High degree precision decomposition method for the evolution problem with an operator under a split form

Mod´elisation Math´ematique et Analyse Num´erique M2AN, Vol. 36, N^o4, 2002, pp. 693–704 DOI: 10.1051/m2an:2002030

HIGH DEGREE PRECISION DECOMPOSITION METHOD FOR THE EVOLUTION PROBLEM WITH AN OPERATOR UNDER

A SPLIT FORM

Zurab Gegechkori

, Jemal Rogava

and Mikheil Tsiklauri

Abstract. In the present work the symmetrized sequential-parallel decomposition method of the third degree precision for the solution of Cauchy abstract problem with an operator under a split form, is presented. The third degree precision is reached by introducing a complex coefficient with the positive real part. For the considered schema the explicita prioriestimation is obtained.

Mathematics Subject Classification. 65M12, 65M15, 65M55.

Received: June 20, 2001. Revised: October 8, 2001.

1. Introduction

The study of the approximated schemas of a solution of evolution problems leads to the conclusion that to each approximated schema there corresponds a definite operator (solving operator of a discrete problem), which approximates a solving operator (semigroup) of a source continuous problem. The opposite is also true:

constructing approximation of a continuous semigroup, we build an approximated schema of a solution of an evolution problem.

For example, if we apply Rotte’s method for a solution of an evolution problem, a solving operator of the obtained difference problem will be a discrete semigroup and we come to a problem of approximating a continuous semigroup with the help of discrete semigroups (in this case see Kato [14], Chap. IX).

In case of applying a decomposition method, the solving operator of the applicable decomposed problem generates the Trotter formula [23], or the Chernoff formula [1, 2], or a formula, which is a combination of these formulas. Therefore, the error estimation of a decomposition method is equivalent to a problem of approximating of a continuous semigroup using Trotter type formulas. Papers [12, 18] (see also [19], Chap. II) are dedicated to the error estimations of Trotter type formulas.

The schema of decomposition, associated with the Trotter formula, allows us to split Cauchy problem for an evolution equation with an operator A= A1+A2+...+Am to m problems correspondingly with operators A1, A2, ..., Am,which are solved sequentially on each time interval with the lengtht/n.

The decomposition schema, associated with the Chernoff formula, is known as a method of fractional steps (see Ianenko [11]).

Keywords and phrases. Decomposition method, Semigroup, Trotter formula, Cauchy abstract problem.

1 Iv. Javakhishvili Tbilisi State University, Tbilisi 380043, Georgia. e-mail:[email protected]

2 I. Vekua Institute of Applied Mathematics of Iv. Javakhishvili Tbilisi State University, Tbilisi 380043, Georgia.

e-mail:[email protected], [email protected]

c EDP Sciences, SMAI 2002

As it is known, the decomposition method is sufficiently general for obtaining economical schemas for the solution of the multidimensional problems of mathematical physics. They can be divided into two groups: the schemas of sequential account (Ianenko [11], Samarskii [20], Marchuk [17], Samarskii and Vabishchevich [21], Fryazinov [5], Diakonov [4], Temam [22], Gordeziani [7]) and the schemas of parallel account (Gordeziani and Samarskii [10], Gordeziani and Meladze [8, 9], Kuzyk and Makarov [16]). In [19] (see Chap. II) the explicit estimations for decomposition schemas of the parallel account are obtained, which were considered in [9]. At present, there are many works dedicated to the decomposition method (see [17, 21]).

In the above-stated works the schemas considered are of the first or second degree precision. As far as we know, high degree precision decomposition formulas in case of two addends for the first time were obtained in [3].

In the present work, a symmetrized sequential-parallel decomposition method of the third degree precision for the solution of the Cauchy abstract problem with operatorA=A1+A2+...+Am, is presented. For the considered schema the explicita priori estimation is obtained. Under explicit estimations we understand such a prioriestimations for an error of solution, where the constants of a right member do not depend on a solution of an initial continuous problem,i.e. are absolute.

2. Setting of the problem

Let us consider the Cauchy abstract problem in the Banach space X:

du(t)

dt +Au(t) = 0, t >0, u(0) =ϕ. (1)

HereAis a closed linear operator with the domainD(A), which is everywhere dense inX,ϕis a given element fromD(A).

