Consistency Analysis of Operator Splitting Methods for C0-Semigroups Expression

Consistency analysis of operator splitting methods for

C0-semigroups

Istvan Farago∗ Agnes Havasi†

Abstract

The consistency of different splitting schemes has mostly been studied for bounded op-erators by means of the traditional Taylor series expansion. In this paper second orderof consistency is proved for the symmetrically weighted sequential (SWS) splitting and theMarchuk-Strang (MS) splitting for unbounded generators of strongly continuous semigroups.

Keywords: Taylor series, strongly continuous semigroup, symmetrically weighted sequentialsplitting, Marchuk-Strang splitting, consistency order.

1 Introduction

Operator splitting is a widely used procedure in the numerical solution of large systems of partialdifferential equations. It allows us to replace an initial value problem with a sequence of simplerproblems, solved successively in each time step. Fields of application where splitting is useful orindispensable to apply include air pollution meteorology [17, 3], fluid dynamic models [9], cloudphysics [12] or biomathematics [7].

Operator splitting can be considered as a time-discretization method. Hence, it is naturalto raise the question: if the different sub-problems are solved exactly, under what conditiondoes the discretized solution (i.e., the solution obtained by splitting) converge to the exactsolution when the time step tends to zero? The answer is based on Lax’s equivalence theorem[10], which roughly speaking states that consistency and stability together imply convergence.Moreover, higher order of consistency yields faster convergence of the discretized solution to theexact solution. This motivates the theoretical investigation of the consistency order of splittingmethods.

The consistency of different splitting schemes has been thoroughly investigated in termsof the local splitting error [5, 15, 4]. These studies are based on the traditional power seriesexpansion of the exact solution and of the solution obtained by splitting. This method is notusually applicable for unbounded operators. However, for a special class of unbounded operators,the so-called generators of strongly continuous semigroups (or C0-semigroups) the Taylor seriesstill have a convenient form. By means of this formula, the consistency analysis of the sequentialsplitting has been performed for generators of C0-semigroups by Bjørhus [2]. The aim of ourpaper is to prove second order of consistency in a similar way for the symmetrically weighted

∗Eotvos Lorand University, Pazmany P. s. 1/c, 1117 Budapest, Hungary ([email protected]).†Eotvos Lorand University, Pazmany P. s. 1/a, 1117 Budapest, Hungary ([email protected]).

1

sequential (SWS) splitting and the Marchuk–Strang (MS) splitting for unbounded generators ofC0-semigroups.

The structure of the paper is as follows. In Section 2 we give an overview of the basicconcepts of semigroup theory as applied in the analysis of operator splitting. In Section 3 and 4the second order of consistency is proved for the SWS and MS splittings, respectively. Finally,in Section 5 we summarize the results.

2 Motivation, basic concepts

Consider the abstract Cauchy problem in a Banach space X

{u′(t) = A0u(t) t ∈ (0, T ],u(0) = u0,

(2.1)

where A0 : X → X is a closed, densely defined linear operator. Assume that A0 generates aC0-semigroup {S0(t)}t≥0. Then there exist constants ω0 ∈ IR and M0 ≥ 1 such that

‖S0(t)‖ ≤ M0eω0t t ≥ 0. (2.2)

Moreover, for any u0 ∈ D(A0), (2.1) has the unique solution (see e.g. [6])

u(t) = S0(t)u0, t ≥ 0. (2.3)

Assume thatA0 = A1 + A2, (2.4)

where A1 and A2 are generators of such C0-semigroups {S1(t)}t≥0 and {S2(t)}t≥0, which can beapproximated more easily than {S0(t)}t≥0 (Details about the approximation of semigoups canbe found in [1].) Furthermore, let

Dk = D(Ak1)

⋂D(Ak

2)⋂

D(Ak0) k = 1, 2, 3 dense in X

and Aki |Dk

, i = 0, 1, 2, k = 1, 2, 3 closed operators. (2.5)

Remark 2.1 If we assume that D(Ak1) = D(Ak

2) = D(Ak0), k = 1, 2, 3, and the resolvent sets

ρ(Ai), i = 0, 1, 2 are not empty, as it is assumed for k = 1, 2 in [2], then (2.5) are automaticallysatisfied. (See [8] and also [6], Appendix B, B.14).

