Higher order approximation for stochastic wave equation
aa r X i v : . [ m a t h . NA ] J u l HIGHER ORDER APPROXIMATION FOR STOCHASTIC SPACEFRACTIONAL WAVE EQUATION FORCED BY AN ADDITIVESPACE-TIME GAUSSIAN NOISE ∗ XING LIU † AND
WEIHUA DENG ‡ Abstract.
The infinitesimal generator (fractional Laplacian) of a process obtained by subordi-nating a killed Brownian motion catches the power-law attenuation of wave propagation. This paperstudies the numerical schemes for the stochastic wave equation with fractional Laplacian as the spaceoperator, the noise term of which is an infinite dimensional Brownian motion or fractional Brownianmotion (fBm). Firstly, we establish the regularity of the mild solution of the stochastic fractionalwave equation. Then a spectral Galerkin method is used for the approximation in space, and thespace convergence rate is improved by postprocessing the infinite dimensional Gaussian noise. Inthe temporal direction, when the time derivative of the mild solution is bounded in the sense ofmean-squared L p -norm, we propose a modified stochastic trigonometric method, getting a higherstrong convergence rate than the existing results, i.e., the time convergence rate is bigger than 1.Particularly, for time discretization, the provided method can achieve an order of 2 at the expensesof requiring some extra regularity to the mild solution. The theoretical error estimates are confirmedby numerical experiments. Key words. spectral Galerkin method, modified stochastic trigonometric method, higher strongconvergence rate, extra regularity
AMS subject classifications.
1. Introduction.
The wave propagation in ideal medium is well described bythe classical wave equation ∂ u ( x, t ) /∂t = ∆ u ( x, t ). However, sometimes the clas-sical wave equation fails to model the wave propagations in complex inhomogeneousmedia (e.g., viscous damping in the seismic isolation of buildings, medical ultrasound,and seismic wave propagation [4, 10, 18, 25]), because of their power-law attenuations.One of the most effective ways to characterize the wave propagation with power-lawattenuations is to resort to the nonlocal operator — the infinitesimal generator (frac-tional Laplacian) of a process obtained by subordinating a killed Brownian motion.Currently, two stochastic processes are very popular: one is killed subordinateBrownian motion, and the other is subordinate killed Brownian motion. Let D bea bounded region, B ( t ) be a Brownian motion with B (0) ∈ D , and τ D = inf { t > B ( t ) / ∈ D } . Denote T t as an α -stable subordinator. The first stochastic process(killed subordinate Brownian motion) [9] is defined as X ( t ) = (cid:26) B ( T t ) , t < τ D , Θ , t ≥ τ D , where Θ is a coffin state, meaning that the subordinate Brownian motion will be killedwhen first leaving the domain D ; while the second stochastic process (subordinate ∗ Submitted to the editors DATE.
Funding:
This work was supported by the National Natural Science Foundation of China underGrant No. 11671182, and the AI and Big Data Funds under Grant No. 2019620005000775. † School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathemat-ics and Complex Systems, Lanzhou University, Lanzhou 730000, People’s Republic of China([email protected]). ‡ Corresponding author. School of Mathematics and Statistics, Gansu Key Laboratory of AppliedMathematics and Complex Systems, Lanzhou University, Lanzhou 730000, People’s Republic ofChina ([email protected]). 1
XING LIU AND WEIHUA DENG killed Brownian motion) [23] is X ( t ) = (cid:26) B ( T t ) , T t < τ D , Θ , T t ≥ τ D with Θ still being a coffin state, implying to subordinate a killed Brownian motion(when first leaving the domain D ). The infinitesimal generator of X ( t ) has the form( − ∆ ) α u ( x ) = c n,α P . V . Z R n u ( x ) − u ( y ) | x − y | n +2 α dy, α ∈ (0 , , where c n,α = α α Γ( n/ α ) π n/ Γ(1 − α ) , P . V . means the principal value integral, and u ( y ) = 0for y ∈ R n \ D . Denote the infinitesimal generator of X ( t ) as ( − ∆) α and − ∆ theinfinitesimal generator of killed Brownian motion. It shows that [21, 23] if { ( λ i , φ i ) } ∞ i =1 are the eigenpairs of − ∆, then { ( λ αi , φ i ) } ∞ i =1 are the eigenpairs of ( − ∆) α , i.e.,(1.1) (cid:26) − ∆ φ i = λ i φ i , in D,φ i = 0 , on ∂D, and(1.2) (cid:26) ( − ∆) α φ i = λ αi φ i , in D,φ i = 0 , on ∂D. The operator used in this paper is the one defined in (1.2). Moreover, we arealso concerned with the external noises that possibly affect the wave propagation.Two most popular external noises are white noise and fractional Gaussian noise,both of which are considered in this paper. The fractional Gaussian noise is definedas the formal derivative of the fractional Brownian motion (fBm) β H ( t ), which isGaussian process with an index H ∈ (0 , H = , the fBm reduces to astandard Brownian motion. The formal derivative of Brownian motion is white noise.For H = , unlike Brownian motion, the fBm exhibits long-range dependence: thebehavior of the process after a given time t depends on the situation at t and thewhole history of the process up to time t [6]. According to the properties of the fBmand Brownian motion, one can choose the appropriate noise in practical applications.With the above introduction of nonlocal operator and the external noise, themodel we discuss in this paper is the stochastic wave equation(1.3) d ˙ u ( x,t )d t = − ( − ∆) α u ( x, t ) + f ( u ( x, t )) + ˙ B H ( x, t ) in D × (0 , T ] ,u ( x,
0) = u , ˙ u ( x,
0) = v in D,u ( x, t ) = 0 , in ∂D × (0 , T ] , where ˙ u ( x, t ) is the first order time derivative of u ( x, t ), d/dt means the partial de-rivative with respect to t , f is the source term, D ⊂ R d ( d = 1 , , B H ( x, t ) isthe formal derivative of the infinite dimensional space-time Gaussian process B H ( x, t )with 0 < α ≤ ≤ H <
IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION u ( x, t ) is bounded in the sense of mean-squared L p -norm, then by modifying thestochastic trigonometric method for time discretization, we can obtain a high orderconvergence rate. In particular, as H = , we use the independent increment propertyof Brownian motion to obtain the optimal temporal error estimate; for H ∈ (cid:0) , (cid:1) ,by using the covariance of stochastic integral for fBm (Lemma 2.2), we obtain the op-timal error estimate in time. For space approximation, the spectral Galerkin schemeis used; and the space convergence rate is improved by postprocessing the additiveGaussian noise.This paper is organized as follows. In the next section, we introduce some nota-tions and preliminaries, including assumptions and properties of fBm. In Section 3,by using the Dirichlet eigenpairs, we present the regularity of the mild solution u ( x, t )and the time derivative ˙ u ( x, t ) in the sense of mean-squared L p -norm. In Section 4, thespectral Galerkin spatial semidiscretization of (1.3) and the postprocessing approachof the additive space-time Gaussian noise are discussed. In Section 5, we modifythe stochastic trigonometric method to obtain a high order temporal discretizationof (1.3); and the convergence order for the proposed fully discrete scheme is derived.The numerical experiments are performed in Section 6. We end the paper with somediscussions in Section 7.
2. Notations and preliminaries.
In this section, we gather preliminary resultson the Dirichlet eigenpairs and fBm, which are commonly used in the paper.Let U = L ( D ; R ) be a real separable Hilbert space with L inner product h· , ·i and the corresponding induced norm k · k . We define the unbounded linear operator A ν by A ν u = ( − ∆) ν u on the domaindom ( A ν ) = { A ν u ∈ U : u ( x ) = 0 , x ∈ ∂D } . Then Equation (1.2) implies that A ν φ i ( x ) = λ ν i φ i ( x )and A ν u = ∞ X i =1 λ ν i h u, φ i ( x ) i φ i ( x ) , where φ i ( x ), i = 1 , , . . . , denote the normalized eigenfunctions of the fractional Lapla-cian operator ( − ∆) ν , and λ ν i , i = 1 , , . . . , are the corresponding eigenvalues. More-over, we define the Hilbert space ˙ U ν = dom (cid:0) A ν (cid:1) equipped with the inner product h u, v i ν = ∞ X i =1 λ ν i h u, φ i ( x ) i × λ ν i h v, φ i ( x ) i XING LIU AND WEIHUA DENG and norm k u k ν = ∞ X i =1 λ νi h u, φ i ( x ) i ! . In particular, ˙ U = U . Lemma
Let Ω denote a bounded domain in R d , d ∈ { , , } ,and | Ω | the volume of Ω . Let λ i be the i-th eigenvalue of the Dirichlet homogeneousboundary problem for the Laplacian operator − ∆ in Ω . Then C i d ≤ λ i ≤ C i d , where i ∈ N , and the constants C and C are independent of i . Assumption The function f : U → U in (1.3) satisfies k f ( u ) − f ( v ) k . k u − v k for any u, v ∈ U, and k A ν f ( u ) k . k A ν u k for u ∈ ˙ U ν with ν ≥ . For later use, we collect concepts of fBm; for more details, one can refer to [3, 7, 11,14, 19].
