DIMENSION SPLITTING FOR TIME DEPENDENT OPERATORS

. In this paper we are concerned with the convergence analysis of splitting methods for nonautonomous abstract evolution equations. We introduce a framework that allows us to analyze the popular Lie, Peaceman– Rachford and Strang splittings for time dependent operators. Our framework is in particular suited for analyzing dimension splittings. The inﬂuence of boundary conditions is discussed.


ESKIL HANSEN AND ALEXANDER OSTERMANN
which fits into our framework.In Section 3, we prove first order convergence of the Lie splitting and in Section 4, we derive the corresponding result for the exponential Lie splitting.The Peaceman-Rachford splitting is analyzed in Section 5. Here, under an additional compatibility condition which is satisfied by smooth solutions in case of periodic boundary conditions, e.g., the method is shown to be second order convergent.Finally, we show in Section 6 how these results can be generalized to Banach spaces and to variable step sizes.
2. Analytic framework.Throughout the paper, we consider the nonautonomous evolution equation with linear unbounded operators on an arbitrary (complex) Hilbert space (H, • ).
The related operator norm will also be referred to as • .The evolution system of this linear equation will be denoted by U (t, τ ), i.e., u(t) = U (t, τ )u(τ ) for t ≥ τ.
Furthermore, C denotes a generic constant that assumes different values at different occurrences.
As we will conduct an analysis including parabolic problems, the following definition is adequate.Definition 2.1.A one-parameter family of operators {F (t) ; 0 ≤ t ≤ T } is called S-regular, if it satisfies the following properties: (a) For each t, the operator F (t) is maximal dissipative on H.(b) The domain D = D F (t) does not depend on t.
(c) The map t → F (t) is m times continuously differentiable with D F (k) (t) ⊇ D, and there exists a constant C D such that In our analysis below, we require m = 1 or m = 2.
Obviously, S-regular operators fulfill the Lipschitz bound We are now prepared to state the main hypothesis on the operators.
Assumption 1.The families of operators A(t), B(t) and L(t) are S-regular.
Assumption 1 implies that the operators generate C 0 semigroups of contractions and the related resolvents are all nonexpansive on H.Moreover, we infer from (2) the inclusion We refer to [7,10] for a general introduction to the linear semigroup theory and to evolution systems.Our convergence analysis below requires that the solution u(t) of ( 2) is sufficiently smooth in time.We will exploit this assumption here for later use.Differentiating (2) a second time gives Since the operator L (t) I − L(t) −1 is bounded on H, we infer from this identity Differentiating (3) once more gives which implies by (4) the relation In general, however, we cannot deduce from these relations that the solution u(t) belongs to D L(t) 3 .
The following example describes a standard situation where dimension splitting can be of advantage.
Example 2.2.Let Ω ⊂ R d be an open d-dimensional rectangle.We consider the parabolic initial-boundary value problem with initial value u(x, 0) = u 0 (x), x ∈ Ω, and homogeneous Dirichlet boundary conditions.Here L(x, t) denotes the second order uniformly strongly elliptic differential operator given by with smooth coefficients a i (x, t).We consider it as an unbounded operator L(t) = L(x, t) on the Hilbert space L 2 (Ω) with dense domain D L(t) = H 2 (Ω) ∩ H 1 0 (Ω).The dimension splitting L(t) = A(t) + B(t) is defined by It is well-known that the operators L(t), A(t) and B(t) satisfy Assumption 1, see [2, Section 5] and [7,10].
We remark that Assumption 1 is equally satisfied for periodic conditions.For Neumann boundary conditions, however, the domains D L(t) in general depend on time through the boundary conditions.Since this is in conflict with the assumed S-regularity, such problems are not covered by our framework.
