On averaged exponential integrators for semilinear wave equations with solutions of low-regularity

In this paper we introduce a class of second-order exponential schemes for the time integration of semilinear wave equations. They are constructed such that the established error bounds only depend on quantities obtained from a well-posedness result of a classical solution. To compensate missing regularity of the solution the proofs become considerably more involved compared to a standard error analysis. Key tools are appropriate filter functions as well as the integration-by-parts and summation-by-parts formulas. We include numerical examples to illustrate the advantage of the proposed methods.


Introduction
In this paper we are interested in solving abstract wave equations of the form q 00 ðtÞ ¼ ÀLqðtÞ þ Gðt; qðtÞÞ; t 2 ½0; t end ; qð0Þ ¼ q 0 ; q 0 ð0Þ ¼ q 0 0 ; ð1:1Þ in some Hilbert space H where L is a positive, self-adjoint operator and G is a sufficiently This article is part of the topical collection ''Waves 2019 invited papers'' edited by Manfred Kaltenbacher and Markus Melenk. regular nonlinearity (e.g., Fréchet-differentiable). Such equations arise in many physical models. A prominent example is the cubic wave equation o 2 t qðt; xÞ ¼ o 2 x qðt; xÞ þ qðt; xÞ 3 ; ðt; xÞ 2 ½0; t end Â I posed on some interval I R and equipped with appropriate initial and boundary conditions. Our aim is to construct and investigate time integration schemes for (1.1) under physically realistic assumptions, in particular finite energy conditions. Hence, the solution will in general be of low regularity and we thus restrict ourselves to second-order schemes. Clearly, a standard time integrator (e.g., a Runge-Kutta scheme or an exponential integrator) can only be applied to an abstract evolution equation if it is unconditionally stable due to the unbounded operator L.
In the finite dimensional case (dim H\1), unconditionally stable integrators (in the sense that L does not cause any restriction on the time step) for this equation were already considered in [8,12,17,24]. Such exponential (or trigonometric) integrators were shown to be second-order convergent while only assuming a finite-energy condition. This was somewhat surprising since usually, second-order (exponential) schemes need two bounded time derivatives of the solution in the error analysis. The key ingredient are certain matrix functions that act as filters. The effect of these filters is that they remove resonances in the local error, which, in contrast to a standard error analysis, enforce cancellation effects in the global error. In fact one can prove that local and global error are of the same order if the filters are chosen appropriately.
Recently, in [2,3] we presented a completely new technique to prove related results for ordinary differential equations by reformulating a trigonometric integrator as a Strang splitting applied to a modified problem. Using ideas from [20,21], a specific representation of the local error was derived, which allowed us to separate terms of order three (which can be treated in a standard way) and the leading local error term, which is of order two only. A carefully adapted Lady Windermere's fan argument is employed to treat these terms in the global error accumulation.
In this paper we prove error bounds for different classes of exponential integrators applied to an evolution equation (1.1) in a unified way. More precisely, we characterize the structure of the defects and the properties of filter functions which allow second-order convergence under a finite-energy condition in different abstract frameworks (i.e., in different function spaces), which define the assumptions on L, G, and the initial data. Within this framework we can handle various boundary conditions. We point out that our results are not restricted to globally Lipschitz continuous functions G, but also apply to locally Lipschitz ones that satisfy certain growth conditions. Our analysis thus covers the class of nonlinearities for which the existence of a classical solution can be guaranteed. In particular, this includes polynomial nonlinearities up to a certain degree which is determined by the spatial dimension and the corresponding Sobolev embeddings. In the one-dimensional case such equations with periodic boundary conditions and arbitrary high polynomial degree have been studied in [9] for nonlinearities with Lipschitz properties on a whole scale of Sobolev spaces. However, this rich structure is not available in our general framework and, in contrast to our work, well-posedness cannot be guaranteed.
Further work on exponential integration schemes for the time-integration of semilinear wave equations was conducted in [1] where a sine-Gordon equation is studied and the difficulties arise in the proper treatment of a single constant c ! 1 which induces high oscillations in time. In the paper [10] the approach from [9] was extended and in the onedimensional case a quasilinear wave equation with periodic boundary conditions was studied. However, they assume smooth coefficients and high regularity for the analysis. Exponential splitting schemes for linear evolution equations have been analyzed in [15]. The error estimates depend on commutator bounds that are not available in our scenario. Finally, we point out that we do not address long-term behavior as in [4,6].
The paper is organized as follows. In Sect. 2, we give an informal overview over the methods of interest, the main concepts, and the main results and also present numerical examples illustrating the main results of our work. In particular, the necessity of using averaging techniques in the regime of low-regularity is shown.
The informal overview is made rigorous in Sect. 3, where we introduce the analytic framework and a functional calculus which allows us to define the operator-valued filters and ensures well-posedness of the problem as well as of the schemes.
We further state the assumptions on the operator L, the nonlinearity G, and the initial data that on the one hand will guarantee the well-posedness of (1.1) and on the other hand allow to carry out the error analysis.
In Sect. 4 we characterize filter functions which allow to prove that the exact solution of the original problem and the solution of the averaged problem only differ up to terms of order s 2 , where s [ 0 denotes the step size. Section 5 provides a characterization of numerical methods in terms of the structure of their defects, which are necessary to derive error bounds.
Finally, Sects. 6 and 7 contain our main results the error bounds for one-step and for multistep methods, respectively.