Definition Let β H ( t ) be the two-sided one-dimensional fBm with Hurst index H ∈ (0 , and t ∈ R . The stochastic process β H ( t ) is characterized by the properties: (i) β H (0) = 0 ; (ii) E [ β H ( t )] = 0 , t ∈ R ; (iii) E [ β H ( t ) β H ( s )] = (cid:0) | t | H + | s | H − | t − s | H (cid:1) , t, s ∈ R , where E denotes the expectation. As H = , β H ( t ) is a standard Brownian motion,being a process with independent increment. Assumption Let driven stochastic process B H ( x, t ) be a cylindrical fBm withrespect to the normal filtration {F t } t ∈ [0 ,T ] . The infinite dimensional space-time sto-chastic process can be represented by the formal series B H ( x, t ) = ∞ X i =1 σ i β iH ( t ) φ i ( x ) , where | σ i | . λ − ρi ( ρ ≥ , λ i is given in Lemma 2.1), β iH ( t ) , i = 1 , , . . . , are mutuallyindependent real-valued fractional Brownian motions with ≤ H < , and { φ i ( x ) } i ∈ N is an orthonormal basis of U . We define L p ( D, ˙ U ν ) to be the separable Hilbert space of p -times integrable ran-dom variables with norm k u k L p ( D, ˙ U ν ) = (E [ k u k pν ]) p , ν ≥ . Lemma
For f, g ∈ L ( R ; R ) ∩ L ( R ; R ) , as < H < , we have E (cid:20)Z R f ( s )d β H ( s ) (cid:21) = 0 IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION and E (cid:20)Z R f ( s )d β H ( s ) Z R g ( t )d β H ( t ) (cid:21) = H (2 H − Z R Z R E [ f ( s ) g ( t )] | s − t | H − d s d t. For H ∈ (cid:0) , (cid:1) and that R ts f ( r )( r − s ) H − (cid:0) sr (cid:1) − H d r belongs to L ([0 , T ] , R ),an expression of the Wiener integral with respect to fBm is introduced as [1, 26] Z t f ( s )d β H ( s ) = C H Z t Z ts (cid:18) H − (cid:19) f ( r )( r − s ) H − (cid:16) sr (cid:17) − H d r d β ( s ) , s, t ∈ [0 , T ] , where β ( t ) is standard Brownian motion, and C H = (cid:18) H × Γ( − H )Γ( H + )Γ(2 − H ) (cid:19) .
3. Regularity of the solution.
To begin with we can give a system of equationsby coupling (1.3) and d u ( x, t ) = ˙ u ( x, t )d t . The system of equations is beneficial toanalyze the regularity of the mild solution of (1.3), including existence, uniqueness,and time H¨older continuity. Moreover, the system of equations is transformed into anequivalent form, which will be used to obtain the approximation of (1.3).In the interest of brevity and readability, we use the following equation insteadof (1.3)(3.1) d ˙ u ( t ) = − A α u ( t )d t + f ( u ( t )) d t + d B H ( t ) , in D × (0 , T ] ,u (0) = u , ˙ u (0) = v in D,u ( t ) = 0 , in ∂D, where u ( t ) = u ( x, t ) and B H ( t ) = B H ( x, t ). Let v ( t ) = ˙ u ( t ). Then(3.2) d X ( t ) = Λ X ( t )d t + (cid:20) f ( u ( t )) (cid:21) d t + (cid:20) I (cid:21) d B H ( t ) , where X ( t ) = (cid:20) u ( t ) v ( t ) (cid:21) , Λ = (cid:20) I − A α (cid:21) . Then a formal mild solution X ( t ) for (3.2) is given as(3.3) X ( t ) = e Λ t X (0) + Z t e Λ( t − s ) (cid:20) f ( u ( s )) (cid:21) d s + Z t e Λ( t − s ) (cid:20) I (cid:21) d B H ( s ) , where e Λ t can be expressed as(3.4) e Λ t = " cos (cid:0) A α t (cid:1) A − α sin (cid:0) A α t (cid:1) − A α sin (cid:0) A α t (cid:1) cos (cid:0) A α t (cid:1) . The definitions of cosine operator cos (cid:0) A α t (cid:1) and sine operator sin (cid:0) A α t (cid:1) are given inAppendix B. Substituting (3.4) into (3.3), then two components of X ( t ) are obtained XING LIU AND WEIHUA DENG as u ( t ) = cos (cid:0) A α t (cid:1) u + A − α sin (cid:0) A α t (cid:1) v + Z t A − α sin (cid:0) A α ( t − s ) (cid:1) f ( u ( s )) d s + Z t A − α sin (cid:0) A α ( t − s ) (cid:1) d B H ( s ) ,v ( t ) = − A α sin (cid:0) A α t (cid:1) u + cos (cid:0) A α t (cid:1) v + Z t cos (cid:0) A α ( t − s ) (cid:1) f ( u ( s )) d s + Z t cos (cid:0) A α ( t − s ) (cid:1) d B H ( s ) . (3.5)In order to obtain the regularity of u ( t ) and v ( t ), we need to consider the regularityestimate of the stochastic integral in (3.5). For ≤ H < p >
1, the Burkh¨older-Davis-Gundy inequality [20, 22] impliesE "(cid:13)(cid:13)(cid:13)(cid:13)Z t A γ − α sin (cid:0) A α ( t − s ) (cid:1) d B ( s ) (cid:13)(cid:13)(cid:13)(cid:13) p (3.6) ≤ C p Z t ∞ X i =1 (cid:12)(cid:12)(cid:12) λ γ − α − ρ i sin (cid:16) λ α i ( t − s ) (cid:17)(cid:12)(cid:12)(cid:12) d s ! p ≤ C p t ∞ X i =1 i γ − α − ρ ) d ! p andE "(cid:13)(cid:13)(cid:13)(cid:13)Z t A γ − α sin (cid:0) A α ( t − s ) (cid:1) d B H ( s ) (cid:13)(cid:13)(cid:13)(cid:13) p = E "(cid:13)(cid:13)(cid:13)(cid:13) C H (cid:18) H − (cid:19) Z t Z ts A γ − α sin (cid:0) A α ( t − r ) (cid:1) (cid:16) sr (cid:17) − H ( r − s ) H − d r d B ( s ) (cid:13)(cid:13)(cid:13)(cid:13) p ≤ C pH (cid:18) H − (cid:19) p C p Z t ∞ X i =1 λ γ − α − ρi (cid:18)Z ts (cid:16) sr (cid:17) − H ( r − s ) H − d r (cid:19) d s ! p ≤ C pH (cid:18) H − (cid:19) p C p Z t ∞ X i =1 λ γ − α − ρi (cid:16) st (cid:17) − H (cid:18)Z ts ( r − s ) H − d r (cid:19) d s ! p ≤ C pH (cid:18) − H (cid:19) p C p t H ∞ X i =1 i γ − α − ρ ) d ! p . (3.7)When γ − α − ρ ) d < −
1, then the infinite series ∞ P i =1 i γ − α − ρ ) d < ∞ .One can obtain the following regularity results of the mild solution u ( t ) and v ( t )by using the above estimates, Lemma 2.1, and (3.5). Theorem
Suppose that Assumptions 1-2 are satisfied, k u k L p ( D, ˙ U γ ) < ∞ , k v k L p ( D, ˙ U γ − α ) < ∞ , ε > , γ = α + 2 ρ − d + ε , and γ > . Then there exists a unique IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION mild solution X ( t ) for (3.2) and (3.8) k u ( t ) k L p ( D, ˙ U γ ) + k v ( t ) k L p ( D, ˙ U γ − α ) . t H ε + k u k L p ( D, ˙ U γ ) + k v k L p ( D, ˙ U γ − α ) . Furthermore, (i) for γ ≤ α , k u ( t ) − u ( s ) k L ( D,U ) . ( t − s ) γα (cid:18) t H ε + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:19) ;(ii) for γ > α , k u ( t ) − u ( s ) k L ( D,U ) . ( t − s ) (cid:16) t H + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) . Proof.