Suppose that (−A) operator generates a strongly continuous semigroup{exp(−tA)}_t≥0, then the solution of the problem (1) is given by the following formula (see [13, 15]):

u(t) =U(t, A)ϕ, ϕ∈D(A), (2)

whereU(t, A)≡exp(−tA) is a strongly continuous semigroup.

LetA=A1+A2+...+Am, whereAj (j= 1,2, ..., m) are compactly defined, closed linear operators inX. Let us introduce a difference net domain:

ω_τ ={t_k=kτ, k= 1,2, ..., τ >0}·

Along with the problem (1) we consider two sequences of the following problems on each interval [t_k−1, t_k]:

dv¹_k(t)

dt +αA1v_k¹(t) = 0, dw_k¹(t)

dt +αAmw¹_k(t) = 0, v¹_k(t_k−1) =u_k−1(t_k−1), w¹_k(t_k−1) =u_k−1(t_k−1), dv²_k(t)

dt +αA2v²_k(t) = 0, dw²_k(t)

dt +αAm−1w²_k(t) = 0, v_k²(tk−1) =v_k¹(tk), w²_k(tk−1) =w_k¹(tk),

. . . . . . . . . . .

dv^m−1_k (t)

dt +αAm−1v^m−1_k (t) = 0, dw^m−1_k (t)

dt +αA2w_k^m−1(t) = 0, v_k^m−1(t_k−1) =v_k^m−2(t_k), w^m−1_k (t_k−1) =w^m−2_k (t_k),

dv^m_k (t)

dt +Amv^m_k (t) = 0, dw_k^m(t)

dt +A1w^m_k (t) = 0, (3)

v_k^m(t_k−1) =v_k^m−1(t_k), w^m_k (t_k−1) =w^m−1_k (t_k), dv_k^m+1(t)

dt +αAm−1v_k^m−1(t) = 0, dw^m+1_k (t)

dt +αA2w^m+1_k (t) = 0, v^m+1_k (t_k−1) =v^m_k(t_k), w^m+1_k (t_k−1) =w^m_k (t_k),

. . . . . . . . . . . . .

dv_k^2m−2(t)

dt +αA₂v_k^2m−2(t) = 0, dw^2m−2_k (t)

dt +αA_m−1w_k^2m−2(t) = 0, v_k^2m−2(tk−1) =v_k^2m−3(tk), w^2m−2_k (tk−1) =w^2m−3_k (tk), dv^2m−1_k (t)

dt +αA₁v^2m−1_k (t) = 0, dw^2m−1_k (t)

dt +αA_mw^2m−1_k (t) = 0, v_k^2m−1(tk−1) =v^2m−2_k (tk), w^2m−1_k (tk−1) =w_k^2m−2(tk).

Here α is a numerical complex parameter with Re (α) >0,u₀(0) = ϕ. Suppose that (−A_j), (−αA_j) and (−αA_j) (j= 1,2, ..., m) operators generate strongly continuous semigroups.

On each [t_k−1, t_k] (k= 1,2, ...) intervalu_k(t) are defined as follows:

u_k(t) =1 2

v_k^2m−1(t) +w^2m−1_k (t)

. (4)

We consider the functionuk(t) as an approximate solution of the problem (1) on the interval [tk−1, tk].

The above-stated schema in case of m= 2 addends is considered in [6].

We will need natural degrees of the operator A=A1+A2+...+Am (A^s, s= 2,3,4). In case of two addends (m= 2) they are defined as follows:

A²= A²₁+A²₂

+ (A1A2+A2A1), A³= A³₁+A³₂

+ A²₁A₂+...+A²₂A₁

+ (A₁A₂A₁+A₂A₁A₂), A⁴= A⁴₁+A⁴₂

+ A³₁A2+...+A³₂A1

+ A²₁A2A1+...+A²₂A1A2

+ (A1A2A1A2+A2A1A2A1). Analogously are definedA^s (s= 2,3,4) whenm >2.

Obviously, the domainD(A^s) of the operatorA^sis the intersection of the domains of its addends.