Informal overview of methods, concepts and results
Before we present the analytical framework necessary to formulate our results rigorously, we first give an informal overview of the methods of interest, the main concepts, and the main results. In the finite dimensional case dim H\1 (which is not the situation of interest in this paper), all the approximations presented are well-defined and the statements valid. However, for evolution equations posed in appropriate function spaces, this is no longer true unless additional assumptions are imposed. Since some of them are rather technical, we postpone them to Sect. 3.

Problem statement: Second-order differential equation
Let L be a linear, self-adjoint, and positive-definite operator on H and G : ½0; t end Â H ! H. We consider the differential equation q 00 ðtÞ ¼ ÀLqðtÞ þ Gðt; qðtÞÞ; t 2 ½0; t end ; qð0Þ ¼ q 0 ; q 0 ð0Þ ¼ q 0 0 ; and assume that the solution q satisfies the finite-energy condition and inner product Obviously, A is skew-adjoint with respect to hÁ; Ái and hence has a purely imaginary point spectrum.

Methods
In the following we shortly present four different types of methods to discretize equation (2.2) in time with a constant stepsize s [ 0.

Strang splitting
The exact flows u A s and u f s of the two subproblems t 0 u 0 ¼ 1 are given explicitly by We consider the Strang splitting in the variants À A; f ; A Á and À f ; A; f Á given by

Corrected Lie Splitting
Next we consider the second-order corrected Lie splitting given by with the correction term rðt; uÞ : It is inspired by a fourth-order method of this type proposed in [22, 4.9.3 (c)]. Note that in the linear case, where f ðt; uÞ ¼ Fu, the correction term reduces to the commutator Hence, one can consider (2.4) as an approximation to the method which was considered in [26, (3.37)].

Exponential Runge-Kutta methods
General two-stage exponential Runge-Kutta methods are of the form U n ¼ e c2sA u n þ c 2 su 1 ðc 2 sAÞf À t n ; u n Á ; ð2:5Þ where c 2 2 ð0; 1 is a given quadrature node. Recall that the u-functions are defined as u kþ1 ðzÞ :¼ the method is second-order convergent for parabolic problems, see [18,Theorem 4.3.]. Popular choices are c 2 ¼ 1 2 , b 1 ¼ 0 or c 2 ¼ 1, b 2 ðzÞ ¼ u 2 ðzÞ. All our results also apply to the symmetric, but implicit exponential Runge-Kutta scheme from [5, Example 2.1] and to ERKN methods, e.g., those considered in [28]. The necessary modifications are straightforward so that we omit the details.

Exponential multistep methods
The two-step exponential multistep method from [19, (2.7)] u nþ1 ¼ e sA u n þ su 1 ðsAÞf t n ; u n ð Þ þ su 2 ðsAÞ f t n ; u n ð ÞÀf t nÀ1 ; u nÀ1 ð Þ ð Þ ; n ! 1 ; ð2:6Þ is derived from the variation-of-constants formula for the exact solution of (1.1) by approximating the nonlinearity f in the integral term by an interpolation polynomial using the last two approximations u nÀ1 ; u n .
In a similar manner we consider a method that was used in [7, (B 4)], namely u nþ1 ¼ e 2sA u nÀ1 þ 2se sA f ðt n ; u n Þ ; ð2:7Þ

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For A ¼ 0 it reduces to an explicit Nyström method, cf. method (1.13') in [14].