Let us divide the time interval (0, T ] of the problem into N sub-intervals of equal length h =tn+1 − tn. The simplest way of operator splitting is the so-called sequential splitting, whereon each sub-interval (tn, tn+1], n = 0, 1, . . . N − 1 the approximate solution Un+1

s of u(tn+1) iscomputed as

Un+1s = Sspl(h)Un

s , (2.6)

where Sspl(h) = S1(h)S2(h). Several further splitting methods have been developed and appliedin different fields of applied mathematics. We will concentrate on the following two schemes:

1. SWS splitting [13, 4]: Sspl(h) = SSWS(h) = 12(S1(h)S2(h) + S2(h)S1(h)),

2. MS splitting [11, 14]: Sspl(h) = SMS(h) = S1(h/2)S2(h)S1(h/2).

2

In connection with the consistency of a splitting, we recall three definitions from [2].

Definition 2.2 Let Th : X × [0, T − h] → X be defined as

Th(u0, t) = S0(h)u(t)− Sspl(h)u(t). (2.7)

For each u0 and t, Th(u0, t) is called the local truncation error of the corresponding splittingmethod.

Definition 2.3 The splitting method is called consistent on [0, T ] if

limh→0

sup0≤tn≤T−h

‖Th(u0, tn)‖h

= 0 (2.8)

whenever u0 ∈ B, B being some dense subspace of X.

Definition 2.4 If in the consistency relation (2.8) we have

sup0≤tn≤T−h

‖Th(u0, tn)‖h

= O(hp), p > 0, (2.9)

then the method is said to be (consistent) of order p.

In [2] it is shown that the sequential splitting is a first-order consistent splitting scheme. As wewill see, both the SWS and MS splittings have second order. This is advantageous because thehigher consistency order provides faster convergence, and so a more accurate numerical solution.

The following formula will play a basic role in our investigations.

Theorem 2.5 For any C0-semigroup {S(t)}t≥0 of bounded linear operators with correspondinginfinitesimal generator A, we have the Taylor series expansion

S(t)x =n−1∑

j=0

tj

j!Ajx +

1(n− 1)!

∫ t

0(t− s)n−1S(s)Anxds for all x ∈ D(An), (2.10)

see [8], Section 11.8. Particularly, for n = 3, 2 and 1 we get the relations

S(h)x = x + hAx +h2

2A2x +

12

∫ h

0(h− s)2S(s)A3xds, (2.11)

S(h)x = x + hAx +∫ h

0(h− s)S(s)A2xds (2.12)

and

S(h)x = x +∫ h

0S(s)Axds, (2.13)

respectively. The following lemmas will also be helpful (see [16], Chapter II.6, Theorem 2).

Lemma 2.6 Let A and B be closed linear operators from D(A) ⊂ X and D(B) ⊂ X, respec-tively, into X. If D(A) ⊂ D(B), then there exists a constant C such that

‖Bx‖ ≤ C(‖Ax‖+ ‖x‖) for all x ∈ D(A). (2.14)

3

This implies that there exists a universal constant C by which for x ∈ Dk, k = 1, 2, 3

‖Aki x‖ ≤ C(‖Ak

j x‖+ ‖x‖) i, j = 0, 1, 2, (2.15)

where Dk are according to (2.5).

Lemma 2.7 Let A be an infinitesimal generator of a C0-semigroup {S(t)}t≥0. Let T > 0 andn ∈ IN arbitrary. If u0 ∈ D(An), then u(t) = S(t)u0 ∈ D(An) for 0 ≤ t ≤ T , and we have

sup[0,T ]

‖Aku(t)‖ ≤ Ck(T ), k = 0, 1, . . . , n, (2.16)

where Ck(T ) are constants independent of h.