Let us start with the estimate of A γ u ( t ) in L p ( D, U ) norm. Combiningthe triangle inequality, (3.6), (3.7), the expression of u ( t ) in (3.5), and the assumptionof f , we obtain (cid:13)(cid:13)(cid:13) A γ u ( t ) (cid:13)(cid:13)(cid:13) L p ( D,U ) . (cid:13)(cid:13)(cid:13) A γ cos (cid:0) A α t (cid:1) u (cid:13)(cid:13)(cid:13) L p ( D,U ) + (cid:13)(cid:13)(cid:13) A γ − α sin (cid:0) A α t (cid:1) v (cid:13)(cid:13)(cid:13) L p ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t A γ − α sin (cid:0) A α ( t − s ) (cid:1) f ( u ( s )) d s (cid:13)(cid:13)(cid:13)(cid:13) L p ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t A γ − α sin (cid:0) A α ( t − s ) (cid:1) d B H ( s ) (cid:13)(cid:13)(cid:13)(cid:13) L p ( D,U ) . (cid:13)(cid:13)(cid:13) A γ u (cid:13)(cid:13)(cid:13) L p ( D,U ) + (cid:13)(cid:13)(cid:13) A γ − α v (cid:13)(cid:13)(cid:13) L p ( D,U ) + Z t (cid:13)(cid:13)(cid:13) A γ u ( s ) (cid:13)(cid:13)(cid:13) L p ( D,U ) d s + t H ε . The application of Gr¨onwall’s inequality leads to (cid:13)(cid:13)(cid:13) A γ u ( t ) (cid:13)(cid:13)(cid:13) L p ( D,U ) . (cid:13)(cid:13)(cid:13) A γ u (cid:13)(cid:13)(cid:13) L p ( D,U ) + (cid:13)(cid:13)(cid:13) A γ − α v (cid:13)(cid:13)(cid:13) L p ( D,U ) + t H ε . The bound of k v ( t ) k L p ( D, ˙ U γ − α ) can be achieved in the same way, that is (cid:13)(cid:13)(cid:13) A γ − α v ( t ) (cid:13)(cid:13)(cid:13) L p ( D,U ) . (cid:13)(cid:13)(cid:13) A γ u (cid:13)(cid:13)(cid:13) L p ( D,U ) + (cid:13)(cid:13)(cid:13) A γ − α v (cid:13)(cid:13)(cid:13) L p ( D,U ) + t H ε . Then using above estimates leads to k u ( t ) k L p ( D, ˙ U γ ) + k v ( t ) k L p ( D, ˙ U γ − α ) . t H ε + k u k L p ( D, ˙ U γ ) + k v k L p ( D, ˙ U γ − α ) . XING LIU AND WEIHUA DENG
Next, we discuss the time H¨older continuity of u ( t ). Equation (3.5) impliesE h k u ( t ) − u ( s ) k i . E h(cid:13)(cid:13)(cid:0) cos (cid:0) A α t (cid:1) − cos (cid:0) A α s (cid:1)(cid:1) u (cid:13)(cid:13) + (cid:13)(cid:13) A − α (cid:0) sin (cid:0) A α t (cid:1) − sin (cid:0) A α s (cid:1)(cid:1) v (cid:13)(cid:13) i + E "(cid:13)(cid:13)(cid:13)(cid:13)Z ts A − α sin (cid:0) A α ( t − r ) (cid:1) f ( u ( r )) d r (cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)Z ts A − α sin (cid:0) A α ( t − r ) (cid:1) d B H ( r ) (cid:13)(cid:13)(cid:13)(cid:13) + E "(cid:13)(cid:13)(cid:13)(cid:13)Z s A − α (cid:0) sin (cid:0) A α ( t − r ) (cid:1) − sin (cid:0) A α ( s − r ) (cid:1)(cid:1) f ( u ( r )) d r (cid:13)(cid:13)(cid:13)(cid:13) + E "(cid:13)(cid:13)(cid:13)(cid:13)Z s A − α (cid:0) sin (cid:0) A α ( t − r ) (cid:1) − sin (cid:0) A α ( s − r ) (cid:1)(cid:1) d B H ( r ) (cid:13)(cid:13)(cid:13)(cid:13) . I + I + I + I + I + I . For < H <
1, the inequality cos( a ) − cos( b ) . | a − b | θ (0 ≤ θ ≤