Averaged differential equation
Let v ¼ /; w : iR ! R be even (i.e., vðÀzÞ ¼ vðzÞ) and analytic functions satisfying vð0Þ ¼ 1. Then we define and an averaged nonlinearity e Gðt; qÞ :¼ e wGðt; e /qÞ : Using the block diagonal operators we consider the averaged differential equation The averaging is done such that the solution e u of (2.8) also satisfies a finite-energy condition (2.1) (with a modified constant e K , which is independent of s and n, cf., Lemma 4.2 below). In Theorem 4.1, we provide sufficient conditions on w; / such that uðtÞ À e uðtÞ k k Cs 2 ; t 2 ½0; t end ; where Á k k denotes the norm induced by hÁ; Ái.

Remark 2.1
We only consider fixed stepsizes in this paper. Introducing variable stepsizes would require the investigation of a whole family of averaged problems (2.8). In addition, our analysis below would involve a much higher technical effort.

Averaged methods
The main idea is to apply one of the numerical methods to the averaged equation (2.8) instead of the original one (2.2). Equivalently, one could modify the numerical scheme in an appropriate way using filter functions. This is illustrated in Fig. 1.
Since the solutions of (2.2) and (2.8) only differ by terms of order s 2 , one might hope for second-order accuracy if the method is of order two at least. In fact we will later see that in  the case of evolution equations, this intuition is not always justified, i.e., order reduction might appear. The main goal in this paper is to characterize the numerical methods, the assumptions on L and G, and the choice of the filter functions which lead to second-order error bounds.

Main results
Our main results, which are detailed in Theorem 6.2 for exponential one-step methods and in Sect. 7 for exponential multistep methods, are the following error bounds.
(a) The Strang splitting, the exponential Runge-Kutta, and the exponential multistep methods applied to the original equation (2.2) satisfy kuðt n Þ À u n k C 1 s : (b) All methods of Sect. 2.2 applied to the averaged equation (2.8) with appropriate filters /; w satisfy kuðt n Þ À u n k C 2 s 2 : The constants C 1 ; C 2 only depend on the initial value u 0 , the finite energy K, properties of G, and t end , but not on n and s. The strategy to prove these bounds is to split the error into two terms, namely uðt n Þ À u n k k uðt n Þ À e uðt n Þ k k þ e uðt n Þ À u n k k : ð2:9Þ The first term is bounded by Theorem 4.1, the second by Theorem 6.1 or Corollaries 7.1, and 7.2, respectively. A crucial step is to show that the averaged solution inherits the regularity of the original solution, which is done in Lemma 4.2.

Numerical example
In this section we illustrate the effect of averaging within numerical methods by approximating the solution of a variant of the sine-Gordon equation given on the torus T ¼ R=ð2pZÞ by q 00 ðtÞ ¼ DqðtÞ À qðtÞ þ m a sinðm i qðtÞÞ qðtÞ; ð2:10Þ with t 2 ½0; 1 and m i ; m a 2 L 1 ðTÞ. Note that for q 2 L 2 ðTÞ and GðqÞðxÞ :¼ m a ðxÞ sinðm i ðxÞ qÞ q ; we have G(q) in L 2 ðTÞ, but even for q 2 H 1 ðTÞ we cannot expect GðqÞ 2 H ðTÞ for any [ 0. Hence, the analysis of [9,10] does not apply to such non-smooth nonlinearities. For the spatial discretization, we used a Fourier spectral method in order to control the regularity of the solution. The initial values ðq 0 ; v 0 Þ 2 H 1 ðTÞ Â L 2 ðTÞ are constructed such that ðq 0 ; v 0 Þ 2 H 1 ðTÞ Â L 2 ðTÞ n H 1þ ðTÞ Â H ðTÞ for ¼ 10 À6 , see [16] for details.
In Fig. 2 we computed the approximate solution with the Strang splitting variant (2.3a), i.e., and without filters, i.e., / ¼ w ¼ 1, (red, crosses) with N ¼ 2 j , j ¼ 9; 10; 11, spatial grid points. The codes are available from the authors on request. We observe order reduction of the non-averaged scheme to order one in the stiff regime, while in the non-stiff regime, the two errors of both schemes are quite close. The non-stiff regime is characterized by time steps s for which u 1 ðsAÞ is invertible for all s\s 0 . Since A k k % N=2, this is true for s 0 % 4p=N. For abstract evolution equations, only the stiff regime is relevant, i.e., the limit N ! 1.