Proof. Let z(t) = An−1u(t) = An−1S(t)u0 = S(t)An−1u0. Clearly, u0 ∈ D(An) impliesAn−1u0 ∈ D(A). It is known from the theory of C0-semigroups (see [6], Chapter II, Lemma 1.3)that then S(t)An−1u0 ∈ D(A), i.e., An−1u(t) ∈ D(A). Consequently, u(t) ∈ D(An).Moreover,

sup[0,T ]

‖Aku(t)‖ = sup[0,T ]

‖AkS(t)u0‖ = sup[0,T ]

‖S(t)Aku0‖ ≤ Me|ω|T ‖Aku0‖ (2.17)

for k = 0, 1, . . . , n.

3 Consistency of the SWS splitting

Our aim is to show the second-order consistency of the SWS splitting for C0-semigroups. Byusing (2.10) for n = 3, for x ∈ D we have

S2(h)S1(h)x = S1(h)x + hA2S1(h)x +h2

2A2

2S1(h)x +

12

∫ h

0(h− s)2S2(s)A3

2S1(h)xds (3.18)

and similarly,

S1(h)S2(h)x = S2(h)x + hA1S2(h)x +h2

2A2

1S2(h)x +

12

∫ h

0(h− s)2S1(s)A3

1S2(h)xds. (3.19)

Applying (2.11), (2.12) and (2.13) for the semigroups {S1(t)}t≥0 and {S2(t)}t≥0 and substitutingthe corresponding expressions into the first, second and third terms on the right-hand side of(3.18), we get

12[S2(h)S1(h)x + S1(h)S2(h)x] =

x + h(A1 + A2)x +h2

2(A1 + A2)2x

+14

∫ h

0(h− s)2S1(s)A3

1xds +14

∫ h

0(h− s)2S2(s)A3

2xds

4

+12hA2

∫ h

0(h− s)S1(s)A2

1xds +12hA1

∫ h

0(h− s)S2(s)A2

2xds (3.20)

+14h2A2

2

∫ h

0S1(s)A1xds +

14h2A2

1

∫ h

0S2(s)A2xds

+14

∫ h

0(h− s)2S2(s)A3

2S1(h)xds +14

∫ h

0(h− s)2S1(s)A3

1S2(h)xds.

On the other hand, we have

S0(h)x = x + hA0x +h2

2A2

0x +12

∫ h

0(h− s)2S0(s)A3

0xds, (3.21)

so the difference is12[S2(h)S1(h)x + S1(h)S2(h)x]− S0(h)x = (3.22)

+14

∫ h

0(h− s)2S1(s)A3

1xds +14

∫ h

0(h− s)2S2(s)A3

2xds (3.23)

+12hA2

∫ h

0(h− s)S1(s)A2

1xds +12hA1

∫ h

0(h− s)S2(s)A2

2xds (3.24)

+14h2A2

2

∫ h

0S1(s)A1xds +

14h2A2

1

∫ h

0S2(s)A2xds (3.25)

+14

∫ h

0(h− s)2S2(s)A3

2S1(h)xds +14

∫ h

0(h− s)2S1(s)A3

1S2(h)xds (3.26)

−12

∫ h

0(h− s)2S0(s)A3

0xds. (3.27)

Proposition 3.1 Let A0, A1 and A2 be infinitesimal generators of the C0-semigroups {S0(t)}t≥0,{S1(t)}t≥0 and {S2(t)}t≥0, respectively. Assume that (2.4) and (2.5) are satisfied, and let T > 0.Then for all x ∈ D the relation

∥∥∥∥12[S2(h)S1(h)x + S1(h)S2(h)x]− S0(h)x

∥∥∥∥ ≤ h3C(T )(‖A30x‖+ ‖A2

0x‖+ ‖A0x‖+ ‖x‖) (3.28)

holds for h ∈ [0, T ], where C(T ) is a constant independent of h.