1) implies I = E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i (cid:16) cos (cid:16) λ α i t (cid:17) − cos (cid:16) λ α i s (cid:17)(cid:17) h u , φ i ( x ) i φ i ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . E "X i λ γi ( t − s ) · min { γα , } h u , φ i ( x ) i . ( t − s ) min { γα , } E (cid:20)(cid:13)(cid:13)(cid:13) A γ u (cid:13)(cid:13)(cid:13) (cid:21) . Similar to the derivation of I , it holds that I . ( t − s ) min { γα , } E (cid:20)(cid:13)(cid:13)(cid:13) A γ − α v (cid:13)(cid:13)(cid:13) (cid:21) . For the H¨older regularity of the third term, using Equation (3.8) and Assumption 1leads to I . ( t − s ) Z ts E h(cid:13)(cid:13) A − α sin (cid:0) A α ( t − r ) (cid:1) f ( u ( r )) (cid:13)(cid:13) i d r . ( t − s ) Z ts E h k u ( r ) k + 1 i d r . ( t − s ) (cid:18) E (cid:20)(cid:13)(cid:13)(cid:13) A γ u (cid:13)(cid:13)(cid:13) (cid:21) + E (cid:20)(cid:13)(cid:13)(cid:13) A γ − α v (cid:13)(cid:13)(cid:13) (cid:21) + 1 (cid:19) . Take θ = min (cid:8) γ α , (cid:9) . Combining the fact that | sin( t ) | . | t | θ ( t ∈ R ) and Lemma2.2 leads to I = E "(cid:13)(cid:13)(cid:13)(cid:13)Z ts A − α sin (cid:0) A α ( t − r ) (cid:1) d B H ( r ) (cid:13)(cid:13)(cid:13)(cid:13) . X i λ − α − ρi Z ts Z ts (cid:12)(cid:12)(cid:12) sin (cid:16) λ α i ( t − r ) (cid:17)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) sin (cid:16) λ α i ( t − r ) (cid:17)(cid:12)(cid:12)(cid:12) × | r − r | H − d r d r . ( t − s ) H +min { γα , } X i λ γ − α − ρi . ( t − s ) H +min { γα , } . IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION a ) − sin( b ) . | a − b | θ and the assumption of f with θ = min (cid:8) γα , (cid:9) , wehave I . ( t − s ) min { γα , } E "(cid:13)(cid:13)(cid:13)(cid:13)Z s A γ − α f ( u ( r )) d r (cid:13)(cid:13)(cid:13)(cid:13) . ( t − s ) min { γα , } (cid:18) E (cid:20)(cid:13)(cid:13)(cid:13) A γ u (cid:13)(cid:13)(cid:13) (cid:21) + E (cid:20)(cid:13)(cid:13)(cid:13) A γ − α v (cid:13)(cid:13)(cid:13) (cid:21) + t H (cid:19) . For γ > α , by using Lemma 2.2 and the inequality sin (cid:16) λ α i ( t − r ) (cid:17) − sin (cid:16) λ α i ( s − r ) (cid:17) . λ α i ( t − s ), we get the bound of I , i.e., I . X i λ − α − ρi Z s Z s (cid:12)(cid:12)(cid:12) sin (cid:16) λ α i ( t − r ) (cid:17) − sin (cid:16) λ α i ( s − r ) (cid:17)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) sin (cid:16) λ α i ( t − r ) (cid:17) − sin (cid:16) λ α i ( s − r ) (cid:17)(cid:12)(cid:12)(cid:12) × | r − r | H − d r d r . t H ( t − s ) X i λ α − γ + γ − α − ρi . t H ( t − s ) . When γ ≤ α , using sin (cid:16) λ α i ( t − r ) (cid:17) − sin (cid:16) λ α i ( s − r ) (cid:17) . λ γ i ( t − s ) γα leads to I . t H ( t − s ) γα X i λ γ − α − ρi . t H ε ( t − s ) γα . Then collecting the above estimates arrives atE h k u ( t ) − u ( s ) k i . ( t − s ) (cid:18) t H + E (cid:20)(cid:13)(cid:13)(cid:13) A γ u (cid:13)(cid:13)(cid:13) (cid:21) + E (cid:20)(cid:13)(cid:13)(cid:13) A γ − α v (cid:13)(cid:13)(cid:13) (cid:21)(cid:19) , γ > α andE h k u ( t ) − u ( s ) k i . ( t − s ) γα (cid:18) t H ε + E (cid:20)(cid:13)(cid:13)(cid:13) A γ u (cid:13)(cid:13)(cid:13) (cid:21) + E (cid:20)(cid:13)(cid:13)(cid:13) A γ − α v (cid:13)(cid:13)(cid:13) (cid:21)(cid:19) , γ ≤ α. When H = , the above H¨older regularity results still hold. The proof is completed.In fact, one can get an equivalent form of (3.2) by using variable substitution, theregularity of whose solution is better than the one of the solution of (3.2). Let(3.9) Z ( t ) = X ( t ) − Z t e Λ( t − s ) (cid:20) I (cid:21) d B H ( s ) , where Z ( t ) = " z ( t )˙ z ( t ) . If X ( t ) is the unique mild solution of (3.2), then Z ( t ) is the unique mild solution ofthe