Analytical framework
We fix some notation for the rest of the paper. For Hilbert spaces X, Y, hÁ; Ái ¼ hÁ; Ái X denotes the scalar product on X and BðX; YÞ the set of all bounded operators T : X ! Y equipped with the standard operator norm kTk Y X . Further, C k ðX; YÞ is the space of all k-times Fréchet-differentiable functions from X to Y. We write W k;p ðXÞ, k 2 N 0 , 1 p 1, for the Sobolev space of order k with all (weak) derivatives in L p ðXÞ and abbreviate H k ðXÞ :

Second-order equation
Let H be a real, separable Hilbert space and L : DðLÞ H ! H be a positive, self-adjoint operator with compact resolvent. We consider the abstract second-order evolution equation To reformulate it as a first-order system we use the intermediate space We exemplify the abstract framework considered in the rest of the paper by a class of semilinear wave equations.
In the following we recall sufficient conditions on the nonlinearity G to guarantee wellposedness of the equation and to establish the error analysis presented in Sects. 4, 5, 6, and 7.
The most subtle assumption is given now. It states the necessary regularity for G evaluated at a sufficiently smooth function.
The next assumption states bounds of G and J G . We point out that the dependency of the constants arising from different radii is crucial for the error analysis.
Assumption 3.4 (Regularity of G) There are constants C ¼ CðrÞ such that for given r V ; r L [ 0 and q with q k k V r V , q k k DðLÞ r L , p 2 V, and t 2 ½0; t end the following inequalities are satisfied: For the corrected Lie Splitting (2.4) we assume in addition for p i k k V r V ; i ¼ 1; 2; Remark 3.5 Note that shifting G to G þ cI for some c 2 R does not affect the validity of Assumptions 3.2 to 3.4. Hence, we can also treat positive semidefinite operators L by applying a shift.
(a) In Example 3.1 the additional regularity q 2 Cð½0; t end ; DðLÞÞ is sufficient to verify the Assumption (A1). (b) For G 2 C 1 ð½0; t end Â V; VÞ the chain rule immediately yields Assumption (A1).
However, in Example 3.1 with H ¼ H À1 ðXÞ and V ¼ L 2 ðXÞ, this would imply that G is already affine-linear, see [11,Sect. 3]. Hence, not even the function q 7 ! sinðqÞ would be covered by the analysis.

First-order equation
We consider the first-order formulation (2.2) of equation (1.1) on the separable Hilbert space X ¼ V Â H. The skew-adjoint operator A is given on its domain DðAÞ ¼ DðLÞ Â V. Hence, A is the generator of a unitary group e tA ð Þ t2R . We call u a classical solution of (2.2) on ½0; t Ã Þ if u solves (2.2), uð0Þ ¼ u 0 , and u 2 C 1 ð½0; t end ; XÞ \ Cð½0; t end ; DðAÞÞ ð3:3Þ for any t end \t Ã . The Assumptions 3.2 to 3.4 are translated into this setting by means of the following three lemmas. The first one provides a classical solution of (2.2) by standard semigroup theory. All statements in the lemmas directly follow from the special structure of f and the assumptions in Sect. 3.1. 2) satisfies f 2 C 1 ð½0; t end Â X; XÞ with Fre´chet derivative J f À t; u Á 2 B ½0; t end Â X; X ð Þfor all u 2 X and t 2 ½0; t end .
The following lemma shows differentiability of f in the stronger DðAÞ norm.

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The next Lemma contains two Lipschitz properties of f which easily follow from the corresponding bound on the derivative. They are crucial for the forthcoming error analysis.
Lemma 3.9 (Regularity of f ) Let G satisfy Assumption 3.4. Then there are constants C ¼ CðrÞ such that for given r X ; r A [ 0 and u i with u i k k r X , u i k k DðAÞ r A , i ¼ 1; 2, v 2 X, and t 2 ½0; t end the following inequalities are satisfied: In the following we refer to (3.4) as the generalized finite-energy condition.

Remark 3.11
(a) Note that for u ¼ À q; q 0 Á in the situation of Example 3.1 with H ¼ H À1 ðXÞ, the generalized finite energy condition implies This corresponds to the finite energy condition used in [8,12,17,24]. (b) The bound (3.4) also implies

Filter
From the compact resolvent property of L and the compact embeddings we can infer that also A has a compact resolvent. Hence, A admits an orthonormal basis of eigenvectors where M N and k k 2 R. Any x 2 X can thus be represented as with the equivalence x 2 DðAÞ () X k2M jk k a k j 2 \1 : This enables us to define the following functional calculus on the set satisfies the following properties: (a) W A is linear (b) hðAÞ k k X X khk 1 (c) ðghÞðAÞ ¼ gðAÞhðAÞ (d) For x 2 DðAÞ it holds hðAÞx 2 DðAÞ and AhðAÞx ¼ hðAÞAx For the construction of the integrators we make use of filter functions.

