γA − I) = I − γA (I + A)−1 .
From this for any natural k we can get the following expansion:
(I + γA)−1 =
k−1∑
i=0
(−1)i γiAi + γkAk (I + γA)−1 . (2.3)
Let us rewrite W (τ, A) in the following form:
W (τ, A) = S (τ, A) − 1
2
τAS (τ, A) +
1
12
τ2A2S (τ, A)
where
S (τ, A) =
(
I +
α
2
τA
)−1 (
I +
α
2
τA
)−1
.
Let us decompose S (τ, A) by means of the formula (2.3)), we obtain the
following recurrent relation:
S (τ, A) = I − α
2
τA
(
I +
α
2
τA
)−1
− α
2
τAS (τ, A) . (2.4)
392 The fourth order...
Let us decompose the rational approximation W (τ, A) according to
the formula (2.4) up to the first order, we obtain:
W (τ, A) = I − RW,1 (τ, A) , (2.5)
where
RW,1 (τ, A) = τA
(
α
2
(
I +
α
2
τA
)−1
− α + 1
2
S (τ, A)
)
+
1
12
τ2A2S (τ, A) .
Since (I + λτA)−1 is bounded according to the condition (a) of the
Theorem 1.1, therefore:
‖RW,1(τ, A)ϕ‖ ≤ c0e
ω0ττ ‖Aϕ‖ , ϕ ∈ D (A) . (2.6)
Let us decompose the rational approximation W (τ, A) according to
the formula (2.4) up to the second order:
W (τ, A) = I − τA
(α
2
I − α2
4
τA
(
I +
α
2
τA
)−1
+
1 + α
2
I
− α + αα
4
τA
(
I +
α
2
τA
)−1
− α + α2
4
τAS (τ, A)
)
+
1
12
τ2A2S (τ, A) = I − τA + RW,2(τ, A)
where
RW,2 (τ, A) =
α2 + α + αα
4
τA
(
I +
α
2
τA
)−1
+
3α + 3α2 + 1
12
S (τ, A)
=
α − 1
3 + α + 1
3
4
τA
(
I +
α
2
τA
)−1
+
3α + 3α − 1 + 1
12
S (τ, A)
= τ2A2
(
α
2
(
I +
α
2
τA
)−1
+
α
2
S (τ, A)
)
.
According to the condition (a) of the Theorem 1.1 we have:
‖RW,2(τ, A)ϕ‖ ≤ c0e
ω0ττ2
∥∥A2ϕ
∥∥ , ϕ ∈ D
(
A2
)
. (2.7)
Let us decompose the rational approximation W (τ, A) according to
the formula (2.4) up to the third order:
W (τ, A) = I − τA + τ2A2
(α
2
I − α2
4
τA
(
I +
α
2
τA
)−1
+
α
2
(
I − α
2
τA
(
I +
α
2
τA
)−1
− α
2
τAS (τ, A)
))
J. Rogava, M. Tsiklauri 393
= I − τA +
1
2
τ2A2 + RW,3 (τ, A) , (2.8)
where
RW,3(τ, A) = −τ3A3
(1 + 3α2
12
(
I +
α
2
τA
)−1
+
α2
4
R(τ, A)
)
= −τ3A3
(α
4
(
I +
α
2
τA
)−1
+
α2
4
R(τ, A)
)
.
