An exponential integrator scheme for time discretization of nonlinear stochastic wave equation
aa r X i v : . [ m a t h . NA ] N ov Journal of Scientific Computing manuscript No. (will be inserted by the editor)
An exponential integrator scheme for time discretization of nonlinearstochastic wave equation
Xiaojie Wang
Received: date / Accepted: date
Abstract
This work is devoted to convergence analysis of an exponential integrator scheme for semi-discretization in time of nonlinear stochastic wave equation. A unified framework is first set forth, whichcovers important cases of additive and multiplicative noises. Within this framework, the proposed schemeis shown to converge uniformly in the strong L p -sense with precise convergence rates given. The abstractresults are then applied to several concrete examples. Further, weak convergence rates of the scheme areexamined for the case of additive noise. To analyze the weak error for the nonlinear case, techniques basedon the Malliavin calculus were usually exploited in the literature. Under certain appropriate assumptions onthe nonlinearity, this paper provides a weak error analysis, which does not rely on the Malliavin calculus.The rates of weak convergence can, as expected, be improved in comparison with the strong rates. Bothstrong and weak convergence results obtained here show that the proposed method achieves higher conver-gence rates than the implicit Euler and Crank-Nicolson time discretizations. Numerical results are finallyreported to confirm our theoretical findings. Keywords nonlinear stochastic wave equation · multiplicative noise · exponential Euler scheme · strongconvergence · weak convergence Mathematics Subject Classification (2000) · · In the present paper, we consider stochastic evolution equation of Itˆo type in a separable Hilbert space ( U , h· , ·i U , k · k U ) , given by (cid:26) d ˙ u ( t ) = − L u ( t ) d t + F ( u ( t )) d t + G ( u ( t )) d W ( t ) , t ∈ ( , T ] , u ( ) = u , ˙ u ( ) = v , (1.1)where T ∈ ( , ¥ ) and L : D ( L ) ⊂ U → U is a densely defined, linear unbounded, positive self-adjointoperator with compact inverse (e.g., L = − D with homogeneous Dirichlet boundary condition). Under thisassumption, there exists an increasing sequence of real numbers { l i } ¥ i = and an orthonormal basis { e i } ¥ i = such that L e i = l i e i and 0 < l ≤ l ≤ · · · ≤ l n ( → ¥ ) . (1.2)In Equation (1.1), { u ( t ) } t ∈ [ , T ] is regarded as a U -valued stochastic process and { ˙ u ( t ) } t ∈ [ , T ] stands forthe time derivative of u . Moreover, F : U → U , G ( u ) : U → U for u ∈ U are deterministic mappings and Xiaojie Wang (Corresponding author)School of Mathematics and Statistics and School of Geosciences and Info-Physics, Central South University, Changsha 410083,Hunan, PR ChinaE-mail: [email protected], [email protected] Xiaojie Wang { W ( t ) } t ∈ [ , T ] is a cylindrical Q -Wiener process on a given probability space ( W , F , P ) with normal filtration { F t } t ∈ [ , T ] .The abstract equation (1.1) includes many stochastic wave equations (SWEs) in applications [5,11,35].Since their true solutions are rarely known explicitly, numerical simulations are often used to understandthe behavior of the solutions. To do this, one has to discretize both the time interval [ , T ] and the infinite di-mensional space U . As pointed out in [32], the main difficulty in studying numerical schemes for stochasticpartial differential equations (SPDEs) of evolutionary type lies in the treatment of the time discretization. Onthe one hand, development of effective high order time-stepping schemes are important since better timeintegration technology gives significant performance improvements in the numerical solution of SPDEs.This issue is, however, essentially difficult, even in the context of finite dimensional stochastic ordinarydifferential equations (SODEs) [21]. Time discretizations of SPDEs encounter all the difficulties that arisein the time approximations of both deterministic PDEs and finite dimensional SODEs as well as many moredue to the infinite dimensional nature of the driving noise processes. On the other hand, as one can see inour forthcoming work, the arguments used in the following analysis of pure time discretization can be ex-tended to the analysis of fully discretized scheme when combined with arguments used in the deterministictheory. For instance, optimal error estimates for the deterministic wave equation obtained in [24] allow usto analyze a fully discretized scheme with finite element spatial discretization. Therefore, and also for thesake of simplicity, we first concentrate on the time discretization of (1.1) in this paper.On the interval [ , T ] , we construct a uniform mesh T M = { t , t , · · · , t M } satisfying T M : 0 = t < t < t < · · · < t M = T (1.3)with t = T / M , M ∈ N , being the time stepsize. This article is concerned with the following scheme for thetime discretization of (1.1): ( u m + = C ( t ) u m + L − S ( t ) v m + tL − S ( t ) F ( u m ) + L − S ( t ) G ( u m ) D W m , v m + = − L S ( t ) u m + C ( t ) v m + t C ( t ) F ( u m ) + C ( t ) G ( u m ) D W m , (1.4)where C ( t ) = cos ( t L ) and S ( t ) = sin ( t L ) for t ∈ [ , T ] are the cosine and sine operators and D W m = W ( t m + ) − W ( t m ) is the Wiener increment. Here u m and v m are, respectively, temporal approximations of u ( t ) and ˙ u ( t ) at the grid points t m ∈ T M . Rewriting (1.4) in another abstract form (see (2.18)), one canobserve that the scheme (1.4) can be identified with a version of the stochastic exponential Euler scheme inthe literature [16,19,20,27,28], where the schemes were designed for parabolic SPDEs. By choosing par-ticular filter functions, the stochastic exponential Euler scheme can be viewed as a stochastic trigonometricmethod for nonlinear problems [7, Example 2]. Recently, Cohen, Larsson and Sigg [8] applied the stochastictrigonometric method to the linear SWE with additive noise, which coincides with the scheme (1.4) whenapplied to the linear case. In the first part of this article, we establish the uniform L p -convergence of thescheme (1.4) in a general framework for nonlinear problems, which of course extends the L -convergenceresults in [8] for the linear case.To get started, we make the following assumptions. Assumption 1.1
Assume F : U → U and G ( u ) : U → U for u ∈ U are deterministic mappings satisfying k F ( u ) k U + (cid:13)(cid:13) L d − G ( u ) (cid:13)(cid:13) L ≤ L ( k u k U + ) , (1.5) k F ( u ) − F ( u ) k U + (cid:13)(cid:13) L d − (cid:0) G ( u ) − G ( u ) (cid:1)(cid:13)(cid:13) L ≤ L k u − u k U (1.6) for all u , u , u ∈ U and some positive constants d , L ∈ ( , ¥ ) . For notations, we refer to the next section. Throughout this article, we choose L ≥ n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 3 result, Theorem 3.1, which shows that the scheme (1.4) is convergent uniformly in the strong L p -sense, p ∈ [ , ¥ ) . In particular, we show (cid:13)(cid:13)(cid:13) sup t m ∈ T M (cid:13)(cid:13) u ( t m ) − u m (cid:13)(cid:13) U (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ C t min ( d , ) . (1.7)Here, and throughout this work, C is a constant that may vary from one place to another and dependson T , d , L , p , l and the initial data u , v , but is independent of M . To derive the uniform bound (1.7), acontinuous-time extension of u m is introduced and the group property of the operator family E ( t ) = e tA is exploited. Then, several concrete examples including SWEs with general additive noise, multiplicativetrace class noise and multiplicative space-time white noise are presented to fit in the above setting. For thethree different cases, (1.7) is then applied with precise convergence rates given. It is shown that the spatialregularity of the noise term determines the order of strong convergence. Moreover, unlike the parabolic case,the exponential Euler scheme achieves higher strong order than the implicit Euler and Crank-Nicolson timediscretizations do (see also [23]). Another interesting finding is that, for the particular case of multiplicativetrace-class noise the scheme (1.4) can achieve the strong convergence order of one, which is, as one can seelater, the ultimate limit on the rate one can achieve for (1.4). This upper limit on convergence order can bealso found in [23].For the case of additive noise satisfying (cid:13)(cid:13) L − + b Q L − (cid:13)(cid:13) L ( U ) < ¥ , (1.8)we also measure the weak error (cid:12)(cid:12) E [ j ( u ( T ))] − E [ j ( u M )] (cid:12)(cid:12) for a class of test functions j : U → R with j ∈ C b ( U , R ) . We mention that the essential condition (1.8) has been used in [22,23] to carry out the weakconvergence analysis of numerical schemes for the linear SWE. Also, we emphasize that the presence of thenonlinear term leads to non-trivial technical difficulties in the analysis of weak error. Indeed, in the weakerror analysis for linear SPDEs with additive noise [13,22,23,33], whose solution can be written downexplicitly, one can get rid of the irregular term involving the unbounded operator by a transformation ofvariables. This transformation, however, does not work for the nonlinear heat equation. Since the operatorfamily E ( t ) = e tA is a group (see the next section), one can adapt the basic line of [12] to make the weakerror analysis easier. Instead of imposing very strong spatial regularity conditions on the nonlinearities [12,Hypothesis 5.7, 5.8], we make certain relatively mild assumptions on the nonlinearity F (c.f. Theorem 5.1and Corollary 5.2). For instance, the assumptions do not require F to be twice Fr´echet differentiable in U ,which is in general not fulfilled for Nemytskij operators but was commonly demanded in the weak erroranalysis [1,12,14,37]. Moreover, a key condition on F (see (5.3)) is used to avoid techniques involving theMalliavin calculus. Under such assumptions we obtain (cid:12)(cid:12) E [ j ( u ( T ))] − E [ j ( u M )] (cid:12)(cid:12) ≤ C e t min ( b , + b − e , ) (1.9)with arbitrarily small e >
0. As expected, the rate of weak convergence is, up to an arbitrarily small e > Xiaojie Wang thesis [25,26] for numerous references. The numerical research of stochastic wave equation, however, is inits beginning. Available results [6,8,15,24,22,23,30,34,36,38] are much less compared with the numericalanalysis of stochastic parabolic problems, which also partly motivates this work.A brief outline of this paper is as follows. In the next section, we collect some preliminaries and formu-late an abstract framework. In Section 3, we analyze the strong approximation error arising from the timediscretization. Then in Section 4, several examples are included, which fit in the abstract setting, to illustrateour abstract results. Weak convergence of the scheme is studied in Section 5 for the case of additive noise.Numerical results are presented at the end of this article.