We close this section with some notation.For the generator F of a C 0 semigroup on H and a real number h ≥ 0, we define the bounded operators ϕ 0 (hF ) = e hF and These operators satisfy the recurrence relation For a time dependent linear operator F (t) and a fixed time t = t j , we henceforth denote F j = F (t j ) for short.
3. Lie splitting.For a first order time discretization of (2) we consider the socalled Lie splitting.This is a one step method that, given a numerical approximation u n to the exact solution u(t) at time t = t n , defines an approximation u n+1 at time For its analysis, it is convenient to rewrite this scheme in a more compact form as The composition of these split step operators defines the discrete evolution system, denoted here by We note for later use that the stability S n,j ≤ 1 of the splitting trivially holds, as the resolvents in (10b) are both nonexpansive on H.
The convergence analysis further requires some relations between the operators at hand.In particular, we need some compatibility of the domains.Theorem 3.1.Let Assumptions 1 and 2 hold, and let the solution of (2) be twice differentiable with u ∈ L ∞ (0, T ; H).Then the Lie splitting is first order convergent, for 0 ≤ t n ≤ T .The positive constant C can be chosen uniformly on [0, T ] and, in particular, independent of n and h.
Proof.Our estimate relies on the telescopic identity Due to the stability bound S n,j+1 ≤ 1, it remains to bound the terms which are the local errors of the Lie splitting.For this purpose, let denote the propagator of an Euler-type method.By Assumption 1 this is again a bounded operator.The main idea of the proof consists in using the identity For estimating the second term on the right hand side, we consider one step of the auxiliary method with initial value on the exact solution.Inserting the exact solution into (15) gives with defects Subtracting then (15) from ( 16) shows at once the bound In view of ( 14) it thus remains to bound As all operators in S j and E j are evaluated at time t j , we are locally in the situation of constant operators and thus can apply the techniques of our previous paper [2].
To simplify notation we introduce the abbreviations and observe the relations Thus, we get the following equalities on D(L 2 j ) By inserting this into (17) the latter can be bounded in the desired way under the made assumptions.
4. Exponential Lie splitting.Another first order time discretization of ( 2) is achieved by the exponential Lie splitting, where the numerical approximation u n+1 at time t n+1 is defined by The composition S n,j of these split step operators is again defined by (11).The stability S n,j ≤ 1 of the splitting holds trivially, as the resolvents are both nonexpansive on H.For this splitting, we need the following compatibility of domains.
Assumption 3.There exists a constant C such that for all t ∈ [0, T ] as well as Here, [A(t), B(t)] = A(t)B(t) − B(t)A(t) denotes the commutator.For the exponential Lie splitting, the following convergence result holds.
Proof.The proof can be carried out along the lines of the proof of Theorem 3.1.As auxiliary scheme, however, we use an exponential integrator and choose E j = e hLj .
For estimating the local error of the frozen semiflow we start from the identity which is a consequence of (8).On the other hand, Taylor series expansion of the exact solution gives u(t j+1 ) = u(t j ) + hL j u(t j ) + δ j+1 (21) with defects Subtracting (20) from (21) shows at once the desired bound.
In order to bound the difference (17), we start with e hLj u(t j ) = e hBj u(t j ) + h 0 e sBj A j e (h−s)Lj u(t j )ds = e hBj u(t j ) + he hBj A j u(t j ) and use once more e hAj = I + hA j + h 2 ϕ 2 (hA j )A 2 j , to obtain e hBj u(t j ) = e hBj e hAj u(t j ) − he hBj A j u(t j ) − h 2 e hBj ϕ 2 (hA j )A 2 j u(t j ).Inserting this back into (22), the difference (17) can now be bounded as desired.