Proof. We estimate the terms on the right-hand side of (3.22)–(3.27). We will often exploit thefact that the semigroups under consideration are C0-semigroups, and so

‖Si(t)‖ ≤ Mieωit ∀t ≥ 0, i = 0, 1, 2, (3.29)

where Mi ≥ 1, ωi ∈ IR are given constants. In the two terms under (3.23) and that under (3.27)we can make the following estimate:

∥∥∥∥∥∫ h

0(h− s)2Si(s)A3

i xds

∥∥∥∥∥ ≤ Mie|ωi|h‖A3

i x‖h3

3i = 0, 1, 2. (3.30)

For the first term in (3.24) by using Lemma 2.6 we can write∥∥∥∥∥12hA2

∫ h

0(h− s)S1(s)A2

1xds

∥∥∥∥∥ ≤ C

2h

∥∥∥∥∥A1

∫ h

0(h− s)S1(s)A2

1xds

∥∥∥∥∥ (3.31)

+C

2h

∥∥∥∥∥∫ h

0(h− s)S1(s)A2

1xds

∥∥∥∥∥ . (3.32)

5

Using (2.12) twice and the fact that all semigroups commute with their generator, we get

A1

∫ h

0(h− s)S1(s)A2

1xds = A1(S1(h)x− hA1x− x) =

= S1(h)A1x− hA21x−A1x =

∫ h

0(h− s)S1(s)A3

1xds. (3.33)

Hence, for term (3.31) we obtain the estimate

C

2h

∥∥∥∥∥A1

∫ h

0(h− s)S1(s)A2

1xds

∥∥∥∥∥ ≤ M1e|ω1|h‖A3

1x‖h3

4C. (3.34)

Term (3.32) can be estimated by

C

2h

∥∥∥∥∥∫ h

0(h− s)S1(s)A2

1xds

∥∥∥∥∥ ≤ M1e|ω1|h‖A2

1x‖h3

4C. (3.35)

So, ∥∥∥∥∥12hA2

∫ h

0(h− s)S1(s)A2

1xds

∥∥∥∥∥ ≤ M1e|ω1|h(‖A3

1x‖+ ‖A21x‖)

h3

4. (3.36)

Similarly, for the second term in (3.24) the following relation is valid:∥∥∥∥∥12hA1

∫ h

0(h− s)S2(s)A2

2xds

∥∥∥∥∥ ≤ M2e|ω2|h(‖A3

2x‖+ ‖A22x‖)

h3

4. (3.37)

For the estimate of the first term of (3.25) on the base of Lemma 2.6 we can write∥∥∥∥∥14h2A2

2

∫ h

0S1(s)A1xds

∥∥∥∥∥ =C

4h2

∥∥∥∥∥A21

∫ h

0S1(s)A1xds

∥∥∥∥∥ + (3.38)

+C

4h2

∥∥∥∥∥∫ h

0S1(s)A1xds

∥∥∥∥∥ , (3.39)

where for term (3.38) we have

C

4h2

∥∥∥∥∥A21

∫ h

0S1(s)A1xds

∥∥∥∥∥ =C

4h2

∥∥∥∥∥∫ h

0S1(s)A3

1xds

∥∥∥∥∥ ≤C

4h3M1e

|ω1|h‖A31x‖, (3.40)

and for term (3.39):C

4h2

∥∥∥∥∥∫ h

0S1(s)A1xds

∥∥∥∥∥ ≤C

4h3M1e

|ω1|h‖A1x‖. (3.41)

Consequently,∥∥∥∥∥14h2A2

2

∫ h

0S1(s)A1xds

∥∥∥∥∥ ≤ M1e|ω1|hC(‖A3

1x‖+ ‖A1x‖)h3

4. (3.42)

In a similar way, the second term of (3.25) is estimated by∥∥∥∥∥14h2A2

1

∫ h

0S2(s)A2xds

∥∥∥∥∥ ≤ M2e|ω2|hC(‖A3

2x‖+ ‖A2x‖)h3

4. (3.43)

6

For the first term of (3.26) one can write∥∥∥∥∥14

∫ h

0(h− s)2S2(s)A3

2S1(h)xds

∥∥∥∥∥ ≤ M2e|ω2|h‖A3

2S1(h)x‖h3

12≤ (3.44)

≤ M2e|ω2|hC(‖A3

1S1(h)x‖+ ‖S1(h)x‖)h3

12≤ M1e

|ω1|hM2e|ω2|hC(‖A3

1x‖+ ‖x‖)h3

12.