Let ( U , h· , ·i U , k · k U ) and ( H , h· , ·i H , k · k H ) be two separable Hilbert spaces. By L ( U , H ) we denote thespace of bounded linear operators from U to H with the usual operator norm k · k L ( U , H ) and write L ( U ) = L ( U , U ) to lighten the notation. Additionally, we need spaces of nuclear and Hilbert-Schmidt operators [9,31]. The space of nuclear operators from U to H is denoted by L ( U , H ) and we write L ( U ) = L ( U , U ) .If G ∈ L ( U ) is nonnegative and symmetric, then k G k L ( U ) = Tr ( G ) : = ¥ (cid:229) i = h Gy i , y i i U , (2.1)where { y i } i ∈ N is an orthonormal basis of U and the trace of a nuclear operator, namely, Tr ( G ) for G ∈ L ( U ) , is independent of the particular choice of the basis { y i } i ∈ N . By N : = { , , , . . . } we denote thenatural numbers and by L ( U , H ) we denote the space of Hilbert-Schmidt operators from U to H , equippedwith the norm k G k L ( U , H ) = (cid:16) ¥ (cid:229) i = k Gy i k H (cid:17) / , (2.2)also not depending on the particular choice of the basis. Analogously, we write L ( U ) = L ( U , U ) . If G ∈ L ( U , H ) and G ∈ L j ( U ) , j = ,
2, then G G ∈ L j ( U , H ) for j = ,
2, and k G G k L j ( U , H ) ≤ k G k L ( U , H ) · k G k L j ( U ) , j = , . (2.3)Moreover, if G ∈ L ( H ) and G ∈ L ( H ) , then both G G and G G belong to L ( H ) andTr ( G G ) = Tr ( G G ) . (2.4)Let ( W , F , P ) be a probability space with a normal filtration { F t } ≤ t ≤ T and by L p ( W , U ) we denote thespace of U -valued integrable random variables with the norm defined by k X k L p ( W , U ) = (cid:0) E (cid:2) k X k pU (cid:3) (cid:1) p for p ∈ [ , ¥ ) . Furthermore, the driven stochastic process W ( t ) in (1.1) is assumed to be a cylindrical Q -Wienerprocess in the stochastic basis ( W , F , P , { F t } ≤ t ≤ T ) , with a covariance operator Q : U → U , which can berepresented as follows [9,31]: W ( t ) = ¥ (cid:229) i = √ q i b i ( t ) f i , t ∈ [ , T ] , (2.5)where { b i ( t ) } i ∈{ n ∈ N : q n > } are a family of mutually independent real Brownian motions and { f i } i ∈ N forman orthonormal basis of U consisting of eigenfunctions of Q with Q f i = q i f i , q i ≥ , i ∈ N . The covarianceoperator Q ∈ L ( U ) is nonnegative and symmetric, but not necessarily of finite trace. As a result, the seriesin (2.5) may not converge in U , but in some space U into which U can be embedded, see, e.g., [9,31]. Let Q denote the unique positive square root of Q . Now we are able to introduce the separable Hilbert space U : = Q ( U ) endowed with the inner product h u , ˆ u i = h Q − u , Q − ˆ u i U for u , ˆ u ∈ U , where Q − is thepseudo inverse of Q in the case when Q is not one-to-one. For lighter notation, we use L to denote theseparable Hilbert spaces L ( U , U ) and also L ( U , H ) when it causes no confusion. n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 5 At this point, we are ready to discuss the existence and uniqueness of the so-called mild solution of(1.1). To this end, we rewrite (1.1) as a stochastic evolution equation of first order in a product space H .Specifically, introducing a new variable v = ˙ u transforms (1.1) into the following Cauchy problem (cid:26) d X ( t ) = AX ( t ) d t + F ( X ( t )) d t + G ( X ( t )) d W ( t ) , t ∈ ( , T ] , X ( ) = X , (2.6)where we denote X = (cid:20) u v (cid:21) , X = (cid:20) uv (cid:21) , A = (cid:20) I − L (cid:21) , F ( X ) = (cid:20) F ( u ) (cid:21) , G ( X ) = (cid:20) G ( u ) (cid:21) . (2.7)Here and below by I we mean the identity operator in U . In this way, we transfer the existence and unique-ness of the mild solution of (1.1) to the same problem for (2.6). Before proceeding further, we need addi-tional spaces and notations. We introduce the Hilbert space ˙ H g = D ( L g ) for g ∈ R [25], equipped with theinner product h u , w i ˙ H g : = h L g u , L g w i U = ¥ (cid:229) i = l g i h u , e i i U h w , e i i U , g ∈ R , (2.8)and the corresponding norm k u k g = h u , u i ˙ H g . Then ˙ H = U and ˙ H a ⊂ ˙ H b if a ≥ b . Furthermore, weintroduce the product space H g : = ˙ H g × ˙ H g − , g ∈ R , (2.9)endowed with the inner product h X , X i H g = h u , u i ˙ H g + h v , v i ˙ H g − , X = ( u , v ) T , X = ( u , v ) T (2.10)and the corresponding norm k X k H g = h X , X i H g = k u k g + k v k g − , g ∈ R , X = ( u , v ) T . (2.11)For the special case g =
0, we denote H : = H = ˙ H × ˙ H − and ( H , h· , ·i H , k · k H ) is a separable Hilbertspace. Throughout this work we regard L as an operator from ˙ H to ˙ H − , defined by ( L u )( j ) = h (cid:209) u , (cid:209) j i for u , j ∈ ˙ H , and define D ( A ) = (cid:26) X = ( u , v ) T ∈ H : AX = h v − L u i ∈ H = ˙ H × ˙ H − (cid:27) = H = ˙ H × ˙ H . (2.12)In this setting one can rigorously check that A is closed and densely defined in H and the resolvent set of A contains all non-zero real numbers and k ( l I − A ) − k L ( H ) ≤ | l | for any l ∈ R (see, e.g., [26, Section5.3] for more details). Here and below, by I we mean the identity operator in H . As a consequence, theoperator A is an infinitesimal generator of a C -group E ( t ) = e tA , t ∈ R on H . In order to see the exact formof E ( t ) , note that X ( t ) = E ( t ) X is the solution of the deterministic linear equation˙ X = AX ; X ( ) = X = ( u , v ) T . (2.13)We solve it by using an eigenfunction expansion: X ( t ) = e tA X = ¥ (cid:229) i = exp (cid:16) t h I − l i i(cid:17)h h u , e i i U e i h v , e i i U e i i = ¥ (cid:229) i = (cid:20) cos ( √ l i t ) √ l i sin ( √ l i t ) −√ l i sin ( √ l i t ) cos ( √ l i t ) (cid:21) (cid:20) h u , e i i U e i h v , e i i U e i (cid:21) . Hence the first component of X = ( u , v ) T is given by u ( t ) = ¥ (cid:229) i = h cos ( p l i t ) h u , e i i U e i + √ l i sin ( p l i t ) h v , e i i U e i i = cos ( t L ) u + L − sin ( t L ) v , (2.14) Xiaojie Wang and the second component of X = ( u , v ) T by v ( t ) = ˙ u ( t ) = − L sin ( t L ) u + cos ( t L ) v . (2.15)Here we introduced the cosine and sine operators cos ( t L ) and sin ( t L ) . Accordingly, the semigroup E ( t ) = e tA should explicitly take the form as E ( t ) = e tA = (cid:20) C ( t ) L − S ( t ) − L S ( t ) C ( t ) (cid:21) , (2.16)where we further write C ( t ) = cos ( t L ) and S ( t ) = sin ( t L ) for brevity. Obviously, the cosine and sineoperators satisfy a trigonometric identity in the sense that k S ( t ) u k U + k C ( t ) u k U = k u k U for u ∈ U . Usingthe trigonometric identity gives k E ( t ) k L ( H ) ≤ , t ∈ R . (2.17)Also, we will frequently use the fact that S ( t ) , C ( t ) , t ∈ R and L g , g ∈ R commute in the following estimates.In the above setting, the scheme (1.4) can be rewritten as a recurrence equation in H : X m + = E ( t ) (cid:16) X m + t F ( X m ) + G ( X m ) D W m (cid:17) . (2.18)For the convergence analysis, it is convenient to work with continuous processes. Hence we define a con-tinuous extension of (2.18), ˜ X ( t ) = ( ˜ u ( t ) , ˜ v ( t )) T , by˜ X ( t ) = E ( t − t m ) (cid:16) X m + F ( X m )( t − t m ) + G ( X m )( W ( t ) − W ( t m )) (cid:17) (2.19)for t ∈ [ t m , t m + ] . It is obvious that ˜ X ( t ) coincides with X m at the grid-points t m ∈ T M . Note that (2.18)implies X m = E ( t m ) X + t m − (cid:229) k = E ( t m − t k ) F ( X k ) + m − (cid:229) k = E ( t m − t k ) G ( X k ) D W k . (2.20)Similarly, for arbitrary t ∈ [ , T ] , ˜ X ( t ) = ( ˜ u ( t ) , ˜ v ( t )) T defined by (2.19) can be expressed as˜ X ( t ) = E ( t ) X + Z t E ( t − ⌊ s ⌋ t ) F ( X ⌊ s / t ⌋ ) d s + Z t E ( t − ⌊ s ⌋ t ) G ( X ⌊ s / t ⌋ ) d W ( s ) , (2.21)where we define ⌊ s / t ⌋ as an integer number not bigger than s / t and ⌊ s ⌋ t = ⌊ s / t ⌋ · t .Next, we show the result on the existence and uniqueness of the mild solution of (2.6). Theorem 2.1 (Existence and uniqueness of mild solution)
Suppose all conditions in Assumption 1.1 arefulfilled, let W ( t ) , t ∈ [ , T ] be a cylindrical Q-Wiener process on the stochastic basis ( W , F , P , { F t } ≤ t ≤ T ) ,given by (2.5) , and let X = ( u , v ) T be an F -measurable H-valued random variable such that k X k L p ( W , H ) < ¥ for some p ∈ [ , ¥ ) . Then (2.6) has a unique mild solutionX ( t ) = E ( t ) X + Z t E ( t − s ) F ( X ( s )) ds + Z t E ( t − s ) G ( X ( s )) dW ( s ) , a . s . (2.22) for t ∈ [ , T ] . Moreover there exists a constant C p , T ∈ [ , ¥ ) depending on p , T such that sup t ∈ [ , T ] (cid:13)(cid:13) X ( t ) (cid:13)(cid:13) L p ( W , H ) ≤ C p , T (cid:0) k X k L p ( W , H ) + (cid:1) . (2.23) n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 7 Proof of Theorem 2.1.
Owing to (1.5), (1.6) and the definition of k · k H , we infer that k F ( X ) − F ( X ) k H = k L − (cid:0) F ( u ) − F ( u ) (cid:1) k U ≤k L − k L ( U ) · L k u − u k U ≤ ˜ L k X − X k H , (2.24) k F ( X ) k H = k L − F ( u ) k U ≤ ˜ L ( k u k U + ) ≤ ˜ L ( k X k H + ) (2.25)for arbitrary X = ( u , v ) T , X = ( u , v ) T , X = ( u , v ) T . Here ˜ L = k L − k L ( U ) · L = L / √ l . Similarly, onecan obtain that k G ( X ) − G ( X ) k L ( U , H ) = k L − (cid:0) G ( u ) − G ( u ) (cid:1) k L ( U , U ) ≤k L − d k L ( U ) · k L d − (cid:0) G ( u ) − G ( u ) (cid:1) k L ( U , U ) ≤ ˆ L k u − u k U ≤ ˆ L k X − X k H , (2.26) k G ( X ) k L ( U , H ) = k L − G ( u ) k L ( U , U ) ≤ ˆ L ( k u k U + ) ≤ ˆ L ( k X k H + ) (2.27)with ˆ L = L / l d / . In view of Proposition 7.1 and Theorem 7.4 in [9], the existence and uniqueness of themild solution (2.22) follow straightforwardly and (2.23) holds. (cid:3) In component-wise manner, the mild solution (2.22) takes the form u ( t ) = C ( t ) u + L − S ( t ) v + R t L − S ( t − s ) F ( u ( s )) d s + R t L − S ( t − s ) G ( u ( s )) d W ( s ) , v ( t ) = − L S ( t ) u + C ( t ) v + R t C ( t − s ) F ( u ( s )) d s + R t C ( t − s ) G ( u ( s )) d W ( s ) (2.28) a . s . for t ∈ [ , T ] . Therefore, u ( t ) in (2.28) serves as the unique mild solution of (1.1). This section focuses on the strong convergence of the scheme (1.4). We begin by presenting a slightlymodified version of the Burkholder-Davis-Gundy type inequality ([9, Lemma 7.2]).
Lemma 3.1
Let ( V , h· , ·i V , k · k V ) be a separable Hilbert space and let Y : [ , T ] × W → L ( U , V ) be apredictable stochastic process. Then for t ∈ [ , T ] and p ∈ [ , ¥ ) there exists a constant C p such that (cid:13)(cid:13)(cid:13)(cid:13) sup s ∈ [ , t ] (cid:13)(cid:13)(cid:13) s ∫ Y ( r ) dW ( r ) (cid:13)(cid:13)(cid:13) V (cid:13)(cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ C p (cid:18) t ∫ (cid:13)(cid:13)(cid:13) k Y ( r ) k L ( U , V ) (cid:13)(cid:13)(cid:13) L p ( W , R ) dr (cid:19) . (3.1)The following result is on further spatial regularity of the mild solution X ( t ) and its numerical approxima-tions. Proposition 3.1
Assume all conditions in Theorem 2.1 are fulfilled and X ∈ L p ( W , H a ) for some a ∈ [ , ¥ ) and p ∈ [ , ¥ ) . Then it holds for g ∈ [ , min ( a , d , )] that (cid:13)(cid:13)(cid:13) sup t ∈ [ , T ] k X ( t ) k H g (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ C ( k X k L p ( W , H g ) + ) , (3.2) and (cid:13)(cid:13)(cid:13) sup t ∈ [ , T ] k ˜ X ( t ) k H g (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ C ( k X k L p ( W , H g ) + ) . (3.3) Xiaojie Wang
Proof of Proposition 3.1.