Let us introduce the following definitions:

kϕk_A=kA1ϕk+...+kAmϕk, ϕ∈D(A),

kϕk_A2= Xm i,j=1

kA_iA_jϕk, ϕ∈D A² ,

wherek·kis a norm inX,similarly are definedkϕk_As (s= 3,4). Theorem. Let the following conditions be satisfied:

(a) α= ¹₂±i₂^√1₃ i=√

−1

;

(b) (−γA_j), γ = 1, α, α (j= 1,2, ..., m)and (−A) operators generate strongly continuous semigroups, for which the following estimations hold correspondingly:

kU(t, γA_j)k ≤e^ωt,

kU(t, A)k ≤Me^ωt, M, ω= const>0;

(c)U(s, A)ϕ∈D A⁴

for every fixeds≥0. Then the following estimation holds:

kuk(tk)−u(tk)k ≤ce^ω0tktkτ³ sup

s∈[0,tk]kU(s, A)ϕk_A4, wherec, ω₀ are positive constants.

3. Auxiliary lemma

Let us prove the auxiliary lemma on which the proof of the theorem is based.

Lemma (see [6]). If the conditions (a) and (b) of the Theorem are satisfied andm= 2, then 1

2[U(τ, αA1)U(τ, A2)U(τ, αA1) +U(τ, αA2)U(τ, A1)U(τ, αA2)] =I−τA+1

2τ²A²−1

6τ³A³+R₄⁽²⁾(τ), (5) where the following estimation holds for R₄⁽²⁾(τ):

R⁽²⁾₄ (τ)ϕ≤ce^ω0ττ⁴kϕk_A4, ϕ∈D A⁴

. (6)

Here c, ω0 are positive constants.

Proof. According to the formula (see Kato [14], p. 603):

A Zt r

U(s, A) ds=U(r, A)−U(t, A), 0≤r≤t,

we can get the following expansion:

U(t, A) =^k−1X

i=0

(−1)ⁱtⁱ

i!Aⁱ+R_k(t, A), (7)

where

Rk(t, A) = (−A)^k Zt 0

s1

Z

0

...

sZk−1

0

U(s, A)dsdsk−1...ds1. (8)

Let us consider the operator in the left side of equality (5). Let us decompose both its items from right to left according to the formula (7), so that each residual member is of the fourth degree. Then, using elementary algebraic transformations, we will get the right side of equality (5), where for the residual memberR⁽²⁾₄ (τ) the following presentation is true:

R⁽²⁾₄ (τ) = 1

2[R1,2(τ) +R2,1(τ)], where

Ri,j(τ) =R4(τ, αAi)−τR3(τ, αAi)Aj+1

2τ²R2(τ, αAi)A²_j−1

6τ³R1(τ, αAi)A³_j +U(τ, αAi)R4(τ, Aj)−ατR3(τ, αAi)Ai+ατ²R2(τ, αAi)AjAi−1

2ατ³R1(τ, αAi)A²_jAi

−ατU(τ, αAi)R3(τ, Aj)Ai+1

2α²τ²R2(τ, αAi)A²_i −1

2α²τ³R1(τ, αAi)AjA²_i +1

2α²τ²U(τ, αAi)R2(τ, Aj)A²_i −1

6α³τ³R1(t, αAi)A³_i −1

6α³τ³U(τ, αAi)R1(τ, Aj)A³_i +U(τ, αA_i)U(τ, A_j)R₄(τ, αA_i), i, j= 1,2.

Hence, according to the formula (8) and condition (b) of the Theorem we obtain the estimation (6).

4. Proof of the theorem

Let us get back to the proof of the Theorem.

It is obvious, that according to the formula (2) for the system (3) we have:

v_k^j(tk) =U(τ, αAj)v_k^j−1(tk), j= 1,2, ..., m−1, v_k^m(t_k) =U(τ, A_m)v_k^m−1(t_k),

v^m+j_k (t_k) =U(τ, αA_m−j)v_k^m+j−1(t_k), j= 1,2, ..., m−1, wherek= 1,2, ...,

v_k⁰(tk) =uk−1(tk−1), u0(0) =ϕ.

Hence we have:

v_k^2m−1(tk) =V1(τ)uk−1(tk−1),

where

V1(τ) =U(τ, αA1)...U(τ, αAm−1)U(τ, Am)U(τ, αAm−1)...U(τ, αA1). Analogously we obtain that:

w^2m−1_k (tk) =V2(τ)uk−1(tk−1), where

V₂(τ) =U(τ, αA_m)...U(τ, αA₂)U(τ, A₁)U(τ, αA₂)...U(τ, αA_m). So according to the formula (4) we obtain:

u_k(t_k) =V(τ)u_k−1(t_k−1) =V^k(τ)ϕ, (9) where

V(τ) = 1

2(V1(τ) +V2(τ)).