Here we have used that

‖A32S1(h)x‖ ≤ C(‖A3

1S1(h)x‖+ ‖S1(h)x‖). (3.45)

Finally, in a similar manner, the second term of (3.26) is estimated by∥∥∥∥∥14

∫ h

0(h− s)2S1(s)A3

1S2(h)xds

∥∥∥∥∥ ≤ M1e|ω1|hM2e

|ω2|hC(‖A32x‖+ ‖x‖)h

3

12. (3.46)

To prove the second-order consistency of the SWS splitting, we need a uniform bound, propor-tional to h3 on ∥∥∥∥

12[S2(h)S1(h)u(t) + S1(h)S2(h)u(t)]− S0(h)u(t)

∥∥∥∥ (3.47)

as t runs from 0 to T−h, where u(t) = S0(t)u0 is the exact solution of the original problem (2.1).

Proposition 3.1, (2.15) and Lemma 2.7 imply the following

Theorem 3.2 Let the conditions of Proposition 3.1 be satisfied. Then for any u0 ∈ D we havea uniform bound

∥∥∥∥12[S2(h)S1(h)u(t) + S1(h)S2(h)u(t)]− S0(h)u(t)

∥∥∥∥ ≤ h3C(T ), (3.48)

where C(T ) is a constant independent of h.

4 Consistency of the Marchuk-Strang splitting

Let us introduce the notation y = S2(h)S1(h/2)x, and let x ∈ D according to (2.5). Then wecan write

S1(h/2)S2(h)S1(h/2)x = S1(h/2)y, (4.49)

and by (2.10) we have

S1(h/2)y = y +h

2A1y +

h2

8A2

1y +12

∫ h/2

0

(h

2− s

)2

S1(s)A31yds. (4.50)

Substituting

y = S1(h/2)x + hA2S1(h/2)x +h2

2A2

2S1(h/2)x +12

∫ h

0(h− s)2S2(s)A3

2S1(h/2)xds, (4.51)

y = S1(h/2)x + hA2S1(h/2)x +∫ h

0(h− s)S2(s)A2

2S1(h/2)xds, (4.52)

7

y = S1(h/2)x +∫ h

0S2(s)A2S1(h/2)xds (4.53)

andy = S2(h)S1(h/2)x (4.54)

successively into the terms on the right-hand side of (4.50), and the expressions

S1(h/2)x = x +h

2A1x +

h2

8A2

1x +12

∫ h/2

0

(h

2− s

)2

S1(s)A31xds, (4.55)

S1(h/2)x = x +h

2A1x +

∫ h/2

0

(h

2− s

)S1(s)A2

1xds (4.56)

and

S1(h/2)x = x +∫ h/2

0S1(s)A1xds, (4.57)

we get

S1(h/2)S2(h)S1(h/2)x =

x + h(A1 + A2)x +h2

2(A1 + A2)2x

+12

∫ h/2

0

(h

2− s

)2

S1(s)A31xds + hA2

∫ h/2

0

(h

2− s

)S1(s)A2

1xds

+h2

2A2

2

∫ h/2

0S1(s)A1xds +

12

∫ h

0(h− s)2S2(s)A3

2S1(h/2)xds (4.58)

+h

2A1

∫ h/2

0

(h

2− s

)S1(s)A2

1xds +h2

2A1A2

∫ h/2

0S1(s)A1xds

+h

2A1

∫ h

0(h− s)S2(s)A2

2S1(h/2)xds +h2

8A2

1

∫ h/2

0S1(s)A1xds

+h2

8A2

1

∫ h

0S2(s)A2S1(h/2)xds +

12

∫ h/2

0

(h

2− s

)2

S1(s)A31S2(h)S1(h/2)xds.