For the first step, we prove (3.2). From (2.22) we deduce thatsup t ∈ [ , T ] k X ( t ) k H g ≤ sup t ∈ [ , T ] k E ( t ) X k H g + sup t ∈ [ , T ] k E ( t − T ) k L ( H ) · sup t ∈ [ , T ] (cid:13)(cid:13)(cid:13) t ∫ E ( T − s ) F ( X ( s )) d s (cid:13)(cid:13)(cid:13) H g + sup t ∈ [ , T ] k E ( t − T ) k L ( H ) · sup t ∈ [ , T ] (cid:13)(cid:13)(cid:13) t ∫ E ( T − s ) G ( X ( s )) d W ( s ) (cid:13)(cid:13)(cid:13) H g . Using (2.17) and elementary inequalities therefore results in (cid:13)(cid:13)(cid:13) sup t ∈ [ , T ] k X ( t ) k H g (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤k X k L p ( W , H g ) + T ∫ k E ( T − s ) F ( X ( s )) k L p ( W , H g ) d s + (cid:13)(cid:13)(cid:13) sup t ∈ [ , T ] (cid:13)(cid:13)(cid:13) t ∫ E ( T − s ) G ( X ( s )) d W ( s ) (cid:13)(cid:13)(cid:13) H g (cid:13)(cid:13)(cid:13) L p ( W , R ) = k X k L p ( W , H g ) + I + I . (3.4)Subsequently we will estimate I and I separately. First, (1.5), (2.23), the fact that g ≤ min ( a , d , ) , thetrigonometric identity and elementary inequalities enable us to get I = T ∫ (cid:13)(cid:13)(cid:0)(cid:13)(cid:13) L − S ( T − s ) F ( u ( s )) (cid:13)(cid:13) g + k C ( T − s ) F ( u ( s )) k g − (cid:1) / (cid:13)(cid:13) L p ( W , R ) d s = T ∫ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L g − F ( u ( s )) (cid:13)(cid:13) U (cid:13)(cid:13)(cid:13) L p ( W , R ) d s ≤ l g − L T ∫ (cid:0) k u ( s ) k L p ( W , U ) + (cid:1) d s ≤ C ( k X k L p ( W , H ) + ) . (3.5)Next, we note thatsup t ∈ [ , T ] (cid:13)(cid:13)(cid:13) t ∫ E ( T − s ) G ( X ( s )) d W ( s ) (cid:13)(cid:13)(cid:13) H g = sup t ∈ [ , T ] (cid:16)(cid:13)(cid:13)(cid:13) t ∫ L − S ( T − s ) G ( u ( s )) d W ( s ) (cid:13)(cid:13)(cid:13) g + (cid:13)(cid:13)(cid:13) t ∫ C ( T − s ) G ( u ( s )) d W ( s ) (cid:13)(cid:13)(cid:13) g − (cid:17) ≤ sup t ∈ [ , T ] (cid:13)(cid:13)(cid:13) t ∫ L g − S ( T − s ) G ( u ( s )) d W ( s ) (cid:13)(cid:13)(cid:13) + sup t ∈ [ , T ] (cid:13)(cid:13)(cid:13) t ∫ L g − C ( T − s ) G ( u ( s )) d W ( s ) (cid:13)(cid:13)(cid:13) . n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 9 Finally, we use the Burkholder-Davis-Gundy type inequality (3.1) and (2.3) to arrive at | I | ≤ C p T ∫ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L g − S ( T − s ) G ( u ( s )) (cid:13)(cid:13) L (cid:13)(cid:13)(cid:13) L p ( W , R ) d s + C p T ∫ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L g − C ( T − s ) G ( u ( s )) (cid:13)(cid:13) L (cid:13)(cid:13)(cid:13) L p ( W , R ) d s ≤ C p T ∫ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L g − G ( u ( s )) (cid:13)(cid:13) L (cid:13)(cid:13)(cid:13) L p ( W , R ) d s ≤ C p T ∫ (cid:13)(cid:13) L g − d (cid:13)(cid:13) L ( U ) · (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L d − G ( u ( s )) (cid:13)(cid:13) L (cid:13)(cid:13)(cid:13) L p ( W , R ) d s ≤ l g − d C p T ∫ (cid:13)(cid:13) L ( k u ( s ) k U + ) (cid:13)(cid:13) L p ( W , R ) d s ≤ L l g − d C p T ∫ (cid:0) k u ( s ) k L p ( W , U ) + (cid:1) d s ≤ C ( k X k L p ( W , H ) + ) . (3.6)Here the stability properties of S ( t ) , C ( t ) , L g − d for t ∈ [ , T ] , g ≤ min ( a , d , ) and some elementary in-equalities were also used. Inserting the above estimates of I and I into (3.4) and taking the assumption X ∈ L p ( W , H a ) into account yield (3.2). To obtain (3.3), we first derive from (2.20) that k X m k L p ( W , H ) ≤ k E ( t m ) X k L p ( W , H ) + m t m − (cid:229) k = k E (cid:0) t m − t k (cid:1) F ( X k ) k L p ( W , H ) + C p m − (cid:229) k = t k + ∫ t k (cid:13)(cid:13) k E (cid:0) t m − t k (cid:1) G ( X k ) k L (cid:13)(cid:13) L p ( W , R ) d s ≤ k X k L p ( W , H ) + T t m − (cid:229) k = k F ( X k ) k L p ( W , H ) + C p t m − (cid:229) k = (cid:13)(cid:13) k G ( X k ) k L (cid:13)(cid:13) L p ( W , R ) ≤ ˆ C p , T (cid:0) k X k L p ( W , H ) + (cid:1) + ¯ C p , T t m − (cid:229) k = k X k k L p ( W , H ) , (3.7)where (2.17), (2.25) and (2.27) were employed. The discrete version of Gronwall’s inequality applied to(3.7) and taking square roots show for all m ∈ { , , · · · , M } that k X m k L p ( W , H ) ≤ C p , T (cid:0) k X k L p ( W , H ) + (cid:1) . (3.8)With this and (2.21), the proof of (3.3) goes along the same way as before. (cid:3) Assumption 1.1 and Proposition 3.1 together immediately imply two corollaries as follows.
Corollary 3.1
Assume all conditions in Theorem 2.1 are fulfilled and X ∈ L p ( W , H ) for p ∈ [ , ¥ ) . Letu ( t ) be given by (2.28) . Then there exists a constant C, depending on p , L , T , k X k L p ( W , H ) , such that sup ≤ t ≤ T (cid:13)(cid:13) F ( u ( t )) (cid:13)(cid:13) L p ( W , U ) ≤ C , and sup ≤ t ≤ T (cid:13)(cid:13) L d − G ( u ( t )) (cid:13)(cid:13) L p ( W , L ) ≤ C . (3.9) Corollary 3.2
Assume all conditions in Theorem 2.1 are fulfilled and X ∈ L p ( W , H ) for p ∈ [ , ¥ ) . Let u m be given by (1.4) . Then there exists a constant C, depending on p , L , T , k X k L p ( W , H ) , such that sup ≤ m ≤ M (cid:13)(cid:13) F ( u m ) (cid:13)(cid:13) L p ( W , U ) ≤ C , and sup ≤ m ≤ M (cid:13)(cid:13) L d − G ( u m ) (cid:13)(cid:13) L p ( W , L ) ≤ C . (3.10) In what follows we present a useful lemma, which can be found in [8,23]. The following H¨older continuityof E ( t ) will put the ultimate limit 1 on the convergence order one can achieve for the time-stepping scheme(1.4). Lemma 3.2
Assume that S ( t ) and C ( t ) are the sine and cosine operators and E ( t ) is the group as definedabove. Then for all g ∈ [ , ] there exists some constant c g such that (cid:13)(cid:13)(cid:0) S ( t ) − S ( s ) (cid:1) L − g (cid:13)(cid:13) L ( U ) ≤ c g ( t − s ) g , (cid:13)(cid:13)(cid:0) C ( t ) − C ( s ) (cid:1) L − g (cid:13)(cid:13) L ( U ) ≤ c g ( t − s ) g (3.11) and k (cid:0) E ( t ) − E ( s ) (cid:1) X k H ≤ c g ( t − s ) g k X k H g (3.12) for all t ≥ s ≥ . Equipped with this lemma, one can investigate the H¨older regularity in time of the mild solution (2.28),which is crucial in analyzing the approximation error of the time-discretization.
Lemma 3.3
Assume conditions in Theorem 2.1 are all fulfilled and X ∈ L p ( W , H a ) for some a ∈ [ , ¥ ) and p ∈ [ , ¥ ) . Then it holds for ≤ s ≤ t ≤ T that k u ( t ) − u ( s ) k L p ( W , U ) ≤ C ( t − s ) min ( a , d , ) . (3.13) Proof of Lemma 3.3.
To get (3.13), we first write X ( t ) − X ( s ) = (cid:0) E ( t − s ) − I (cid:1) X ( s ) + t ∫ s E ( t − r ) F ( X ( r )) d r + t ∫ s E ( t − r ) G ( X ( r )) d W ( r ) , which admits u ( t ) − u ( s ) = (cid:0) C ( t − s ) − I (cid:1) u ( s ) + L − S ( t − s ) v ( s )+ t ∫ s L − S ( t − r ) F ( u ( r )) d r + t ∫ s L − S ( t − r ) G ( u ( r )) d W ( r ) . (3.14)In the course of the proof, we assign r = min ( a , d , ) for simplicity of presentation. The Burkholder-Davis-Gundy type inequality applied to the previous equality shows that k u ( t ) − u ( s ) k L p ( W , U ) ≤ (cid:13)(cid:13)(cid:0) C ( t − s ) − I (cid:1) u ( s ) (cid:13)(cid:13) L p ( W , U ) + (cid:13)(cid:13) L − S ( t − s ) v ( s ) (cid:13)(cid:13) L p ( W , U ) + t ∫ s (cid:13)(cid:13) L − S ( t − r ) F ( u ( r )) (cid:13)(cid:13) L p ( W , U ) d r + C p (cid:16) t ∫ s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L − S ( t − r ) G ( u ( r )) (cid:13)(cid:13) L (cid:13)(cid:13)(cid:13) L p ( W , R ) d r (cid:17) ≤ (cid:13)(cid:13)(cid:0) C ( t − s ) − I (cid:1) L − r (cid:13)(cid:13) L ( U ) · k u ( s ) k L p ( W , ˙ H r ) + (cid:13)(cid:13) L − r S ( t − s ) (cid:13)(cid:13) L ( U ) · (cid:13)(cid:13) v ( s ) (cid:13)(cid:13) L p ( W , ˙ H r − ) + t ∫ s (cid:13)(cid:13) L − S ( t − r ) (cid:13)(cid:13) L ( U ) · (cid:13)(cid:13) F ( u ( r )) (cid:13)(cid:13) L p ( W , U ) d r + C p (cid:16) t ∫ s (cid:13)(cid:13) L − d S ( t − r ) (cid:13)(cid:13) L ( U ) · (cid:13)(cid:13) L d − G ( u ( r )) (cid:13)(cid:13) L p ( W , L ) d r (cid:17) (3.15)for 0 ≤ s ≤ t ≤ T and p ∈ [ , ¥ ) . Note that Proposition 3.1 ensures k u ( s ) k L p ( W , ˙ H r ) ≤ k X ( s ) k L p ( W , H r ) < ¥ and k v ( s ) k L p ( W , ˙ H r − ) ≤ k X ( s ) k L p ( W , H r ) < ¥ for any s ∈ [ , T ] . Combining these bounds with Corollary 3.1 n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 11 and Lemma 3.2, one can easily deduce from (3.15) that k u ( t ) − u ( s ) k L p ( W , U ) ≤ C | t − s | r + C t ∫ s ( t − r ) k F ( u ( r )) k L p ( W , U ) d r + C (cid:16) t ∫ s ( t − r ) ( d , ) (cid:13)(cid:13) L d − G ( u ( r )) (cid:13)(cid:13) L p ( W , L ) d r (cid:17) ≤ C | t − s | r (3.16)for 0 ≤ s ≤ t ≤ T and p ∈ [ , ¥ ) . This finishes the proof of Lemma 3.3. (cid:3) It is worthwhile to remark that one can similarly work with X ( t ) and examine its H¨older regularity inthe product space H . But this leads to reduced H¨older regularity in time and one can only obtain k X ( t ) − X ( s ) k L p ( W , H ) ≤ C ( t − s ) min ( a , d , ) , which puts an ultimate limit on the strong convergence order of thescheme (1.4). Now we formulate the main result in this section as follows. Theorem 3.1
Suppose that all conditions in Theorem 2.1 are fulfilled and assume X ∈ L p ( W , H ) for somep ∈ [ , ¥ ) . Then it holds that (cid:13)(cid:13)(cid:13) sup t ∈ [ , T ] (cid:13)(cid:13) X ( t ) − ˜ X ( t ) (cid:13)(cid:13) H (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ C t min ( d , ) , (3.17) where X ( t ) is the mild solution of (2.6) and ˜ X ( t ) is a continuous-time extension of X m , given by (2.19) . As an immediate consequence, we have the following strong convergence result.