Remark. The operatorV^k(τ) is a solving operator of the above considered decomposed problem. It is obvious that according to the condition of the Theorem (U(t, γA_i)≤e^ωt)

V^k(τ)≤e^ω1tk, (10) where ω₁ = (2m−1)ω. From here it follows the stability of the above-stated decomposition schema on each finite time interval.

Let us suppose that W(τ) is a combination (sum, product) of semigroups, generated by operators (−γAi) (i= 1,2, ..., m). Let us decompose all semigroups including in the operatorW(τ) according to the formula (7), multiply these decompositions, group together the similar members and define the coefficients of the mem- bers (−τA_i), τ²A_iA_j

and τ³A_iA_jA_k

(i, j, k= 1,2, ..., m) to be correspondingly [W(τ)]_i,[W(τ)]_i,j and [W(τ)]_i,j,k in the obtained decomposition.

If we decompose all semigroups in the V(τ) from right to left according to the formula (7) so that each residual member is of the fourth degree, we get the following formula:

V(τ) =I−τX^m

i=1

[V(τ)]_iAi+τ² X^m

i,j=1

[V(τ)]_i,jAiAj−τ³ X^m

i,j,k=1

[V(τ)]_i,j,kAiAjAk+R₄^(m)(τ). (11)

Similarly toR⁽²⁾₄ , according to the first inequality of the condition (b) of the theorem the following estimation is true forR^(m)₄ (τ) (m >2):

R^(m)₄ (τ)ϕ≤ce^ω2ττ⁴kϕk_A4, ϕ∈D A⁴

, (12)

wherec, ω2are positive constants.

It is obvious that:

[V (τ)]_i=1

2([V1(τ)]_i+ [V2(τ)]_i), i= 1,2, ..., m, [V (τ)]_i,j= 1

2

[V₁(τ)]_i,j+ [V₂(τ)]_i,j

, i, j= 1,2, ..., m, [V (τ)]_i,j,k=1

2

[V1(τ)]_i,j,k+ [V2(τ)]_i,j,k

, i, j, k= 1,2, ..., m.

Let us compute coefficients [V1(τ)]_i. Obviously, we get the corresponding members of these coefficients from decomposition of only those multipliers (semigroups) of the operator V1(τ) which are generated by opera- tors (−γAi). From decomposition of other semigroups only first addends (identical operators) will be used. So we have:

[V1(τ)]_i = [U(τ, Ai)]_i= 1. Analogously

[V2(τ)]_i = [U(τ, Ai)]_i= 1. So we have

[V(τ)]_i = 1, i= 1,2, ..., m.

Let us compute coefficients [V1(τ)]_i,j. Obviously, we get the corresponding members of these coefficients from decomposition of only those multipliers (semigroups) of the operator V1(τ) which are generated by opera- tors (−γAi) and (−γAj). From decomposition of other semigroups only first addends (identical operators) will be used. So we have:

[V1(τ)]_i,j= [U(τ, αAi1)U(τ, Ai2)U(τ, αAi1)]_i,j. Analogously

[V2(τ)]_i,j= [U(τ, αAi2)U(τ, Ai1)U(τ, αAi2)]_i,j,

where (i₁, i₂) is a pair ofiandj indices, arranged in an increasing order. According to the lemma we have:

1 2

[U(τ, αA_i1)U(τ, A_i2)U(τ, αA_i1)]_i,j+ [U(τ, αA_i2)U(τ, A_i1)U(τ, αA_i2)]_i,j

=1 2· So we have

[V(τ)]_i,j= 1

2, i, j= 1,2, ..., m.

Let us compute coefficients [V1(τ)]_i,j,k. Obviously, we get the corresponding members of these coefficients from decomposition of only those multipliers (semigroups) of the operator V₁(τ), which are generated by operators (−γA_i),(−γA_j) and (−γA_k). From decomposition of other semigroups only first addends (identical operators) will be used. So we have:

[V₁(τ)]_i,j,k= [U(τ, αA_i1)U(τ, αA_i2)U(τ, A_i3)U(τ, αA_i2)U(τ, αA_i1)]_i,j,k.