Taking into account (3.21), we obtain that

S1(h/2)S2(h)S1(h/2)x− S0(h)x = (4.59)12

∫ h/2

0

(h

2− s

)2

S1(s)A31xds + hA2

∫ h/2

0

(h

2− s

)S1(s)A2

1xds (4.60)

+h2

2A2

2

∫ h/2

0S1(s)A1xds +

12

∫ h

0(h− s)2S2(s)A3

2S1(h/2)xds (4.61)

+h

2A1

∫ h/2

0

(h

2− s

)S1(s)A2

1xds +h2

2A1A2

∫ h/2

0S1(s)A1xds (4.62)

+h

2A1

∫ h

0(h− s)S2(s)A2

2S1(h/2)xds +h2

8A2

1

∫ h/2

0S1(s)A1xds (4.63)

+h2

8A2

1

∫ h

0S2(s)A2S1(h/2)xds +

12

∫ h/2

0

(h

2− s

)2

S1(s)A31S2(h)S1(h/2)xds (4.64)

−12

∫ h

0(h− s)2S0(s)A3

0xds. (4.65)

8

Proposition 4.1 Let A0, A1 and A2 be infinitesinal generators of the C0-semigroups {S0(t)}t≥0,{S1(t)}t≥0 and {S2(t)}t≥0, respectively. Assume that (2.4) and (2.5) are satisfied, and let T > 0.Then for all x ∈ D

‖S1(h/2)S2(h)S1(h/2)x− S0(h)x‖ ≤ h3C(T )(‖A30x‖+ ‖A2

0x‖+ ‖A0x‖+ ‖x‖) (4.66)

for h ∈ [0, T ], where C(T ) is a constant independent of h.

Proof. We estimate the terms on the right-hand side of (4.59)–(4.65) one by one. The firstterm of (4.60) can be directly estimated as

12

∥∥∥∥∥∫ h/2

0

(h

2− s

)2

S1(s)A31xds

∥∥∥∥∥ ≤ M1e|ω1|h/2‖A3

1x‖h3

48. (4.67)

For the second term of (4.60) by the use of Lemma 2.6 we can write∥∥∥∥∥hA2

∫ h/2

0

(h

2− s

)S1(s)A2

1xds

∥∥∥∥∥ ≤ Ch

∥∥∥∥∥A1

∫ h/2

0

(h

2− s

)S1(s)A2

1xds

∥∥∥∥∥ (4.68)

+Ch

∥∥∥∥∥∫ h/2

0

(h

2− s

)S1(s)A2

1xds

∥∥∥∥∥ . (4.69)

Here, by using (2.12) twice, the first integral can be written as

A1

∫ h/2

0

(h

2− s

)S1(s)A2

1xds =∫ h/2

0

(h

2− s

)S1(s)A3

1xds, (4.70)

and so term (4.68) can be estimated by

M1e|ω1|h/2‖A3

1x‖h3

8C. (4.71)

For term (4.69) the following estimate holds:

M1e|ω1|h/2‖A2

1x‖h3

8C. (4.72)

The first term of (4.61) is similar to the first term of (3.25), so we have∥∥∥∥∥h2

2A2

2

∫ h/2

0S1(s)A1xds

∥∥∥∥∥ ≤ M1e|ω1|h/2C(‖A3

1x‖+ ‖A1x‖)h3

4. (4.73)

The second term of (4.61) can be estimated as

12

∥∥∥∥∥∫ h

0(h− s)2S2(s)A3

2S1(h/2)xds

∥∥∥∥∥ ≤12M2e

|ω2|h‖A32S1(h/2)x‖h3

3. (4.74)

Since ‖A32S1(h/2)x‖ ≤ C(‖A3

1S1(h/2)x‖ + ‖S1(h/2)x‖) by Lemma 2.6, the right-hand side of(4.74) is estimated by

M1e|ω1|h/2M2e

|ω2|hC(‖A31x‖+ ‖x‖)h

3

6. (4.75)