Corollary 3.3
Suppose that all conditions in Theorem 3.1 are fulfilled. Then (cid:13)(cid:13)(cid:13) sup t ∈ [ , T ] (cid:13)(cid:13) u ( t ) − ˜ u ( t ) (cid:13)(cid:13) U (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ C t min ( d , ) , (3.18) where u ( t ) and ˜ u ( t ) are the first components of X ( t ) and ˜ X ( t ) , respectively. The above results reveal that the order of strong convergence is essentially governed by the spatial regularityof the noise term and is in accordance with the exponents of the H¨older regularity in time as stated in Lemma3.3. In addition, one can easily observe that the upper limit on strong order can be achieved only when d ∈ [ , ¥ ) . Furthermore, as indicated in [8] for the linear SWE, the exponential scheme (1.4) for nonlinearSWE also allows for higher strong order than the implicit Euler and Crank-Nicolson time discretizations do(compare Corollary 3.3 with [23, Theorem 4.6]). Proof of Theorem 3.1.
Combining (2.21) and (2.22) gives X ( s ) − ˜ X ( s ) = s ∫ (cid:16) E ( s − r ) F ( X ( r )) − E ( s − ⌊ r ⌋ t ) F ( X ⌊ r / t ⌋ ) (cid:17) d r + s ∫ (cid:16) E ( s − r ) G ( X ( r )) − E ( s − ⌊ r ⌋ t ) G ( X ⌊ r / t ⌋ ) (cid:17) d W ( r ) . (3.19)Thanks to the same arguments as used in (3.4), one can similarly get (cid:13)(cid:13)(cid:13) sup s ∈ [ , t ] (cid:13)(cid:13) X ( s ) − ˜ X ( s ) (cid:13)(cid:13) H (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ t ∫ (cid:13)(cid:13)(cid:13) E ( t − r ) F ( X ( r )) − E ( t − ⌊ r ⌋ t ) F ( X ⌊ r / t ⌋ ) (cid:13)(cid:13)(cid:13) L p ( W , H ) d r + (cid:13)(cid:13)(cid:13)(cid:13) sup s ∈ [ , t ] (cid:13)(cid:13)(cid:13) s ∫ (cid:16) E ( t − r ) G ( X ( r )) − E ( t − ⌊ r ⌋ t ) G ( X ⌊ r / t ⌋ ) (cid:17) d W ( r ) (cid:13)(cid:13)(cid:13) H (cid:13)(cid:13)(cid:13)(cid:13) L p ( W , R ) = I + I . (3.20) Noting that ˜ X ( ⌊ r ⌋ t ) = X ⌊ r / t ⌋ and using (2.17), (2.24), (3.9), (3.13) and Lemma 3.2, we find that I ≤ t ∫ (cid:13)(cid:13)(cid:13)(cid:16) E ( t − r ) − E ( t − ⌊ r ⌋ t ) (cid:17) F ( X ( r )) (cid:13)(cid:13)(cid:13) L p ( W , H ) d r + t ∫ (cid:13)(cid:13)(cid:13) E ( t − ⌊ r ⌋ t ) (cid:16) F ( X ( r )) − F ( X ( ⌊ r ⌋ t )) (cid:17)(cid:13)(cid:13)(cid:13) L p ( W , H ) d r + t ∫ (cid:13)(cid:13)(cid:13) E ( t − ⌊ r ⌋ t ) (cid:16) F ( X ( ⌊ r ⌋ t )) − F ( X ⌊ r / t ⌋ ) (cid:17)(cid:13)(cid:13)(cid:13) L p ( W , H ) d r ≤ c t t ∫ (cid:13)(cid:13) F ( X ( r )) (cid:13)(cid:13) L p ( W , H ) d r + t ∫ (cid:13)(cid:13) F ( X ( r )) − F ( X ( ⌊ r ⌋ t )) (cid:13)(cid:13) L p ( W , H ) d r + t ∫ (cid:13)(cid:13) F ( X ( ⌊ r ⌋ t )) − F ( ˜ X ( ⌊ r ⌋ t )) (cid:13)(cid:13) L p ( W , H ) d r ≤ c t t ∫ k F ( u ( r )) k L p ( W , U ) d r + L / p l t ∫ (cid:13)(cid:13) u ( r ) − u ( ⌊ r ⌋ t ) (cid:13)(cid:13) L p ( W , U ) d r + L / p l t ∫ (cid:13)(cid:13) X ( ⌊ r ⌋ t ) − ˜ X ( ⌊ r ⌋ t ) (cid:13)(cid:13) L p ( W , H ) d r ≤ C t + C t min ( d , ) + C t ∫ (cid:13)(cid:13)(cid:13) sup r ∈ [ , s ] (cid:13)(cid:13) X ( r ) − ˜ X ( r ) (cid:13)(cid:13) H (cid:13)(cid:13)(cid:13) L p ( W , R ) d s ≤ C t min ( d , ) + C t ∫ (cid:13)(cid:13)(cid:13) sup r ∈ [ , s ] (cid:13)(cid:13) X ( r ) − ˜ X ( r ) (cid:13)(cid:13) H (cid:13)(cid:13)(cid:13) L p ( W , R ) d s . (3.21)With regard to I , we apply the Burkholder-Davis-Gundy type inequality (3.1) to get | I | ≤ C p t ∫ (cid:13)(cid:13)(cid:13) E ( t − r ) G ( X ( r )) − E ( t − ⌊ r ⌋ t ) G ( X ⌊ r / t ⌋ ) (cid:13)(cid:13)(cid:13) L p ( W , L ) d r ≤ C p t ∫ (cid:13)(cid:13)(cid:13)(cid:16) E ( t − r ) − E ( t − ⌊ r ⌋ t ) (cid:17) G ( X ( r )) (cid:13)(cid:13)(cid:13) L p ( W , L ) d r + C p t ∫ (cid:13)(cid:13)(cid:13) E ( t − ⌊ r ⌋ t ) (cid:16) G ( X ( r )) − G ( X ( ⌊ r ⌋ t )) (cid:17)(cid:13)(cid:13)(cid:13) L p ( W , L ) d r + C p t ∫ (cid:13)(cid:13)(cid:13) E ( t − ⌊ r ⌋ t ) (cid:16) G ( X ( ⌊ r ⌋ t )) − G ( X ⌊ r / t ⌋ ) (cid:17)(cid:13)(cid:13)(cid:13) L p ( W , L ) d r . (3.22)Further, employing (2.17), (2.26), (3.9), (3.13) and Lemma 3.2 shows that | I | ≤ c d t ( d , ) t ∫ (cid:13)(cid:13)(cid:13) k G ( X ( r )) k L ( U , H d ) (cid:13)(cid:13)(cid:13) L p ( W , R ) d r + C p t ∫ k G ( X ( r )) − G ( X ( ⌊ r ⌋ t )) k L p ( W , L ) d r + C p t ∫ (cid:13)(cid:13) G ( X ( ⌊ r ⌋ t )) − G ( X ⌊ r / t ⌋ ) (cid:13)(cid:13) L p ( W , L ) d r ≤ c d t ( d , ) t ∫ (cid:13)(cid:13)(cid:13) k L d − G ( u ( r )) k L ( U , U ) (cid:13)(cid:13)(cid:13) L p ( W , R ) d r + C p L / l d t ∫ k u ( r ) − u ( ⌊ r ⌋ t ) k L p ( W , U ) d r + C p L / l d t ∫ (cid:13)(cid:13) X ( ⌊ r ⌋ t ) − ˜ X ( ⌊ r ⌋ t ) (cid:13)(cid:13) L p ( W , H ) d r ≤ C t ( d , ) + C t ∫ (cid:13)(cid:13)(cid:13) sup r ∈ [ , s ] (cid:13)(cid:13) X ( r ) − ˜ X ( r ) (cid:13)(cid:13) H (cid:13)(cid:13)(cid:13) L p ( W , R ) d s . (3.23) n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 13 Defining a non-decreasing function J : [ , T ] → R by J ( t ) = (cid:13)(cid:13) sup s ∈ [ , t ] k X ( s ) − ˜ X ( s ) k H (cid:13)(cid:13) L p ( W , R ) and takingthe estimates (3.21) and (3.23) into account we derive from (3.20) that J ( t ) = (cid:13)(cid:13)(cid:13) sup s ∈ [ , t ] (cid:13)(cid:13) X ( s ) − ˜ X ( s ) (cid:13)(cid:13) H (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ C t ( d , ) + C Z t J ( s ) d s . (3.24)Proposition 3.1 guarantees the boundedness of J ( t ) for t ∈ [ , T ] , which enables us to apply Gronwall’slemma (see, e.g., [29, Theorem 8.1]) to show for all t ∈ [ , T ] that J ( t ) = (cid:13)(cid:13)(cid:13) sup s ∈ [ , t ] (cid:13)(cid:13) X ( s ) − ˜ X ( s ) (cid:13)(cid:13) H (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ C t ( d , ) . (3.25)Finally, taking square roots of (3.25) completes the proof of Theorem 3.1. (cid:3) The aim of this section is to include several concrete examples which fit in the abstract setting formulatedabove. To this end, let O ⊂ R d , d = , ,
3, be a bounded open set with smooth boundary ¶ O . A nonlinearwhite noise driven stochastic wave equation (SWE) with Dirichlet boundary condition is usually describedby ¶ u ¶ t = D u + f ( x , u ) + g ( x , u ) ˙ W ( t ) , t ∈ ( , T ] , x ∈ O , u ( , x ) = u ( x ) , ¶ u ¶ t ( , x ) = v ( x ) , x ∈ O , u | ¶ O = , t ∈ ( , T ] , (4.1)where T ∈ ( , ¥ ) , D = (cid:229) dk = ¶ ¶x k is the Laplace operator and f , g : O × R → R are deterministic functions.Moreover, the initial data u , v : O × W → R are random variables and W is a noise process, which will bespecified later. Numerics of such equation has been considered in [6,15,34,36,38].Let U : = L (cid:0) O , R (cid:1) be the separable Hilbert space of real-valued square integrable functions from O to R , with the scalar product and the norm h u , u i U = Z O u ( x ) u ( x ) d x , k u k U = (cid:18) Z O | u ( x ) | d x (cid:19) / for all u , u , u ∈ U . Moreover, let { W ( t ) } t ∈ [ , T ] be a cylindrical Q -Wiener process on a stochastic basis (cid:0) W , F , P , { F t } t ∈ [ , T ] (cid:1) , given by (2.5). It is a classical result that the covariance operator Q = Q ◦ Q ∈ L ( U ) is a nonnegative, symmetric operator so that Q f i = q i f i , q i ≥ , i ∈ N . (4.2)Then one can rewrite (4.1) in an abstract Itˆo form as (1.1), namely, (cid:26) d ˙ u = − L u d t + F ( u ) d t + G ( u ) d W ( t ) , t ∈ ( , T ] , u ( ) = u , ˙ u ( ) = v , where − L : D ( L ) ⊂ U → U denotes the Laplacian with homogeneous Dirichlet boundary condition, andwhere F : U → U , G ( u ) : U → U are the Nemytskij operators given by F ( u )( x ) = f ( x , u ( x )) , (cid:0) G ( u )( j ) (cid:1) ( x ) = g ( x , u ( x )) · j ( x ) , x ∈ O . (4.3)With the above setting, we take a close look at conditions in Assumption 1.1. To do this we considerseparately several cases as follows. g ( x , u ) ≡ x ∈ O , u ∈ R . Fur-thermore, assume for some b ∈ ( , ¥ ) that k L b − Q k L ( U ) = (cid:18) ¥ (cid:229) i = q i k L b − f i k U (cid:19) < ¥ . (4.4)Concerning f , we assume f : O × R → R in (4.1) satisfies | f ( x , u ) | ≤ L ( | u | + ) , | f ( x , u ) − f ( x , u ) | ≤ L | u − u | (4.5)for all x ∈ O , u , u , u ∈ R . Then the Nemytskij operator F satisfies k F ( u ) k U = Z O (cid:12)(cid:12) f ( x , u ( x )) (cid:12)(cid:12) d x ≤ L Z O (cid:0) | u ( x ) | + (cid:1) d x = L (cid:0) k u k U + r ( O ) (cid:1) , (4.6)where r ( O ) , the measure of the set O , is bounded by assumption. In a similar way, (cid:13)(cid:13) F ( u ) − F ( u ) (cid:13)(cid:13) U ≤ L k u − u k U . (4.7)Moreover, G ( u ) ≡ I in this setting and thus G ( u ) − G ( u ) ≡ (cid:13)(cid:13) L b − G ( u ) (cid:13)(cid:13) L = (cid:13)(cid:13) L b − (cid:13)(cid:13) L = k L b − Q k L ( U ) < ¥ . (4.8)Therefore, Assumption 1.1 is fulfilled with d = b . An immediate consequence of Corollary 3.3 reads: Corollary 4.1