According to the condition (a) of the Theorem 1.1 we have:
‖RW,3(τ, A)ϕ‖ ≤ c0e
ω0ττ3
∥∥A3ϕ
∥∥ , ϕ ∈ D
(
A3
)
. (2.9)
Let us decompose the rational approximation W (τ, A) according to
the formula (2.4) up to the fourth order:
W (τ, A) = I − τA +
1
2
τ2A2 − τ3A3
(α
4
I − α2
8
τA
(
I +
α
2
τA
)−1
+
α2
4
(
I − α
2
τA
(
I +
α
2
τA
)−1
− α
2
τAS (τ, A)
))
= I − τA +
1
2
τ2A2 − 1
6
τ3A3 + RW,4(τ, A), (2.10)
where
RW,4(τ, A) = τ4A4
(α2 + αα2
8
(
I +
α
2
τA
)−1
+
α3
8
S (τ, A)
)
= τ4A4
( α
12
(
I +
α
2
τA
)−1
+
α3
8
S (τ, A)
)
According to the condition (a) of the Theorem 1.1 we have:
‖RW,4(τ, A)ϕ‖ ≤ c0e
ω0ττ4
∥∥A4ϕ
∥∥ , ϕ ∈ D
(
A4
)
. (2.11)
Let us decompose the rational approximation W (τ, A) according to the
formula (2.4) up to the fifth order:
W (τ, A) = I − τA +
1
2
τ2A2 − 1
6
τ3A3
− τ4A4
( α
12
− α2
24
τA
(
I +
α
2
τA
)−1
+
α3
8
(
I − α
2
τA
(
I +
α
2
τA
)−1
− α
2
τAS (τ, A)
))
= I − τA +
1
2
τ2A2 − 1
6
τ3A3 − 1
24
τ4A4 + RW,5(τ, A), (2.12)
394 The fourth order...
where
RW,5(τ, A) = τ5A5
(2α2 + 3α3α
48
(
I +
α
2
τA
)−1
+
α4
16
S (τ, A)
)
= τ5A5
( α
24
(
I +
α
2
τA
)−1
+
α4
16
S (τ, A)
)
According to the condition (a) of the Theorem 1.1 we have:
‖RW,5(τ, A)ϕ‖ ≤ c0e
ω0ττ5
∥∥A5ϕ
∥∥ , ϕ ∈ D
(
A5
)
(2.13)
Lemma 2.2. If the conditions (a), (b) and (c) of the Theorem 1.1 are
satisfied, then the following estimate holds:
∥∥[
Uk (τ, A) − V k (τ)
]
ϕ
∥∥ ≤ ceω0tktkτ
4 sup
s∈[0,tk]
‖U(s, A)ϕ‖A5 , (2.14)
where c and ω0 are positive constants.
Proof. The following formula is true (see T. Kato [9, p. 603]):
A
t∫
r
U (s, A) ds = U (r, A) − U (t, A) , 0 ≤ r ≤ t. (2.15)
Hence we get the following expansion:
U(t, A) =
k−1∑
i=0
(−1)i t
i
i!
Ai + Rk(t, A), (2.16)
where
Rk(t, A) = (−A)k
t∫
0
s1∫
0
. . .
sk−1∫
0
U(s, A) ds dsk−1 · · · ds1. (2.17)
Let us decompose W operators in the expression of V (τ) according
to the formula (2.1) from right to left, so that each residual member be
of the fifth order. We shall have:
V (τ) = I − τA +
1
2
τ2A2 − 1
6
τ3A3 +
1
24
τ4A4 + RV,5 (τ) , (2.18)
where for the residual member according to the condition (a) of the
Theorem 1.1 we have the following estimate:
‖RV,5 (τ)ϕ‖ ≤ ceω0ττ5 ‖ϕ‖A5 , ϕ ∈ D
(
A5
)
. (2.19)
J. Rogava, M. Tsiklauri 395
From the (2.15) and (2.17) it follows:
U (τ, A) − V (τ) = R5 (τ, A) − RV,5 (τ) .
From here according to inequalities (2.16) and (2.18) we obtain the
following estimate:
‖[U (τ, A) − V (τ)] ϕ‖ ≤ ceω0ττ5 ‖ϕ‖A5 , ϕ ∈ D
(
A5
)
. (2.20)
The following representation is obvious:
[Uk(τ, A) − V k(τ)]ϕ =
k∑
i=1
V k−i(τ)[U(τ, A) − V (τ)]U i−1(τ, A)ϕ.
Hence, according to the conditions (a), (b), (c) of the Theorem 1.1
and inequality (2.19), we have the sought estimate
Lemma 2.3. Let the following conditions be satisfied:
(a) The operator A satisfies the conditions of the Theorem 1.1;
(b) f(t) ∈ C4([0,∞); X), and f(t) ∈ D
(
A4
)
, f (k)(t) ∈ D
(
A4−k
)
(k = 1, 2, 3) for every fixed t ≥ 0.