9

For the first term of (4.62) we can write

h

2

∥∥∥∥∥A1

∫ h/2

0

(h

2− s

)S1(s)A2

1xds

∥∥∥∥∥ ≤ M1e|ω1|h/2‖A3

1x‖h3

16, (4.76)

see (4.70). For the second term one has∥∥∥∥∥h2

2A1A2

∫ h/2

0S1(s)A1xds

∥∥∥∥∥ ≤ Ch2

2‖A2

2

∫ h/2

0S1(s)A1xds‖+ C

h2

2‖A2

∫ h/2

0S1(s)A1xds‖,

(4.77)which, by Lemma 2.6, is less than or equal to

M1e|ω1|h/2C2(‖A3

1x‖+ ‖A21x‖+ 2‖A1x‖)h

3

4. (4.78)

Using again Lemma 2.6, the first term of (4.63) can be estimated as

h

2

∥∥∥∥∥A1

∫ h

0(h− s)S2(s)A2

2S1(h/2)xds

∥∥∥∥∥ ≤

≤ h

2C(‖A2

∫ h

0(h− s)S2(s)A2

2S1(h/2)xds‖+ ‖∫ h

0(h− s)S2(s)A2

2S1(h/2)xds‖) ≤

≤ M2e|ω2|hM1e

|ω1|h/2C2(‖A31x‖+ ‖A2

1x‖+ 2‖x‖)h3

4. (4.79)

For the second term of (4.63) we have

h2

8

∥∥∥∥∥A21

∫ h/2

0S1(s)A1xds

∥∥∥∥∥ ≤ M1e|ω1|h/2‖A3

1x‖h3

16. (4.80)

We estimate the first term of (4.64) as

h2

8

∥∥∥∥∥A21

∫ h

0S2(s)A2S1(h/2)xds

∥∥∥∥∥ ≤

h2

8C

∥∥∥∥∥A22

∫ h

0S2(s)A2S1(h/2)xds

∥∥∥∥∥ +h2

8C

∥∥∥∥∥∫ h

0S2(s)A2S1(h/2)xds

∥∥∥∥∥ ≤ (4.81)

h3

8C2M2e

|ω2|h‖A32S1(h/2)x‖+

h3

8C2M2e

|ω2|h‖A2S1(h/2)x‖ ≤

M1e|ω1|h/2M2e

|ω2|hC2(‖A31x‖+ ‖A1x‖+ 2‖x‖)h

3

8,

and the second term as

12

∥∥∥∥∥∫ h/2

0

(h

2− s

)2

S1(s)A31S2(h)S1(h/2)xds

∥∥∥∥∥ ≤

≤ h3

48M1e

|ω1|h/2C(‖A32S2(h)S1(h/2)x‖+ ‖S2(h)S1(h/2)x‖). (4.82)

Here

‖A32S2(h)S1(h/2)x‖ ≤ M2e

|ω2|h‖A32S1(h/2)x‖

≤ M2e|ω2|hC(‖A3

1S1(h/2)x‖+ ‖S1(h/2)x‖). (4.83)

10

Therefore

12

∥∥∥∥∥∫ h/2

0

(h

2− s

)2

S1(s)A31S2(h)S1(h/2)xds

∥∥∥∥∥ ≤

M1e|ω1|h/2M2e

|ω2|hM1e|ω1|h/2[C2(‖A3

1x‖+ ‖x‖) + C‖x‖]h3

48. (4.84)

Finally, for term (4.65) we have

12

∥∥∥∥∥∫ h

0(h− s)2S0(s)A3

0xds

∥∥∥∥∥ ≤12M0e

|ω0|h‖A30x‖

h3

3. (4.85)

Proposition 4.1, (2.15) and Lemma 2.7 imply the following

Theorem 4.2 Let the conditions of Proposition 4.1 be satisfied. Then for any u0 ∈ D we havea uniform bound

‖S1(h/2)S2(h)S1(h/2)u(t)− S0(h)u(t)‖ ≤ h3C(T ) (4.86)

where C(T ) is a constant independent of h.