Suppose that g ( x , u ) ≡ for all x ∈ O , u ∈ R and f : O × R → R satisfies (4.5) . Let W ( t ) be acylindrical Q-Wiener process on the stochastic basis ( W , F , P , { F t } ≤ t ≤ T ) with (4.4) fulfilled. Additionally,we assume u , v ∈ F and u ∈ L p ( W , ˙ H ) , v ∈ L p ( W , U ) for some p ∈ [ , ¥ ) . Then the problem (4.1) hasa unique mild solution. Moreover, there exists a generic constant C ∈ [ , ¥ ) depending on T , L , p , b and theinitial data, such that (cid:13)(cid:13)(cid:13) sup t m ∈ T M (cid:13)(cid:13) u ( t m ) − u m (cid:13)(cid:13) U (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ C t min ( b , ) , (4.9) where u ( t ) is the mild solution of (4.1) and u m is the numerical solution produced by (1.4) . f : O × R → R in (4.1) satisfies (4.5) and g : O × R → R in (4.1) satisfies | g ( x , u ) | ≤ L ( | u | + ) , | g ( x , u ) − g ( x , u ) | ≤ L | u − u | (4.10)for all x ∈ O , u , u , u ∈ R . Furthermore, we assume in this subsectionTr ( Q ) = (cid:229) i ∈ N q i < ¥ , sup i ∈ N sup x ∈ ¯ O | f i ( x ) | < ¥ . (4.11)Then it is not difficult to see that k G ( u ) k L = (cid:13)(cid:13) G ( u ) Q (cid:13)(cid:13) L ( U ) = ¥ (cid:229) i = (cid:13)(cid:13) G ( u ) Q f i (cid:13)(cid:13) U = ¥ (cid:229) i = q i (cid:13)(cid:13) G ( u ) f i (cid:13)(cid:13) U = ¥ (cid:229) i = q i Z O (cid:12)(cid:12) g ( x , u ( x )) f i ( x ) (cid:12)(cid:12) d x ≤ sup i ∈ N sup x ∈ ¯ O | f i ( x ) | · Tr ( Q ) · L (cid:0) k u k U + r ( O ) (cid:1) . (4.12) n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 15 In the same way, one can obtain (cid:13)(cid:13)(cid:0) G ( u ) − G ( u ) (cid:1) Q (cid:13)(cid:13) L ( U ) ≤ sup i ∈ N sup x ∈ ¯ O | f i ( x ) | · Tr ( Q ) · L k u − u k U . (4.13)Consequently, Assumption 1.1 is fulfilled with d = Corollary 4.2
Suppose that f , g : O × R → R satisfy (4.5) and (4.10) . Let W ( t ) be a standard U-valuedQ-Wiener process on a stochastic basis ( W , F , P , { F t } ≤ t ≤ T ) with (4.11) fulfilled. Additionally, we assumeu , v ∈ F and u ∈ L p ( W , ˙ H ) , v ∈ L p ( W , U ) for some p ∈ [ , ¥ ) . Then the problem (4.1) has a uniquemild solution and it holds that (cid:13)(cid:13)(cid:13) sup t m ∈ T M (cid:13)(cid:13) u ( t m ) − u m (cid:13)(cid:13) U (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ C t , (4.14) where u ( t ) is the mild solution of (4.1) and u m is produced by (1.4) . d = O = ( , ) and let Q = I . Then W ( t ) becomes a cylindrical I -Wiener process,which can be given by W ( t ) = (cid:229) i ∈ N b i ( t ) e i , t ∈ [ , T ] , (4.15)where { b i ( t ) } i ∈ N are a family of mutually independent real Brownian motions and { e i = √ ( i p x ) , x ∈ ( , ) } i ∈ N form an orthonormal basis of U consisting of eigenfunctions of L with L e i = l i e i , l i = p i , i ∈ N . It is then obvious for any e > (cid:13)(cid:13) L − e + (cid:13)(cid:13) L ( U ) = p − ( e + ) ¥ (cid:229) i = i − ( e + ) < ¥ , and sup i ∈ N sup x ∈ [ , ] | e i ( x ) | ≤ √ . (4.16)Therefore, it follows that (cid:13)(cid:13) L − e + G ( u ) (cid:13)(cid:13) L ( U ) = ¥ (cid:229) i = (cid:13)(cid:13) L − e + G ( u ) e i (cid:13)(cid:13) U = ¥ (cid:229) i = ¥ (cid:229) j = (cid:12)(cid:12) h L − e + G ( u ) e i , e j i (cid:12)(cid:12) = ¥ (cid:229) i = ¥ (cid:229) j = (cid:12)(cid:12) h G ( u ) e i , l − e + j e j i (cid:12)(cid:12) = ¥ (cid:229) i = ¥ (cid:229) j = l − e + j (cid:12)(cid:12)(cid:12) Z O g ( x , u ( x )) e i ( x ) e j ( x ) d x (cid:12)(cid:12)(cid:12) ≤ ¥ (cid:229) i = ¥ (cid:229) j = l − e + j (cid:12)(cid:12)(cid:12) Z O g ( x , u ( x )) e i ( x ) d x (cid:12)(cid:12)(cid:12) = p − ( e + ) ¥ (cid:229) j = j − ( e + ) · k g ( · , u ( · )) k U ≤ ˆ C e (cid:0) k u k U + r ( O ) (cid:1) (4.17)for arbitrarily small e >
0. Similarly, one can show for arbitrarily small e > (cid:13)(cid:13) L − e + (cid:0) G ( u ) − G ( u ) (cid:1)(cid:13)(cid:13) L ( U ) ≤ ˇ C e k u − u k U . (4.18)Hence, Assumption 1.1 is satisfied with d = − e for arbitrarily small e > Corollary 4.3
Suppose that f , g : ( , ) × R → R satisfy (4.5) and (4.10) . Additionally, we assume u , v ∈ F and u ∈ L p ( W , ˙ H ) , v ∈ L p ( W , U ) for some p ∈ [ , ¥ ) . Moreover, let W ( t ) be a cylindrical I-Wienerprocess given by (4.15) . Then the problem (4.1) has a unique mild solution. Moreover, there exists a constantC ∈ [ , ¥ ) depending on T , L , p , e and the initial data, such that (cid:13)(cid:13)(cid:13) sup t m ∈ T M (cid:13)(cid:13) u ( t m ) − u m (cid:13)(cid:13) U (cid:13)(cid:13)(cid:13) L p ( W , R ) ≤ C e t − e , (4.19) where u ( t ) is the mild solution of (4.1) and u m is the numerical solution produced by (1.4) . In many applications, a key task is to approximate the quantity E [ j ( u ( T ))] , where j is a functional ofthe mild solution to (1.1). This leads to another important notion of convergence for a numerical scheme,the weak convergence, which is concerned with the approximation of law. In this section, let us turn ourattention to this topic. As is well known that the rate of the weak error can, in some situations, be improvedcompared to that of the strong error. Below, we shall show this fact for nonlinear SWE driven by additivenoise. Since the treatment of multiplicative noise case causes additional technical difficulties, it will beaddressed elsewhere in our future work.For the additive case when G ( u ) ≡ I for u ∈ U , the abstract equation (2.6) reduces to (cid:26) d X ( t ) = AX ( t ) d t + F ( X ) d t + B d W ( t ) , t ∈ ( , T ] , X ( ) = X , (5.1)where we adopt the same notations as in (2.7) and additionally define here B : U → H by B = ( , I ) T .To begin with, we need the following assumption. Assumption 5.1
Assume G ( u ) ≡ I for all u ∈ U, and let F : U → U be a twice differentiable mappingsatisfying k F ( u ) k U ≤ L ( k u k U + ) , k F ′ ( u ) y k U ≤ L k y k U , k L − F ′′ ( u )( y , y ) k U ≤ L k y k U k y k U (5.2) for all u , y , y , y ∈ U. Furthermore we assume (1.8) holds for some b ∈ ( , ) . Several comments should be added concerning the above assumption. We first mention that the essentialcondition (1.8) has been also used in [22,23], to study weak errors for the linear SWE ( F ≡
0) with addi-tive noise. Secondly, it should be pointed out that F in the previous assumption is not required to be twiceFr´echet differentiable in U , which is in general not fulfilled for the Nemytskij operators (see Example 5.1below). Besides, recall that (cid:13)(cid:13) L b − Q (cid:13)(cid:13) L ( U ) ≤ (cid:13)(cid:13) L − + b Q L − (cid:13)(cid:13) L ( U ) (see Theorem 2.1 in [23]). Accord-ingly, it is easy to see that Assumption 1.1 is fulfilled with d = b under Assumption 5.1. In other words,Assumption 5.1 suffices to guarantee a unique mild solution of (5.1) and the strong convergence order ofmin ( b , ) for the underlying scheme (1.4). When b ∈ [ , ¥ ) , Theorem 3.1 shows that the scheme possessesthe strong order of 1. This trivially implies weak convergence order of 1, which can not be improved furtheras shown in the later weak convergence result (Theorem 5.1). Based on the above observations, we takecondition (1.8) with b ∈ ( , ) here, instead of b ∈ ( , ¥ ) . Theorem 5.1