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Definition 3. 13 Let v 2 C b iR ð Þ. We call v a filter of order m, m ¼ 1; 2, if the following properties are satisfied. There exist #; H 2 C b iR ð Þ such that for all z 2 iR 1 À vðzÞ ¼ z m #ðzÞ ; ðF2Þ In addition, for m ¼ 2, v is symmetric, i.e.
By Theorem 3.12 we can define a corresponding class of filter operators that we later use in the averaged schemes.
Here, p i : X ! X denotes the projection onto the i-th component.
Proof The properties (OF1), (OF3), (OF4) directly follow from the functional calculus and (OF2) is a direct consequence of (OF4). To prove (OF5), we use the fact that we can approximate v uniformly on iR by even rational functions as lim x!AE1 vðixÞ ¼ 0, see [27,Sect. 1.6]. Hence, the assertion is true since it is easily verified for functions of the type (a) An example for m ¼ 2 is the short average filter proposed in [8] that we used in (2.11). We note that in this example vðixÞ ¼ sincð x 2 Þ holds for all x 2 R, which relates our filters to the ones considered in [13,Chapter XIII.].

Averaged problem
In this section we bound the difference between the solution e u of the averaged equation (2.8) and the solution u of (2.2). Note that by Proposition 3.10, a unique classical solution e u of (2.8) exists since the assumptions on f also hold for e f . In order to apply (A5a 0 ) we define r X via max t2½0;t end uðtÞ k k C emb K ¼: The constant C av and s 0 depend on r X , u 0 , t end , the finite energy K defined in (2.1), the filter functions, and the embedding constant C emb , but not on s.
In particular, e u exists on ½0; t end and is bounded by max t2½0;tend e uðtÞ k k 3 4 r X : Proof Let e t Ã [ 0 be the maximal existence time of e u and define t 0 :¼ supfs 2 ð0; e t Ã Þ j max t2½0;s e uðtÞ k k r X g : We first observe that for t minft 0 ; t end g the variation-of-constants formula yields since these bounds are sufficient to apply a Gronwall lemma which shows the assertion for all t minft 0 ; t end g.
To bound I 1 we use integration-by-parts and (OF3) to obtain where we used that f À s; uðsÞ Á is differentiable in X. By Assumptions (A3 0 ), (A4b 0 ), and the bound (3.4) on u 0 we have This proves (4.3) for j ¼ 1.
By assumption (A1 0 ) we also have Again integration-by-parts and Assumptions (A1 0 ) and (A4b 0 ) yield the desired bound This proves t 0 ! t end and hence (4.1) holds on ½0; t end for all s s 0 . h In the next lemma we show that e u inherits the regularity of u uniformly in s. where s 0 and the constants d C av and e K depend on r X , u 0 , t end , the finite energy K defined in (2.1), the filter functions, and the embedding constant C emb , but not on s.
Proof We proceed as in the proof of Theorem 4.1 and define t 0 by  With Remark 3.15, similar arguments as before yield OðsÞ bounds for AI 1 ðtÞ k k and AI 2 ðtÞ k k. By possibly reducing s 0 we obtain the result for 0 t t end . This immediately implies the first bound in (4.5) and the second bound is then obtained from (2.8). h