Then the following estimate holds
‖R5,k (τ)‖ ≤ ceω0ττ5
4∑
i=0
max
s∈[tk−1,tk]
∥∥f (i) (s)
∥∥
A4−i , (2.21)
where R5,k (τ) is a residual member of simpson formula,
R5,k (τ) =
tk∫
tk−1
U (tk − s, A) f (s) ds
− τ
6
(
f (tk) + 4U
(τ
2
, A
)
f
(
tk−1/2
)
+ U (τ, A) f (tk−1)
)
(2.22)
and where c and ω0 are positive constants, and f (0) (s) = f (s).
Proof. By means of changing variables, the integral in the equality (2.21)
takes the following form:
tk∫
tk−1
U (tk − s, A) f (s) ds =
τ∫
0
U (τ − s, A) f (tk−1 + s) ds.
396 The fourth order...
If we decompose the function f (tk−1 + s) into the Taylor series, and
expand the semigroup U (τ − s, A) according to formula (2.15), we ob-
tain:
U (τ − s, A) f (tk−1 + s) = P3,k (s) + R̃4,k (τ, s) , (2.23)
where
P3,k (s) =
(
I − (τ − s) A +
(τ − s)2
2
A2 − (τ − s)3
6
A3
)
f (tk−1)
+ s
(
I − (τ − s)A +
(τ − s)2
2
A2
)
f ′ (tk−1)
+
s2
2
(I − (τ − s)A) f ′′ (tk−1) +
s3
6
f ′′′ (tk−1) ,
R̃4,k (τ, s) =
1
6
U (τ − s, A)
s∫
0
(s − ξ)3 f (IV ) (tk−1 + ξ) dξ
+ R4 (τ − s, A) f (tk−1) + (τ − s) AR3 (τ − s, A) f ′ (tk−1)
+
(τ − s)2
2
A2R2 (τ − s, A) f ′′ (tk−1)
+
(τ − s)3
6
A3R1 (τ − s, A) f ′′′ (tk−1) .
Hence according condition (b) and (d) of the Theorem 1.1 we obtain the
following estimate:
R̃4,k (τ, s) ≤ ceω0ττ4
4∑
i=0
max
s∈[tk−1,tk]
∥∥f (i) (s)
∥∥
A4−i . (2.24)
From equality (2.21) with account of formula (2.22), we have:
R5,k (τ) =
τ∫
0
U (τ − s, A) f (tk−1 + s) ds
− τ
6
(
f (tk) + 4U
(τ
2
, A
)
f
(
tk−1/2
)
+ U (τ, A) f (tk−1)
)
=
τ∫
0
P3,k (s) ds +
τ∫
0
R̃4,k (τ, s) ds
− τ
6
(
P3,k (τ) + 4P3,k
(τ
2
)
+ P3,k (0)
)
− τ
6
R̃4,k (τ, 0) + 4R̃4,k
(
τ,
τ
2
)
+ R̃4,k (τ, τ) , (2.25)
J. Rogava, M. Tsiklauri 397
Because of Simpson’s formula is exact for the third order polynomial, for
R5,k (τ) we have:
R5,k (τ) =
τ∫
0
R̃4,k (τ, s) ds − τ
6
(
R̃4,k (τ, 0) + 4R̃4,k
(
τ,
τ
2
)
+ R̃4,k (τ, τ)
)
.
hence according to inequality (2.22), we have:
‖Rk,5 (τ)‖ ≤ ceω0ττ5
4∑
i=0
max
s∈[tk−1,tk]
∥∥f (i) (s)
∥∥
A4−i (2.26)
3. Proof of the theorem
Let us return to the proof of the Theorem 1.1.