Suppose that all conditions in Assumption 5.1 are satisfied and assume additionally that for k = min ( b , − e ) with arbitrarily small e ∈ ( , ) , k F ′ ( u ) z k − ≤ C k z k − k · (cid:0) k u k k + (cid:1) , u ∈ ˙ H k , z ∈ U , (5.3) and X = ( u , v ) T ∈ H is deterministic. Then there exists a constant C e depending on T , b , e , F , L and theinitial data, such that (cid:12)(cid:12) E [ F ( X ( T ))] − E [ F ( X M )] (cid:12)(cid:12) ≤ C e t min ( b , + b − e , ) (5.4) n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 17 for F ∈ C b ( H ; R ) , and where X ( t ) , t ∈ [ , T ] is the mild solution of (5.1) and X m , m = , , ... M are producedby the recurrence equation (2.18) . The proof of Theorem 5.1 is postponed after some preparations. Here and below, by C kb ( V , V ′ ) we denotethe space of not necessarily bounded mappings from a Banach space V to the other Banach space V ′ thathave continuous and bounded Fr´echet derivatives up to order k . For the particular case when V is a Hilbertspace and V ′ = R , one can identify the first derivative D Y ( X ) ∈ L ( V , R ) of a function Y : V → R with anelement in V due to the Riesz representation theorem, i.e., D Y ( X ) Z = h D Y ( X ) , Z i V , ∀ X , Z ∈ V , and the second derivative D Y ( X ) with a bounded linear operator such that D Y ( X )( Z , Z ) = h D Y ( X ) Z , Z i V , ∀ X , Z , Z ∈ V . Next, we make some comments concerning the new key condition (5.3). Such condition imposed onthe derivative operator F ′ ( u ) , u ∈ U can be fulfilled in a concrete setting established in Section 4. Morespecifically, we set U : = L (cid:0) ( , ) , R (cid:1) and let F : U → U be the Nemytskij operator given by F ( u )( x ) = f ( x , u ( x )) , u ∈ U , x ∈ ( , ) with f : O × R → R being a real-valued function. When f has bounded partialderivatives up to order two as required in (5.51), the condition (5.3) can be fulfilled (see Example 5.1 belowfor more details on the verification). Of course, the condition (5.3) is hard to be satisfied if the associatedfunction f grows super-linearly or is non-Lipschitz.An immediate consequence of Theorem 5.1 gives the following result. Corollary 5.2
Assume all conditions in Theorem 5.1 are fulfilled. Then it holds for arbitrarily small e ∈ ( , ) that (cid:12)(cid:12) E [ j ( u ( T ))] − E [ j ( u M )] (cid:12)(cid:12) ≤ C e t min ( b , + b − e , ) (5.5) for j ∈ C b ( U ; R ) . Here u ( T ) and u M are the first components of X ( T ) and X M , respectively. Comparing the weak error bound (5.5) with the strong one (4.9), one can find that the rate of weakconvergence is, as expected, improved. To see this, let us look at the special interesting case of space-time white noise ( Q = I ). In this case, (4.9) admits a strong convergence order b < , while (5.5) givesa weak convergence order 1 − e for arbitrarily small e >
0. A closely related work on weak convergenceof numerical schemes for SWEs can be found in [23], where only the linear SWE was considered andtime discretizations were done by rational approximation to the exponential function. The correspondingweak rates in [23] are min ( pp + b , ) , with p ≥ p =
1) admits weak rate of min ( b , ) and the Crank-Nicolson method ( p =
2) admits weakrate of min ( b , ) . Apparently, the exponential scheme attains better weak convergence rates than timediscretization schemes in [23].To carry out the weak error analysis, we first introduce a function m : [ , T ] × H → R , defined by m ( t , x ) = E [ F ( X ( t , x ))] , t ∈ [ , T ] , x ∈ H , (5.6)where F ∈ C b ( H ; R ) and X ( t , x ) is the unique mild solution of (5.1) with the initial value x ∈ H . Owing to(5.2), one can readily infer that k F ′ ( X ) Z k H = k L − F ′ ( u ) u z k U ≤ l − L k u z k U ≤ l − L k Z k H , k F ′′ ( X )( Z , Z ) k H = k L − F ′′ ( u )( u z , u z ) k U ≤ L k u z k U k u z k U ≤ L k Z k H k Z k H (5.7)for all X = ( u , v ) T ∈ H , Z = ( u z , v z ) T ∈ H , Z = ( u z , v z ) T ∈ H and Z = ( u z , v z ) T ∈ H . This straightfor-wardly implies F ∈ C b ( H , H ) . By [9, Theorem 9.4], one knows that the solution X ( t , x ) is twice differen-tiable with respect to the initial value x . More formally, the process z h ( t ) = ¶ X ¶ x ( t , x ) h for t ∈ [ , T ] , h ∈ H is the mild solution of the following equation (cid:26) d z h = ( A z h + F ′ ( X ( t , x )) z h ) dt , t ∈ ( , T ] , z h ( ) = h . (5.8) And h h , g ( t ) = ¶ X ¶ x ( t , x )( h , g ) for t ∈ [ , T ] , h , g ∈ H is the mild solution of ( d h h , g = (cid:16) A h h , g + F ′ ( X ( t , x )) h h , g + F ′′ ( X ( t , x ))( z h , z g ) (cid:17) d t , t ∈ ( , T ] , h h , g ( ) = . (5.9)Bearing these facts in mind and differentiating (5.6) with respect to x yield D m ( t , x ) h = E h D F ( X ( t , x )) z h ( t ) i , t ∈ [ , T ] , x , h ∈ H . (5.10)Differentiating (5.10) further shows D m ( t , x )( h , g ) = E h D F ( X ( t , x ))( z h ( t ) , z g ( t )) i + E h D F ( X ( t , x )) h h , g ( t ) i , t ∈ [ , T ] , x , h , g ∈ H , (5.11)where z h ( t ) , z g ( t ) and h h , g ( t ) for t ∈ [ , T ] , h , g ∈ H are processes defined as above. Then it is well-known(see [9, Theorem 9.17], [10, Theorem 5.4.2]) that m ( t , x ) defined by (5.6) is a unique strict solution to thefollowing deterministic PDE: ( ¶m¶ t ( t , x ) = (cid:10) Ax + F ( x ) , D m ( t , x ) (cid:11) H + Tr h D m ( t , x ) BQ (cid:0) BQ (cid:1) ∗ i , t ∈ ( , T ] , x ∈ D ( A ) , m ( , x ) = F ( x ) , x ∈ H . (5.12)In order to eliminate the operator A , we introduce another process n : [ , T ] × H → R , defined by n ( t , y ) = m ( t , E ( − t ) y ) , t ∈ [ , T ] , y ∈ H . (5.13)By virtue of (5.12), and also noticing that ¶¶ t (cid:0) E ( − t ) y (cid:1) = − AE ( − t ) y for y ∈ D ( A ) , one can derive that ¶n¶ t ( t , y ) = ¶m¶ t ( t , E ( − t ) y ) + (cid:10) D m ( t , E ( − t ) y ) , − AE ( − t ) y (cid:11) H = (cid:10) AE ( − t ) y + F ( E ( − t ) y ) , D m ( t , E ( − t ) y ) (cid:11) H + Tr h D m ( t , E ( − t ) y ) BQ (cid:0) BQ (cid:1) ∗ i − (cid:10) D m ( t , E ( − t ) y ) , AE ( − t ) y (cid:11) H = (cid:10) F ( E ( − t ) y ) , D m ( t , E ( − t ) y ) (cid:11) H + Tr h D m ( t , E ( − t ) y ) BQ (cid:0) BQ (cid:1) ∗ i . (5.14)Note that D n ( t , y ) z = D m ( t , E ( − t ) y ) E ( − t ) z , t ∈ [ , T ] , y , z ∈ H , (5.15)and D n ( t , y )( z , z ) = D m ( t , E ( − t ) y )( E ( − t ) z , E ( − t ) z ) , t ∈ [ , T ] , y , z , z ∈ H . (5.16)The above three identities together with (2.4) immediately imply that ( ¶n¶ t ( t , y ) = Tr h D n ( t , y ) E ( t ) BQ (cid:0) E ( t ) BQ (cid:1) ∗ i + (cid:10) E ( t ) F ( E ( − t ) y ) , D n ( t , y ) (cid:11) H , t ∈ ( , T ] , y ∈ D ( A ) n ( , y ) = F ( y ) , y ∈ H . (5.17)Before proceeding further, we show regularity results on D n ( t , y ) and D n ( t , y ) . Lemma 5.1
Assume Assumption 5.1 holds. Then D n ( t , y ) : H → R given by (5.15) and D n ( t , y ) : H × H → R given by (5.16) for any t ∈ [ , T ] and y ∈ H satisfy k D n ( t , y ) k H ≤ C , and k D n ( t , y ) k L ( H ) ≤ C (5.18) for all t ∈ [ , T ] and y ∈ H. n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 19 Proof of Lemma 5.1.
First of all, we shall prove for all t ∈ [ , T ] and x ∈ H that k D m ( t , x ) k H ≤ C , and k D m ( t , x ) k L ( H ) ≤ C . (5.19)Then (5.18) is an easy consequence of (5.19) thanks to (5.15), (5.16) and (2.17). Hence we focus on theproof of (5.19). Recall that z h ( t ) = ¶ X ( t , x ) ¶ x h is the mild solution of (5.8), given by z h ( t ) = E ( t ) h + Z t E ( t − s ) F ′ ( X ( s , x )) z h ( s ) d s , t ∈ [ , T ] , h ∈ H . (5.20)Therefore, (2.17) and (5.7) guarantee that k z h ( t ) k H ≤ k E ( t ) h k H + l − L Z t k z h ( s ) k H d s , t ∈ [ , T ] . (5.21)The Gronwall inequality thus shows for all t ∈ [ , T ] that k z h ( t ) k H ≤ C k h k H . (5.22)Since F ∈ C b ( H , R ) and (5.22) holds, one can derive from (5.10) that | D m ( t , x ) h | ≤ E (cid:2) k D F ( X ( t , x )) k H · k z h ( t ) k H (cid:3) ≤ C k h k H (5.23)for all t ∈ [ , T ] , h ∈ H . Likewise, h h , g ( t ) = ¶ X ( t , x ) ¶ x ( h , g ) for t ∈ [ , T ] , h , g ∈ H is given by h h , g ( t ) = Z t E ( t − s ) (cid:16) F ′ ( X ( s , x )) h h , g ( s ) + F ′′ ( X ( s , x ))( z h ( s ) , z g ( s )) (cid:17) d s . (5.24)With (5.7) and (5.22) at hand, we first obtain k h h , g ( t ) k H ≤ C k h k H · k g k H + l − L Z t k h h , g ( s ) k H d s . (5.25)Once again, we use the Gronwall inequality to get k h h , g ( t ) k H ≤ C k h k H · k g k H , t ∈ [ , T ] , h , g ∈ H , (5.26)which together with (5.11), (5.22) implies D m ( t , x )( h , g ) ≤ C k h k H · k g k H , t ∈ [ , T ] , x , h , g ∈ H . (5.27)This and (5.23) together finish the proof of (5.19). (cid:3) As opposed to the parabolic case [1,14,37], D m ( t , y ) and D m ( t , y ) defined as above only admit spatialregularity in the space H and this also remains true for D n ( t , y ) and D n ( t , y ) . This is due to the lack of thesmoothing property of the associated group E ( t ) . Lemma 5.2
Suppose that u ∈ ˙ H , v ∈ U and Assumption 5.1 is fulfilled. Then it holds for any g ∈ [ , ) and p ∈ [ , ¥ ) that k ˜ u ( t ) − u m k L p ( W , ˙ H − g ) ≤ C t min ( b + g , ) (5.28) for t ∈ [ t m , t m + ] , m = , , · · · , M − , and where ˜ u ( t ) is the first component of ˜ X ( t ) given by (2.19) and u m comes from (1.4) . Proof of Lemma 5.2.