Abstract assumptions on the methods
In this section we characterize the classes of methods which are covered by our error analysis. We recall that u denotes the solution of the original problem (2.2) and e u the solution of the averaged problem (2.8). Further, we denote the numerical flow by S s and the defect by d n , i.e., a one-step method is given by u nþ1 ¼ S s ðt n ; u n Þ; d n ¼ S s À t n ; e uðt n Þ Á À e uðt nþ1 Þ: ð5:1Þ We start with an assumption on the stability of the method. where J : R Â DðAÞ Â X ! X is bounded by Next, we consider the consistency. where C [ 0 is independent of s and n.
For second-order methods, our analysis requires a particular structure of the defect. Before we state this in an abstract way, we briefly motivate it. Most of the methods we consider are constructed from the variation-of-constants formula (a) If /; w are filters of order 2, then there exist w n 2 X and a linear map W n : X ! DðAÞ which satisfy w n k k C; ð5:9aÞ W n k k X X C; ð5:9bÞ AW n k k X X C; ð5:9cÞ with a constant C which is independent of s and n such that d ðiÞ n can be written as  The constant C e and s 0 depend on u 0 , t end , the finite energy K defined in (2.1), the filter functions, and the embedding constant C emb , but are independent of s and n.
Proof The proof makes use of the error recursion from [12] and adapts techniques from Theorem 5.3 in [2]. Due to definition (5.1) of the defect d n , the global error e e n ¼ e uðt n Þ À u n can be written as e e nþ1 ¼ S s ðt n ; e uðt n ÞÞ À S s ðt n ; u n Þ À d n : By with a constant C d being independent of s and n. The proof is done by induction on n. For n ¼ 0, the statement is obviously true. Hence we assume that for all 0 k n it holds u k k k r X ; u k À e uðt k Þ k k C e s 2 ; C e :¼ C d e C J ð e K ;r X Þt end : To bound E j ðe sA À IÞ we exploit a telescopic sum to get E j ðe sA À IÞ ¼ X j k¼0 e ksA ðe sA À IÞ ¼ e ðjþ1ÞsA À I 2: Together with (5.9a) and (OF4) this yields (6.2) for d The terms can be estimated by (5.9b) and (5.9c) since u 1 ðzÞ j j 1 for z 2 iR. Next we consider ðe sA À IÞAF j for j n. After adding the exact solution we apply the variation-of-constants formula, (A3 0 ), and (4.5), which gives ðe sA e uðt k Þ À e uðt k þ sÞÞ þ A X j k¼0 ðe uðt k þ sÞ À e uðt k ÞÞ  if the method is applied to the averaged equation (2.8).
The constants C 1 ; C 2 and s 0 depend on u 0 , t end , the finite energy K defined in (2.1), the filter functions, and the embedding constant C emb , but are independent of s and n.
Proof Part (a) follows directly from Assumption 5.2 and equation (6.1). For part (b), we simply combine Theorem 4.1 and Theorem 6.1 by the triangle inequality (2.9). h 7 Main result for exponential multistep methods We briefly indicate how to extend the developed theory to the exponential multistep methods of Sect. 2.2.4. The first-order convergence as in part (a) of Theorem 6.2 is easily shown. To get second order, Assumption 5.1 needs to be modified. For method (2.6), we denote the numerical flow by S s ðt; v n ; v nÀ1 Þ and obtain S s ðt; v n ; v nÀ1 Þ À S s ðt; w n ; w nÀ1 Þ ¼ e sA À v n À w n Á þ sJ n ; where J n ¼ J À t; v n ; v nÀ1 ; w n ; w nÀ1 Á is bounded by This yields the following convergence result. where C and s 0 depend on u 0 , t end , the finite energy K defined in (2.1), the filter functions, and the embedding constant C emb , but are independent of s and n.
Proof We first employ Theorem 4.1 and Lemma 4.2, so again it remains to prove the error in approximating the filtered solution. As in the proof of [19,Thm. 4.3] the defect stems from a quadrature error that yields the dominant terms as in (5.6). Considering the defect d n ¼ S s À t n ; e uðt n Þ; e uðt nÀ1 Þ Á À e uðt nþ1 Þ ; For method (2.7) we have S s ðt; v n ; v nÀ1 Þ À S s ðt; w n ; w nÀ1 Þ ¼ e 2sA À v nÀ1 À w nÀ1 Á þ sJ n where J n ¼ J t; v n ; w n ð Þis bounded by J n k k C J À v n k k; w n k k Á v n À w n k k ; 8t 2 ½0; t end : In order to apply the techniques from above we define the modification and can state the following result.
Corollary 7.2 Let Assumptions 3.2 to 3.4 be valid and u be the classical solution of (2.2). Consider the numerical approximations ðu n Þ n from (2.7) applied to the averaged equation (2.8) with filters v 2 w; v 2 / where w; / are filters of order 2. Then there is a s 0 [ 0 and a constant C [ 0 such that for all s s 0 uðt n Þ À u n k k Cs 2 ; 0 t n ¼ ns t end ; where C and s 0 depend on u 0 , t end , the finite energy K defined in (2.1), the filter functions, and the embedding constant C emb , but are independent of s and n.
Proof Since the method stems from a midpoint rule applied to the variation-of-constants formula the defect is again given with dominant terms similar to (5.6) and (5.7). If we resolve the error recursion, we only obtain every second defect and the propagation is driven by e 2sA . As e z in (F3) is replaced by e 2z , this can be combined to conclude the assertion similar to the proof of Theorem 6.1. h