Analogously

[V2(τ)]_i,j,k= [U(τ, αAi3)U(τ, αAi2)U(τ, Ai1)U(τ, αAi2)U(τ, αAi3)]_i,j,k, where (i₁, i₂, i₃) is a triple ofi, j andkindices, arranged in an increasing order.

Firstly let us consider the case wheni=j=k,we have:

[V1(τ)]_i,j,k= [U(τ, Ai)]_i,i,i= 1 6 and

[V2(τ)]_i,j,k= [U(τ, Ai)]_i,i,i= 1 6·

Now let us consider the case when only two ofi, j, kindices are different. In this case we have:

[V₁(τ)]_i,j,k= [U(τ, αA_i1)U(τ, A_i2)U(τ, αA_i1)]_i,j,k and

[V₂(τ)]_i,j,k= [U(τ, αA_i2)U(τ, A_i1)U(τ, αA_i2)]_i,j,k.

where (i1, i2) is pair of different indices ofi, j and ktriple, arranged in an increasing order. According to the lemma we have:

[V(τ)]_i,j,k= 1 6·

Now let us consider the case when i, j, k indices are different. We have six variants. Let us consider each one separately:

Case 1. Ifi < j < k, then

[V1(τ)]_i,j,k= [U(τ, αAi)U(τ, αAj)U(τ, Ak)U(τ, αAj)U(τ, αAi)]_i,j,k

= [U(τ, αA_i)]_i[U(τ, αA_j)]_j[U(τ, A_k)]_k =α² and

[V2(τ)]_i,j,k= [U(τ, αAk)U(τ, αAj)U(τ, Ai)U(τ, αAj)U(τ, αAk)]_i,j,k

= [U(τ, A_i)]_i[U(τ, αA_j)]_j[U(τ, αA_k)]_k =α². So we have

[V(τ)]_i,j,k= 1

2 α²+α²

= 1 6· Case 2. Ifi < k < j,then

[V1(τ)]_i,j,k= [U(τ, αAi)U(τ, αAk)U(τ, Aj)U(τ, αAk)U(τ, αAi)]_i,j,k

= [U(τ, αA_i)]_i[U(τ, A_j)]_j[U(τ, αA_k)]_k=αα

and

[V2(τ)]_i,j,k= [U(τ, αAj)U(τ, αAk)U(τ, Ai)U(τ, αAk)U(τ, αAj)]_i,j,k= 0. So we have

[V(τ)]_i,j,k= 1 2αα= 1

6· Case 3. Ifj < i < k, then

[V1(τ)]_i,j,k= [U(τ, αAj)U(τ, αAi)U(τ, Ak)U(τ, αAi)U(τ, αAj)]_i,j,k= 0 and

[V2(τ)]_i,j,k= [U(τ, αAk)U(τ, αAi)U(τ, Aj)U(τ, αAi)U(τ, αAk)]_i,j,k

= [U(τ, αA_i)]_i[U(τ, A_j)]_j[U(τ, αA_k)]_k=αα.

So we have

[V(τ)]_i,j,k= 1 2αα= 1

6· Case 4. Ifj < k < i,then

[V₁(τ)]_i,j,k= [U(τ, αA_j)U(τ, αA_k)U(τ, A_i)U(τ, αA_k)U(τ, αA_j)]_i,j,k= 0 and

[V₂(τ)]_i,j,k= [U(τ, αA_i)U(τ, αA_k)U(τ, A_j)U(τ, αA_k)U(τ, αA_i)]_i,j,k

= [U(τ, αA_i)]_i[U(τ, A_j)]_j[U(τ, αA_k)]_k=αα.

So we have

[V(τ)]_i,j,k= 1 2αα= 1

6· Case 5. Ifk < i < j,then

[V1(τ)]_i,j,k= [U(τ, αAk)U(τ, αAi)U(τ, Aj)U(τ, αAi)U(τ, αAk)]_i,j,k

= [U(τ, αAi)]_i[U(τ, Aj)]_j[U(τ, αAk)]_k=αα and

[V2(τ)]_i,j,k= [U(τ, αAj)U(τ, αAi)U(τ, Ak)U(τ, αAi)U(τ, αAj)]_i,j,k= 0.