Equality (2.19) shows for t ∈ [ t m , t m + ] that˜ X ( t ) − X m = (cid:0) E ( t − t m ) − I (cid:1) X m + E ( t − t m ) (cid:16) F ( X m )( t − t m ) + B ( W ( t ) − W ( t m )) (cid:17) , (5.29)which implies for t ∈ [ t m , t m + ] that˜ u ( t ) − u m = (cid:0) C ( t − t m ) − I (cid:1) u m + L − S ( t − t m ) v m + L − S ( t − t m ) F ( u m )( t − t m ) + L − S ( t − t m )( W ( t ) − W ( t m )) . (5.30)Further, using Lemma 3.1, Proposition 3.1, Corollary 3.2 and (3.11) yields for t ∈ [ t m , t m + ] , p ∈ [ , ¥ ) and g ∈ [ , ) that (cid:13)(cid:13) ˜ u ( t ) − u m (cid:13)(cid:13) L p ( W , ˙ H − g ) ≤ (cid:13)(cid:13) L − g (cid:0) C ( t − t m ) − I (cid:1) u m (cid:13)(cid:13) L p ( W , U ) + (cid:13)(cid:13) L − + g S ( t − t m ) v m (cid:13)(cid:13) L p ( W , U ) + ( t − t m ) · (cid:13)(cid:13) L − + g S ( t − t m ) F ( u m ) (cid:13)(cid:13) L p ( W , U ) + C p ( t − t m ) · (cid:13)(cid:13) L − + g S ( t − t m ) Q (cid:13)(cid:13) L ( U ) ≤ C t ( b + g , ) + C t k F ( u m ) k L p ( W , U ) + C | t − t m | · (cid:13)(cid:13) L − b + g S ( t − t m ) (cid:13)(cid:13) L ( U ) · (cid:13)(cid:13) L b − Q (cid:13)(cid:13) L ( U ) ≤ C t ( b + g , ) . The proof of Lemma 5.2 is thus completed. (cid:3)
Roughly speaking, Lemma 5.2 implies that higher H¨older regularity in time of the numerical approxi-mation process ˜ u ( t ) can be achieved when measured in the negative Sobolev space ˙ H − g = D ( L − g ) , g > K , m below). Now we arein a position to start the proof of Theorem 5.1. Proof of Theorem 5.1.
First, we define an auxiliary process ˜ Y ( t ) = E ( T − t ) ˜ X ( t ) , t ∈ [ , T ] , by˜ Y ( t ) = E ( T − t m ) X m + Z tt m E ( T − t m ) F ( X m ) d s + Z tt m E ( T − t m ) B d W ( s ) , t ∈ [ t m , t m + ] . (5.31)The above definition of ˜ Y ( t ) allows for˜ Y ( T ) = ˜ X ( T ) = X M and ˜ Y ( ) = E ( T ) X . (5.32)This together with (5.6) and (5.13) ensures that E (cid:2) F ( X M ) (cid:3) − E (cid:2) F (cid:0) X ( T ) (cid:1)(cid:3) = E (cid:2) F ( ˜ X ( T )) (cid:3) − E (cid:2) m ( T , X ) (cid:3) = E (cid:2) F ( ˜ Y ( T )) (cid:3) − E (cid:2) m ( T , X ) (cid:3) = E (cid:2) n ( , ˜ Y ( T )) (cid:3) − E (cid:2) n ( T , E ( T ) X ) (cid:3) = E (cid:2) n ( , ˜ Y ( T )) (cid:3) − E (cid:2) n ( T , ˜ Y ( )) (cid:3) . (5.33)Before further analysis, we define a finite dimensional subspace U N of U by U N : = span { e , e , · · · , e N } for N ∈ N , and the projection operator P N : ˙ H a → U N by P N f = (cid:229) Ni = h f , e i i U e i , for f ∈ ˙ H a , a ≥ − . Then, wedefine P N X = ( P N u , P N v ) T for X = ( u , v ) T ∈ H . It can be easily verified that P N X ∈ D ( A ) for X ∈ H and n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 21 that lim N → ¥ k P N X − X k H = X ∈ H . At the moment, we apply Itˆo’s formula [9, Theorem 4.17] to n ( T − t , P N ˜ Y ( t )) in each interval [ t m , t m + ] and utilize (5.17) to obtain E (cid:2) n ( , P N ˜ Y ( T )) (cid:3) − E (cid:2) n ( T , P N ˜ Y ( )) (cid:3) = M − (cid:229) m = (cid:0) E (cid:2) n ( T − t m + , P N ˜ Y ( t m + )) (cid:3) − E (cid:2) n ( T − t m , P N ˜ Y ( t m )) (cid:3)(cid:1) = M − (cid:229) m = (cid:18) − Z t m + t m E h ¶n¶ t ( T − t , P N ˜ Y ( t )) i d t + Z t m + t m E (cid:2) D n ( T − t , P N ˜ Y ( t )) P N E ( T − t m ) F ( X m ) (cid:3) d t + E Z t m + t m Tr h D n ( T − t , P N ˜ Y ( t )) P N E ( T − t m ) BQ (cid:0) P N E ( T − t m ) BQ (cid:1) ∗ i d t (cid:19) = M − (cid:229) m = (cid:18) Z t m + t m E hD D n ( T − t , P N ˜ Y ( t )) , P N E ( T − t m ) F ( X m ) − E ( T − t ) F ( E ( t − T ) P N ˜ Y ( t )) E H i d t + E Z t m + t m Tr h D n ( T − t , P N ˜ Y ( t )) (cid:16) P N E ( T − t m ) BQ (cid:0) P N E ( T − t m ) BQ (cid:1) ∗ − E ( T − t ) BQ (cid:0) E ( T − t ) BQ (cid:1) ∗ (cid:17)i d t (cid:19) . (5.34)Note that n ( t , · ) ∈ C b ( H ; R ) for t ∈ [ , T ] and lim N → ¥ P N X = X in H . Therefore, taking limits in (5.34) as N → ¥ shows E (cid:2) n ( , ˜ Y ( T )) (cid:3) − E (cid:2) n ( T , ˜ Y ( )) (cid:3) = M − (cid:229) m = (cid:18) Z t m + t m E hD D n ( T − t , ˜ Y ( t )) , E ( T − t m ) F ( X m ) − E ( T − t ) F ( E ( t − T ) ˜ Y ( t )) E H i d t + E Z t m + t m Tr h D n ( T − t , ˜ Y ( t )) (cid:16) E ( T − t m ) BQ (cid:0) E ( T − t m ) BQ (cid:1) ∗ − E ( T − t ) BQ (cid:0) E ( T − t ) BQ (cid:1) ∗ (cid:17)i d t (cid:19) = M − (cid:229) m = (cid:0) K m + K m (cid:1) . (5.35)Now, we further decompose K m and K m as K m = Z t m + t m E hD D n ( T − t , ˜ Y ( t )) , (cid:0) E ( T − t m ) − E ( T − t ) (cid:1) F ( X m ) E H i d t + Z t m + t m E hD D n ( T − t , ˜ Y ( t )) , E ( T − t ) (cid:0) F ( X m ) − F ( ˜ X ( t )) (cid:1)E H i d t = K , m + K , m , (5.36) and K m = E Z t m + t m Tr h D n ( T − t , ˜ Y ( t )) (cid:0) E ( T − t m ) − E ( T − t ) (cid:1) BQ (cid:0) E ( T − t m ) BQ (cid:1) ∗ i d t + E Z t m + t m Tr h D n ( T − t , ˜ Y ( t )) E ( T − t ) BQ (cid:0) ( E ( T − t m ) − E ( T − t )) BQ (cid:1) ∗ i d t = E Z t m + t m Tr h D n ( T − t , ˜ Y ( t )) (cid:0) E ( T − t m ) − E ( T − t ) (cid:1) BQ (cid:0) E ( T − t m ) B (cid:1) ∗ i d t + E Z t m + t m Tr h D n ( T − t , ˜ Y ( t )) E ( T − t ) BQ (cid:0) ( E ( T − t m ) − E ( T − t )) B (cid:1) ∗ i d t = K , m + K , m . (5.37)In (5.36), the fact was used that E ( t − T ) ˜ Y ( t ) = ˜ X ( t ) by the previous definitions. In the sequel, we shallestimate K , m , K , m , K , m and K , m separately. With the aid of (3.10), (3.12) and (5.18), we first estimate K , m as follows | K , m | ≤ Z t m + t m E (cid:2)(cid:13)(cid:13) D n ( T − t , ˜ Y ( t )) (cid:13)(cid:13) H · c t (cid:13)(cid:13) F ( X m ) (cid:13)(cid:13) H (cid:3) d t ≤ C t Z t m + t m E (cid:2)(cid:13)(cid:13) F ( u m ) (cid:13)(cid:13) U (cid:3) d t ≤ C t . (5.38)In order to estimate K , m , a common choice is to invoke techniques involved with the Malliavin calculus [1,3,4,14,37]. Here we will provide an alternative way. To be precise, by means of (5.18), H¨older’s inequality,the trigonometric identity and Taylor’s formula in Banach space we obtain | K , m | ≤ Z t m + t m E (cid:2)(cid:13)(cid:13) D n ( T − t , ˜ Y ( t )) (cid:13)(cid:13) H · (cid:13)(cid:13) E ( T − t ) (cid:0) F ( X m ) − F ( ˜ X ( t )) (cid:1)(cid:13)(cid:13) H (cid:3) d t ≤ C Z t m + t m (cid:18) E h(cid:13)(cid:13) L − S ( T − t ) (cid:0) F ( u m ) − F ( ˜ u ( t )) (cid:1)(cid:13)(cid:13) U i + E h(cid:13)(cid:13) L − C ( T − t ) (cid:0) F ( u m ) − F ( ˜ u ( t )) (cid:1)(cid:13)(cid:13) U i (cid:19) d t = C Z t m + t m (cid:13)(cid:13) L − (cid:0) F ( ˜ u ( t )) − F ( u m ) (cid:1)(cid:13)(cid:13) L ( W , U ) d t ≤ C Z t m + t m Z (cid:13)(cid:13) L − F ′ (cid:0) u m + r ( ˜ u ( t ) − u m ) (cid:1) · (cid:0) ˜ u ( t ) − u m (cid:1)(cid:13)(cid:13) L ( W , U ) d r d t . (5.39)Proposition 3.1 ensures that, for r ∈ [ , ] , p ∈ [ , ¥ ) , k = min ( b , − e ) with arbitrarily small e > k u m + r ( ˜ u ( t ) − u m ) k L p ( W , ˙ H k ) ≤ sup t ∈ [ , T ] k ˜ u ( t ) k L p ( W , ˙ H k ) ≤ sup t ∈ [ , T ] k ˜ X ( t ) k L p ( W , H k ) ≤ C . (5.40)Accordingly, by (5.3), (5.28), H¨older’s inequality and also taking the preceding estimate into account, wederive from (5.39) that | K , m | ≤ C Z t m + t m Z (cid:13)(cid:13) k ˜ u ( t ) − u m k − k · (cid:0) k u m + r ( ˜ u ( t ) − u m ) k k + (cid:1)(cid:13)(cid:13) L ( W , R ) d r d t ≤ C t + min ( b + k , ) = C t + min ( b , + b − e , ) (5.41)with k = min ( b , − e ) for arbitrarily small e >
0. This together with (5.38) yields | K m | ≤ C t + min ( b , + b − e , ) . (5.42) n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 23 Now we turn to the term K m . Concerning K , m , we use (2.17), (5.18) and (1.8) to get (cid:12)(cid:12)(cid:12) Tr (cid:2) D n ( T − t , ˜ Y ( t )) (cid:0) E ( T − t m ) − E ( T − t ) (cid:1) BQ (cid:0) E ( T − t m ) B (cid:1) ∗ (cid:3)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13) D n ( T − t , ˜ Y ( t )) (cid:0) E ( T − t m ) − E ( T − t ) (cid:1) BQ (cid:0) E ( T − t m ) B (cid:1) ∗ (cid:13)(cid:13) L ( H ) ≤ (cid:13)(cid:13) D n ( T − t , ˜ Y ( t )) E ( T − t ) (cid:0) E ( t − t m ) − I (cid:1) B L − b (cid:13)(cid:13) L ( U , H ) × (cid:13)(cid:13) L − + b Q L − (cid:13)(cid:13) L ( U ) × (cid:13)(cid:13) L (cid:0) E ( T − t m ) B (cid:1) ∗ (cid:13)(cid:13) L ( H , U ) ≤ C (cid:13)(cid:13)(cid:0) E ( t − t m ) − I (cid:1) B L − b (cid:13)(cid:13) L ( U , H ) × (cid:13)(cid:13) E ( T − t m ) B L (cid:13)(cid:13) L ( U , H ) , (5.43)where (3.11) implies (cid:13)(cid:13) ( E ( t − t m ) − I ) B L − b u (cid:13)(cid:13) H = (cid:13)(cid:13) L − b S ( t − t m ) u (cid:13)(cid:13) U + (cid:13)(cid:13) L − b (cid:0) C ( t − t m ) − I (cid:1) u (cid:13)(cid:13) U ≤ C ( t − t m ) ( b , ) · k u k U , (5.44)and the trigonometric identity gives (cid:13)(cid:13)(cid:0) E ( T − t m ) B L (cid:1) u (cid:13)(cid:13) H = (cid:13)(cid:13) S ( T − t m ) u (cid:13)(cid:13) U + (cid:13)(cid:13) C ( T − t m ) u (cid:13)(cid:13) U = k u k U . (5.45)With the aid of these two estimates, we derive from (5.43) that (cid:12)(cid:12)(cid:12) Tr (cid:2) D n ( T − t , ˜ Y ( t )) (cid:0) E ( T − t m ) − E ( T − t ) (cid:1) BQ (cid:0) E ( T − t m ) B (cid:1) ∗ (cid:3)(cid:12)(cid:12)(cid:12) ≤ C t min ( b , ) , (5.46)and hence | K , m | ≤ C t + min ( b , ) . (5.47)Following the same arguments as used in the estimate of K , m and noticing that k L − + b Q L − k L ( U ) = k L − Q L − + b k L ( U ) , one can similarly obtain that | K , m | ≤ C t + min ( b , ) . (5.48)Gathering (5.47) and (5.48) together leads us to | K m | ≤ C t + min ( b , ) . (5.49)Taking (5.42), (5.49) and (5.35) into account, we conclude form (5.33) that (cid:12)(cid:12) E (cid:2) F ( X M ) (cid:3) − E (cid:2) X ( T ) (cid:3)(cid:12)(cid:12) ≤ M − (cid:229) m = (cid:0) | K m | + | K m | (cid:1) ≤ M − (cid:229) m = (cid:16) C t + min ( b , + b − e , ) + C t + min ( b , ) (cid:17) ≤ C t min ( b , + b − e , ) , (5.50)which finishes the proof of Theorem 5.1. (cid:3) To illustrate the previous assumptions, we give a concrete example as follows.