5 Summary

Operator splitting can be considered as a time-discretization method. By Lax’s equivalencetheorem, the consistency and stability of a time-discretization method imply convergence. Inthis paper we investigated the consistency of two splitting schemes in an abstract setting: thesymmetrically weighted sequential (SWS) splitting and the Marchuk–Strang (MS) splitting. Wedid not deal with the numerical solution of the sub-problems, i.e., we assumed that the sub-problems are solved exactly.

The traditional Taylor series expansion, widely used for studying the consistency order ofsplitting schemes, is only applicable for bounded operators. By means of the semigroup theory,the first-order consistency of the sequential splitting has been shown in [2] for unbounded gen-erators of strongly continuous semigroups. In this paper we used the same technique to provesecond order of consistency in the SWS and MS splittings for generators of strongly continuoussemigroups.

The question of stability was not considered. Here we only remark that there are some spe-cial cases where the stability of a splitting method is easy to investigate. For example, if bothsemigoups {S1(t)}t≥0 and {S2(t)}t≥0 are contractive, then the considered splitting methods arestable. For more details in connection with the sequential splitting we refer to [2], Section 3.

Acknowledgements. The authors thank Andras Batkai, Eszter Sikolya and the unknownreferee for their useful suggestions. This work was supported by Hungarian National ResearchFounds (OTKA) N. T043765 and NATO Collaborative Linkage Grant N. 980505. The secondauthor is a grantee of the Bolyai Janos Scholarship.

References

[1] N. Bakaev, Linear Discrete Parabolic Problems, North-Holland Mathematics Studies, 203,Elsevier (2006).

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[2] M. Bjørhus, Operator splitting for abstract Cauchy problems, IMA Journal of NumericalAnalysis 18, 419-443 (1998).

[3] M. Botchev, I. Farago, A. Havasi, Testing weighted splitting schemes on a one-columntransport-chemistry model, Int. J. Env. Pol. 22, Nos. 1/2, 3-16. (2004)

[4] P. Csomos, I. Farago, I., Havasi, A., Weighted sequential splittings and their analysis, Comp.Math. Appl. 50 1017-1031 (2005).

[5] I. Dimov, I. Farago, A. Havasi, Z. Zlatev, L-commutativity of the operators in splittingmethods for air pollution models, Annales Univ. Sci. Sec. Math. 44, 127-148 (2001).

[6] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, GraduateTexts in Mathematics, 194, Springer, New York (2000).

[7] A. Gerisch, J. G. Verwer (2002) Operator splitting and approximate factorization for taxis-diffusion-reaction models. Applied Numerical Mathematics 42, pp. 159-176.

[8] E. Hille, R. S. Phillips (1957) Functional Analysis and Semi-groups, Vol. XXXI of Ameri-can Mathematical Society Colloquium Publications, revised edn. Providence, RI: AmericanMathematical Society.

[9] D. Lanser, J. G. Blom, J. G. Verwer (2001) Time integration of the shallow water equationsin spherical geometry. J. Comput. Phys. 1, pp. 86-98.

[10] P. Lax (2002) Functional Analysis, Wiley Interscience.

[11] G. I. Marchuk (1968). Some application of splitting-up methods to the solution of mathe-matical physics problems. Applik. Mat., 13(2), pp. 103-132.

[12] F. Muller (2001) Splitting error estimation for micro-physical–multiphase chemical systemsin meso-scale air quality models. Atmospheric Environment 35, pp. 5749-5764.

[13] G. Strang, Accurate partial difference methods I: Linear Cauchy problems, Archive forRational Mechanics and Analysis 12, pp. 392-402 (1963).

[14] G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer.Anal. 5, No. 3 (1968).

[15] J. G. Verwer, W. Hundsdorfer, Numerical solution of time-dependent advection-diffusion-reaction equations, Springer (2003).

[16] K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften 123, 6thed., Springer, Berlin (1980).

[17] Z. Zlatev, Computer Treatment of Large Air Pollution Models, Kluwer, 1995.

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