So we have

[V(τ)]_i,j,k= 1 2αα= 1

6·

Case 6. Ifk < j < i,then

[V1(τ)]_i,j,k= [U(τ, αAk)U(τ, αAj)U(τ, Ai)U(τ, αAj)U(τ, αAk)]_i,j,k

= [U(τ, Ai)]_i[U(τ, αAj)]_j[U(τ, αAk)]_k =α² and

[V2(τ)]_i,j,k= [U(τ, αAi)U(τ, αAj)U(τ, Ak)U(τ, αAj)U(τ, αAi)]_i,j,k

= [U(τ, αA_i)]_i[U(τ, αA_j)]_j[U(τ, A_k)]_k =α². So we have

[V(τ)]_i,j,k= 1

2 α²+α²

= 1 6· Finally, for any triple (i, j, k) we have:

[V(τ)]_i,j,k= 1 6· Inserting in (11) the obtained coefficients, we will get:

V(τ) =I−τX^m

i=1

A_i+1 2τ² X^m

i,j=1

A_iA_j−1 6τ³ X^m

i,j,k=1

A_iA_jA_k+R^(m)₄ (τ)

=I−τX^m

i=1

A_i+1

2τ² X^m

i=1

A_i

!₂

−1

6τ³ X^m

i=1

A_i

!₃

+R₄^(m)(τ)

=I−τA+1

2τ²A²−1

6τ³A³+R^(m)₄ (τ). (13)

According to the formula (7) we have:

U(τ, A) =I−τA+1

2τ²A²−1

6τ³A³+R4(τ, A). (14)

According to the second inequality of the condition (b) of the theorem the following estimation is true for R4(τ, A):

kR4(τ, A)ϕk ≤ce^ωττ⁴A⁴ϕ≤ce^ωττ⁴kϕk_A4. (15) According to the formulas (13) and (14) we have:

U(τ, A)−V(τ) =R₄(τ, A)−R^(m)₄ (τ). Hence using inequalities (12) and (15) we can get the following estimation:

k[U(τ, A)−V(τ)]ϕk ≤ce^ω2ττ⁴kϕk_A4. (16)

According to the formulas (2) and (9) we have:

u(tk)−uk(tk) =

U(tk, A)−V^k(τ) ϕ=

U^k(τ, A)−V^k(τ) ϕ

= Xk i=1

V^k−i(τ) [U(τ, A)−V (τ)]U((i−1)τ, A)ϕ.

Hence according to the inequalities (10) and (16) we can obtain the following estimation:

ku(t_k)−u_k(t_k)k ≤X^k

i=1

kV(τ)k^k−ik[U(τ, A)−V (τ)]U((i−1)τ, A)ϕk

≤X^k

i=1

e^ω1(k−i)τce^ω2ττ⁴kU((i−1)τ, A)ϕk_A4

≤ce^ω0tkτ⁴X^k

i=1

kU((i−1)τ, A)ϕk_A4

≤ce^ω0tktkτ³ sup

s∈[o,tk]kU(s, A)ϕk_A4.

5. Conclusion

In the case when operators A_1, A₂, ..., A_m are matrices, it is obvious that conditions of the theorem are automatically satisfied. Also conditions of the theorem are satisfied, ifA₁, A₂, ..., A_m and A are self-adjoint, positive definite operators. The requirement αA operator α= 1/√

3 cos 30⁰+isin 30⁰

must generate a strongly continuous semigroup puts the condition for the spectrum ofA. Namely, the spectrum of Amust be placed within sector with the angle less than 120 degrees, because in case of turning of spectrum by±30 degrees (this is caused by multiplying ofAonαparameter) the spectrum area will stay in the positive (right) half-plane.

Third degree precision is reached by introducing a complex parameter. Because of this, each equation of the given decomposed system is changed by a pair of real equations, unlike lower degree precision schemas. To solve the specific problem, (for example) the matrix factorization may be used, where the coefficients are the matrices of the second order, unlike lower degree precision schemas, where the common factorization may be used.

It must be noted that the sum of the absolute values of coefficients of the addends of transition operatorV(τ) equals to one, unlike the high degree precision decomposition schemas considered in [3]. Hence, the considered schema is stable for any bounded operatorsA₁, A₂, ..., A_m.

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