Example 5.1
Consider a class of SWEs as discussed in Subsection 4.1, where we assign here d = O =( , ) , g ( x , u ) ≡ x ∈ ( , ) , u ∈ R , and f : [ , ] × R → R is assumed to be a smooth nonlinearfunction satisfying | f ( x , u ) | ≤ L ( | u | + ) , (cid:12)(cid:12)(cid:12) ¶ f ¶ u ( x , u ) (cid:12)(cid:12)(cid:12) ≤ L , (cid:12)(cid:12)(cid:12) ¶ f ¶x¶ u ( x , u ) (cid:12)(cid:12)(cid:12) ≤ L , and (cid:12)(cid:12)(cid:12) ¶ f ¶ u ( x , u ) (cid:12)(cid:12)(cid:12) ≤ L (5.51) for all x ∈ ( , ) , u ∈ R . Let U : = L (( , ) , R ) and let F : U → U be the Nemytskij operators, defined by(4.3). Such mappings are, in general, not Fr´echet differentiable in U , but only Gˆateaux differentiable, withthe corresponding derivative operators given by F ′ ( u )( y ) ( x ) = ¶ f ¶ u ( x , u ( x )) · y ( x ) , x ∈ ( , ) , (5.52) F ′′ ( u )( y , y ) ( x ) = ¶ f ¶ u ( x , u ( x )) · y ( x ) · y ( x ) , x ∈ ( , ) (5.53)for all u , y , y , y ∈ U . Thanks to H¨older’s inequality and a Sobolev inequality: ˙ H g = D ( L g ) is continu-ously embedded into L ¥ (( , ) , R ) for g > , we get k L − F ′′ ( u )( y , y ) k U = sup k y k U ≤ (cid:12)(cid:12)(cid:10) L − F ′′ ( u )( y , y ) , y (cid:11) U (cid:12)(cid:12) = sup k y k U ≤ (cid:12)(cid:12)(cid:10) F ′′ ( u )( y , y ) , L − y (cid:11) U (cid:12)(cid:12) ≤ (cid:13)(cid:13) F ′′ ( u )( y , y ) k L (( , ) , R ) · sup k y k U ≤ k L − y k L ¥ (( , ) , R ) ≤ L k y k L (( , ) , R ) · k y k L (( , ) , R ) · C sup k y k U ≤ k y k U ≤ C k y k U k y k U . (5.54)Now it remains to check (5.3). Since in our example F is a Nemytskij operator, the derivative operator F ′ ( u ) , u ∈ U is self-adjoint. Therefore, using the self-adjointness of L and F ′ ( u ) , u ∈ U yields that (cid:13)(cid:13) L − F ′ ( u ) z (cid:13)(cid:13) U = sup k y k U ≤ (cid:12)(cid:12)(cid:10) L − F ′ ( u ) z , y (cid:11) U (cid:12)(cid:12) = sup k y k U ≤ (cid:12)(cid:12)(cid:10) z , (cid:0) F ′ ( u ) (cid:1) ∗ L − y (cid:11) U (cid:12)(cid:12) = sup k y k U ≤ (cid:12)(cid:12)(cid:10) L − k z , L k F ′ ( u ) L − y (cid:11) U (cid:12)(cid:12) ≤ (cid:13)(cid:13) L − k z (cid:13)(cid:13) U · sup k y k U ≤ (cid:13)(cid:13) L k F ′ ( u ) L − y (cid:13)(cid:13) U (5.55)with k = min ( b , − e ) for arbitrarily small e >
0. Further, the setting in Example 5.1 suffices to ensure k F ′ ( u ) j k g ≤ C ( k u k g + ) k j k s (5.56)for any u ∈ ˙ H g , j ∈ ˙ H s , with arbitrarily g ∈ ( , ) and s ∈ ( , ] (see [38, Lemma 4.4] for more details).By (5.56) with g = k = min ( b , − e ) , s = (cid:13)(cid:13) L − F ′ ( u ) z (cid:13)(cid:13) U ≤ (cid:13)(cid:13) L − k z (cid:13)(cid:13) U · C sup k y k U ≤ (cid:0) k u k k + (cid:1) k y k U ≤ C k z k − k · (cid:0) k u k k + (cid:1) . (5.57)This completes the verification of the key condition (5.3). (cid:3) In this section, numerical results are included to demonstrate the above assertions. Consider the Sine-Gordon equation subject to noise as follows ¶ u ¶ t = ¶ u ¶x − sin ( u ) + ( s + s u ) ˙ W , t ∈ ( , ] , x ∈ ( , ) , u ( , x ) = u ( x ) , ¶ u ¶ t ( , x ) = v ( x ) , x ∈ ( , ) , u ( t , ) = u ( t , ) = , t ∈ ( , ] , (6.1) n exponential integrator scheme for time discretization of nonlinear stochastic wave equation 25 where for simplicity we assume that the covariance operator Q has the same eigenfunctions as the Laplacianwith Dirichlet boundary condition, i.e., f i = e i = √ ( i px ) , i ∈ N . Such equation driven by additivenoise ( s =
0) has been considered as a numerical example in [8,38]. Although only semi-discretizationin time has been investigated in this article, spatial discretization needs to be done in order to performthe simulations on the computer. To this end we simply spatially discretize (6.1) via a spectral Galerkinmethod, with N = fixed (see, for example, [38]). Since the true solutions required in the followingare not available, we take numerical solutions produced by the Crank-Nicolson scheme, using very smallstepsize t exact = − , as reference solutions. Furthermore, the expectations are approximated by averagesover 1000 samples in the following numerical tests.First, let us start with tests on the strong convergence rates and examine the strong approximation error (cid:0) E (cid:2) k u ( ) − u M k U (cid:3)(cid:1) = (cid:0) E R | u ( , x ) − u M ( x ) | d x (cid:1) , which arises due to the linear implicit Euler (LIE),Crank-Nicolson (CN) and exponential Euler (EE) time discretizations. We choose a set of parameters as s = , s = u ( x ) = , v ( x ) = cos ( x ) . For the case of space-time white noise ( q i = , i ∈ N ) and trace-class noise ( q i = i − . , i ∈ N ), the left and right plots of Fig.6.1, respectively, depict the strong approximationerrors against M on a log-log scale, with M = k , k = , , ,
8. It turns out that the exponential Euler(EE) scheme (1.4) performs better than the linear implicit Euler (LIE) and Crank-Nicolson (CN) timediscretizations and exhibits the right strong rates, i.e., order for the space-time white noise and order 1for the trace-class noise. −3 −2 −1 −3 −2 −1 S t r ong e rr o r s EECNLIEOrder 1/2 10 −3 −2 −1 −4 −3 −2 −1 S t r ong e rr o r s EECNLIEOrder 1
Fig. 6.1
Strong approximation errors for time-stepping schemes applied to (6.1) with space-time white noise (left) and trace classnoise (right).
To illustrate the weak convergence results, we consider the additive space-time white noise and as-sign s = , s = u ( x ) = cos ( p ( x − )) , v ( x ) =
0. In addition, we choose a particular test function j ( u ) = R u ( x ) sin ( px ) d x and consider the weak errors (5.5) when using time-stepping schemes toapproximate the quantity E [ j ( u ( ))] = E (cid:2) R u ( , x ) sin ( px ) d x (cid:3) . In Table 6.1, numerical approxima-tions of E [ j ( u ( ))] = − . t = − , one can observe a good behavior of the scheme (1.4). Also, weak approximation errors of the threeschemes are plotted in Fig. 6.2, where one can observe an expected weak rate of 1 for the scheme (1.4).Despite larger weak approximation errors, the other two schemes (LIE and CN) exhibit almost weak orderone in this particular example. Table 6.1
Numerical approximations of E [ j ( u ( ))] = − . t Linear implicit Euler Crank-Nicolson Scheme (1.4)2 − -2.77375 -5.07689 -4.898652 − -3.62967 -4.99096 -4.907632 − -4.21349 -4.95601 -4.916202 − -4.55052 -4.93828 -4.91888 −2 −1 −3 −2 −1 W ea k e rr o r s EECNLIEOrder 1
Fig. 6.2
Weak approximation errors for various time-stepping schemes applied to (6.1) with space-time white noise.
Acknowledgment
The author thanks the anonymous referee whose insightful comments and valuable suggestions are crucialto the improvement of the manuscript. The author would like to thank Professor Arnulf Jentzen for hisfinancial support and helpful discussions during the author’s short visit to ETH Z¨urich in 2013. Thanks alsogo to Dr. Fengze Jiang for his careful reading the early version of this manuscript. This work was partiallysupported by National Natural Science Foundations of China under grant numbers 11301550, 11171352,China Postdoctoral Science Foundation under grant numbers 2013M531798, 2014T70779 and ResearchFoundation of Central South University.