Let us write formula (1.6) in the following form:
u(tk) = Uk(τ, A)ϕ +
k∑
i=1
Uk−i(τ, A)
(
F
(1)
i + R5,k (τ)
)
, (3.1)
where
F
(1)
k =
τ
6
(
f (tk) + 4U
(τ
2
, A
)
f
(
tk−1/2
)
+ U (τ, A) f (tk−1)
)
. (3.2)
Analogously let us present uk as follows:
uk = V k(τ)ϕ +
k∑
i=1
V k−i(τ)F
(2)
i , (3.3)
where
F
(2)
i =
τ
6
(
f (tk) + 4V
(τ
2
)
f
(
tk−1/2
)
+ V (τ) f (tk−1)
)
. (3.4)
From equalities (3.1) and (3.3) it follows:
u(tk) − uk =
[
Uk(τ, A) − V k(τ)
]
ϕ
+
k∑
i=0
[
Uk−i(τ, A)F
(1)
i − V k−i(τ)F
(2)
i
]
+
k∑
i=0
Uk−i(τ, A)Rk,5 (τ) =
[
Uk(τ, A) − V k(τ)
]
ϕ
398 The fourth order...
+
k∑
i=1
[(
Uk−i(τ, A) − V k−i(τ)
)
F
(1)
i + V k−i(τ)
(
F
(1)
i − F
(2)
i
)]
+
k∑
i=0
Uk−i(τ, A)R5,k (τ) . (3.5)
From formulas (3.2) and (3.4) we have:
F
(1)
k − F
(2)
k =
τ
6
(
4
(
U
(τ
2
, A
)
− V
(τ
2
))
f
(
tk−1/2
)
+ (U (τ, A) − V (τ)) f (tk−1)
)
(3.6)
From here, according to inequality (2.17) and Lemma 2.1 we obtain
the following estimate:
∥∥F
(1)
k − F
(2)
k
∥∥ ≤ ceω0ττ5 sup
t∈[tk−1,tk]
‖f(t)‖A4 . (3.7)
According to the Lemma 2.1 we have:
∥∥∥∥
k∑
i=1
(
Uk−i(τ, A)−V k−i(τ)
)
F
(1)
i
∥∥∥∥ ≤ ceω0tkt2kτ
4 sup
s,t∈[0,tk]
‖U(s, A)f (t)‖A5 .
(3.8)
From equality (3.5) according to inequalities (3.7), (3.8), (2.20) and
the condition (b) of the Theorem 1.1 we obtain sought estimation.
Remark 3.1. The decomposition scheme constructed in this paper can
be applied for solving heat equation, diffusion-reaction equation and
other multidimensional evolution problems.
References
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Contact information
Jemal Rogava,
Mikheil Tsiklauri
I. Vekua Institute of Applied Mathematics
Tbilisi State University
380043 University Street 2,
Tbilisi,
Georgia
E-Mail: mtsiklauri@gmail.com
|
| id | nasplib_isofts_kiev_ua-123456789-124365 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | English |
| last_indexed | 2025-12-07T16:06:24Z |
| publishDate | 2009 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Rogava, J. Tsiklauri, M. 2017-09-24T13:05:03Z 2017-09-24T13:05:03Z 2009 The fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem / J. Rogava, M. Tsiklauri // Український математичний вісник. — 2009. — Т. 6, № 3. — С. 385-399. — Бібліогр.: 14 назв. — англ. 1810-3200 2000 MSC. 65M12, 65M15, 65M55. https://nasplib.isofts.kiev.ua/handle/123456789/124365 In the present work symmetrized sequential type decomposition scheme of the fourth degree precision for the solution of inhomogeneous evolution problem is constructed. The fourth degree precision is reached by introducing the complex parameter α = 1/2 ± i(1/2√3) and by the approximation of the semigroup through the rational approximation. For the considered scheme the explicit a priori estimation is obtained. en Інститут прикладної математики і механіки НАН України Український математичний вісник The fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem Article published earlier |
| spellingShingle | The fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem Rogava, J. Tsiklauri, M. |
| title | The fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem |
| title_full | The fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem |
| title_fullStr | The fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem |
| title_full_unstemmed | The fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem |
| title_short | The fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem |
| title_sort | fourth order of accuracy sequential type rational splitting of inhomogeneous evolution problem |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/124365 |
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