5.
Peaceman-Rachford splitting.The convergence result for the Lie splitting can be generalized to higher order methods.As an example, we consider here the popular Peaceman-Rachford splitting.A single time step of this method is given as with The operators (23b) define the discrete evolution system For this system, we have the following stability bound.
Lemma 5.1.Under Assumption 1, the stability bound holds with a constant C S that can be chosen uniformly on bounded time intervals 0 ≤ nh ≤ T and, in particular, independent of j, n and h.
Proof.It is well known that operators of the form are nonexpansive under Assumption 1.In order to bound the transition operator that arises between two consecutive steps, we employ once more Assumption 1. Altogether, we get the estimate which is the desired stability bound.
For a second order convergence result, we require the additional compatibility condition D L(t) ⊆ D B(t) and B(t) I − L(t) −1 ≤ C (24) as well as u(t) ∈ D L(t) 3 .
(25) The latter condition is satisfied for smooth solutions of the parabolic problem in Example 2.2, if the problem is equipped with periodic boundary conditions.We note, however, that temporal smoothness of the solution might not be sufficient to guarantee (25) in general.
For time-invariant operators, stability and convergence of the Peaceman-Rachford splitting have already been analyzed in [2,9].Theorem 5.2.Let Assumptions 1 and 2, and conditions (24), (25) hold, and let the solution of (2) be three times differentiable with u ∈ L ∞ (0, T ; H).Then the Peaceman-Rachford splitting (23) is second order convergent, The positive constant C can be chosen uniformly on [0, T ].It depends on the solution and its derivatives (as specified in the proof ), but it is independent of n and h.
Proof.Our estimate relies again on the telescopic identity (12).In order to bound the local error of the Peaceman-Rachford splitting, we consider as auxiliary method this time the implicit midpoint rule with propagator (a) Starting from the exact solution at time t j , the implicit midpoint rule is given by the two stage formula In order to analyze its local error, we insert the exact solution into the numerical scheme to obtain with defects Subtracting ( 27) from (28) shows at once the bound (b) In view of (26) it thus remains to bound ).An appropriate bound is again achieved with the techniques introduced in our paper [2].There, we considered time-invariant operators only.For sake of simplicity we introduce the notation and observe the following relations For the resolvent −1 one derives at once the following identities Thus, we get the following equalities on D(L 3 The leading operator β in this expression is combined with S n,j .Recall that such a structure is required for our stability bound in Lemma 5.1. (c) It remains to bound the term αab λ 1 u(t j ).We encounter here a small technical problem as the arising operators and the solution are evaluated at different times.Therefore, we expand with remainders Due to Assumption 1 these remainders satisfy the bound Using expansion (30), we obtain The term (32c) has already the desired form.In order to bound (32b), we write and employ Assumption 1 and (25).Finally, in order to bound (32a), we rewrite The term (32a) with (33b) is bounded at once by using (31), whereas the expression (33a) is rewritten as ) and inserted back into (32a).The arising term is then easily bounded with the help of (25).This concludes our proof.6. Generalizations.In this final section we sketch how the above results can be extended to more general situations.In particular, we discuss the Strang splitting and we indicate extensions to Banach spaces and to variable step sizes.Second order convergence (for problems with periodic boundary conditions) can be shown along the lines of the proof of Theorem 5.2 or, alternatively, as in the proof of Theorem 4.1.For the corresponding time-invariant computations, see also [5].In order to bound the local error S j+ 1  2 − U (t j+1 , t j ) u(t j ), (34) of the Strang splitting, we this time consider a Magnus integrator [1] with propagator M j+ 1 2 = e hnL(t n+1/2 ) as the auxiliary method.The local error of this integrator is analyzed in the same way as the local error of the exponential integrator, see the proof of Theorem 4.1.
6.2.Banach spaces.Our techniques can also be used in the case when H is only a Banach space.However, one has to be careful with the stability bounds.For the Lie splitting, a sufficient hypothesis is that the resolvents (I − hA j ) −1 and (I − hB j ) −1 are nonexpansive or at least bounded by 1 + Ch.Such a hypothesis, however, is in general not sufficient for the Peaceman-Rachford splitting, see [8, Appendix to Section 2] for an example.
For the exponential splittings, the following growth condition on the semigroups e sA(t) ≤ e ωs , e sB(t) ≤ e ωs , t ∈ [0, T ], s ≥ 0 is sufficient for the stability of the splitting methods.Here, ω denotes an arbitrary but fixed real number.
6.3.Variable step sizes.For notational simplicity, we have restricted our analysis so far to constant step sizes.Note, however, that the step size h only plays the role of a parameter in the consistency proofs.Moreover, the stability lemma for the Peaceman-Rachford splitting (Lemma 5.1) holds for variable step sizes, as well.Therefore, all convergence results of this paper extend at once to variable step sizes.

For a verification of Assumption 2
in the context of Example 2.2, see [2, Example 5.1].We are now in the position to formulate our first convergence result.

10 ESKIL
HANSEN AND ALEXANDER OSTERMANNis again straightforward under Assumption 1.