partial differential equation(3.10) dd t Z ( t ) = Λ Z ( t ) + (cid:20) f ( u ( t )) (cid:21) for t ∈ (0 , T ] with Z (0) = X (0) , XING LIU AND WEIHUA DENG where f ( u ( t )) = f (cid:18) z ( t ) + Z t A − α sin (cid:0) A α ( t − s ) (cid:1) d B H ( s ) (cid:19) . The unique mild solution of (3.10) is given by(3.11) Z ( t ) = e Λ t Z (0) + Z t e Λ( t − s ) (cid:20) f ( u ( t )) (cid:21) d s. Then we can obtain z ( t ) and ˙ z ( t ) as z ( t ) = cos (cid:0) A α t (cid:1) u + A − α sin (cid:0) A α t (cid:1) v + Z t A − α sin (cid:0) A α ( t − s ) (cid:1) f ( u ( s )) d s, ˙ z ( t ) = − A α sin (cid:0) A α t (cid:1) u + cos (cid:0) A α t (cid:1) v + Z t cos (cid:0) A α ( t − s ) (cid:1) f ( u ( s )) d s. (3.12)Equation (3.10) will be used to obtain the spatial semi-discretization solution of(3.2). Therefore we give the following estimates, which will be used to discuss thespatial error. Corollary
Suppose that Assumptions 1-2 are satisfied, k u k L p ( D, ˙ U γ + α ) < ∞ , k v k L p ( D, ˙ U γ ) < ∞ , ε > , γ = α + 2 ρ − d + ε , and γ > . Then there exists aunique mild solution Z ( t ) for (3.10) and k z ( t ) k L p ( D, ˙ U γ + α ) + k ˙ z ( t ) k L p ( D, ˙ U γ ) . t H ε + k u k L p ( D, ˙ U γ + α ) + k v k L p ( D, ˙ U γ ) . The proof of this corollary is very similar to the proof of Theorem 3.1.
4. Galerkin approximation for spatial discretization.
The convergencerate of the spectral approximation depends on the regularity of the mild solutionin space. Theorem 3.1 and Corollary 3.2 show that z ( t ) is more regular than u ( t ) inspace; so we obtain the spatial approximation of (3.2) by using the spectral Galerkinmethod to discretize (3.10) and postprocessing the stochastic integral.A finite dimensional subspace of U will be needed to implement the Galerkinspatial approximation of (3.10). Denoting the N dimensional subspace of U by U N ,the sequence { φ ( x ) , . . . , φ i ( x ) , . . . , φ N ( x ) } N ∈ N is an orthonormal basis of U N . Thenwe introduce the projection operator P N : U → U N , for ξ ∈ U , P N ξ = N X i =1 h ξ, φ i ( x ) i φ i ( x )and(4.1) h P N ξ, χ i = h ξ, χ i ∀ χ ∈ U N . To obtain the Galerkin formulation of (3.10), we look for z N ( t ) ∈ U N and ˙ z N ∈ U N such that " (cid:10) d z N ( t ) , χ (cid:11)(cid:10) d ˙ z N ( t ) , χ (cid:11) = Λ " (cid:10) z N ( t ) , χ (cid:11)(cid:10) ˙ z N ( t ) , χ (cid:11) d t + " (cid:10) f (cid:0) u N ( t ) (cid:1) , χ (cid:11) d t (4.2) IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION (cid:10) z N , χ (cid:11) = h z , χ i , (cid:10) ˙ z N , χ (cid:11) = h ˙ z , χ i , where f (cid:0) u N ( t ) (cid:1) = f (cid:18) z N ( t ) + P N Z t A − α sin (cid:0) A α ( t − s ) (cid:1) d B H ( s ) (cid:19) . The Galerkin formulation of (3.10) is obtained by using (4.1) and (4.2), that is " d z N ( t )d ˙ z N ( t ) = Λ " z N ( t )˙ z N ( t ) d t + " f N (cid:0) u N ( t ) (cid:1) d t (4.3)and z N = P N u , ˙ z N = P N v , where f N = P N f . We can obtain the mild solution of (4.3) as z N ( t ) = cos (cid:0) A α t (cid:1) u N + A − α sin (cid:0) A α t (cid:1) v N + Z t A − α sin (cid:0) A α ( t − s ) (cid:1) f N (cid:0) u N ( s ) (cid:1) d s, ˙ z N ( t ) = − A α sin (cid:0) A α t (cid:1) u N + cos (cid:0) A α t (cid:1) v N + Z t cos (cid:0) A α ( t − s ) (cid:1) f N (cid:0) u N ( s ) (cid:1) d s. (4.4)Then the spatial semi-discretization solution of (3.2) is given by u N ( t ) = z N ( t ) + P N Z t A − α sin (cid:0) A α ( t − s ) (cid:1) d B H ( s )(4.5)and v N ( t ) = ˙ z N ( t ) + P N Z t cos (cid:0) A α ( t − s ) (cid:1) d B H ( s ) . (4.6)In fact, Corollary 3.2 shows z ( t ) has better regularity than the stochastic integral R t A − α sin (cid:0) A α ( t − s ) (cid:1) d B H ( s ) in space; so we can improve accuracy of the Galerkinapproximate solution by postprocessing the stochastic integral of (4.5). Let N = (cid:2) N θ (cid:3) and θ ≥
1, with [ y ] being the nearest integer to y . Then the spatial semi-discretization solution of (3.2) can be expressed as u N ( t ) = z N ( t ) + P N Z t A − α sin (cid:0) A α ( t − s ) (cid:1) d B H ( s )(4.7)and v N ( t ) = ˙ z N ( t ) + P N Z t cos (cid:0) A α ( t − s ) (cid:1) d B H ( s ) . (4.8)Using (4.3) leads to˙ z N ( t ) = − A α sin (cid:0) A α ( t − s ) (cid:1) z N ( s ) + cos (cid:0) A α ( t − s ) (cid:1) ˙ z N ( s )(4.9) + Z ts cos (cid:0) A α ( t − r ) (cid:1) f N (cid:0) u N ( r ) (cid:1) d r. XING LIU AND WEIHUA DENG
Then substituting (4.7) and (4.8) into (4.9) leads to v N ( t ) = − A α sin (cid:0) A α ( t − s ) (cid:1) u N ( s ) + cos (cid:0) A α ( t − s ) (cid:1) v N ( s )(4.10) + P N Z ts cos (cid:0) A α ( t − r ) (cid:1) d B H ( r )+ Z ts cos (cid:0) A α ( t − r ) (cid:1) f N (cid:0) u N ( r ) (cid:1) d r, which will be used to prove Theorem 5.2.Combining Theorem 3.1, (4.7) and (4.8), we now deduce the following regularityresults of the spatial semi-discretization solution. Corollary
Suppose that Assumption 1-2 are satisfied; k u k L p ( D, ˙ U γ ) < ∞ , k v k L p ( D, ˙ U γ − α ) < ∞ , ε > , γ = α + 2 ρ − d + ε and γ > . The approximate solution u N ( t ) and v N ( t ) is expressed by (4.7) and (4.8) , respectively. Then (cid:13)(cid:13) u N ( t ) (cid:13)(cid:13) L p ( D, ˙ U γ ) + (cid:13)(cid:13) v N ( t ) (cid:13)(cid:13) L p ( D, ˙ U γ − α ) . t H ε + k u k L p ( D, ˙ U γ ) + k v k L p ( D, ˙ U γ − α ) and (i) for γ ≤ α , (cid:13)(cid:13) u N ( t ) − u N ( s ) (cid:13)(cid:13) L ( D,U ) . ( t − s ) γα (cid:18) t H ε + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:19) . (ii) for γ > α , (cid:13)(cid:13) u N ( t ) − u N ( s ) (cid:13)(cid:13) L ( D,U ) . ( t − s ) (cid:16) t H + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) . The proof of this corollary is done in the same way as Theorem 3.1.Next, the following lemma are given to analyze the error of the approximatesolution u N ( t ) in (4.7). Lemma If E (cid:2) k A ν ξ k (cid:3) < ∞ , ξ ∈ U , then E (cid:2) k ( P N − I ) ξ k (cid:3) . λ − νN +1 E (cid:2) k A ν ξ k (cid:3) . Corollary 3.2 shows z ( t ) ∈ L p ( D, ˙ U γ + α ). From Lemma 4.2, we can infer thatE (cid:2) k z ( t ) − P N z ( t ) k (cid:3) . λ − γ − αN +1 E h k A γ + α z ( t ) k i . Therefore we can obtain an orderof γ + αd for the spatial semi-discretization solution by adjusting N in (4.7). Let N = h N γ + αγ i . Then we get following result. Theorem
Let X ( t ) and Z N ( t ) be the mild solution of (3.2) and (3.10) ,respectively. Suppose that Assumption 1-2 are satisfied. Let k u k L p ( D, ˙ U γ + α ) < ∞ , k v k L p ( D, ˙ U γ ) < ∞ , ε > , γ = α + 2 ρ − d + ε , γ > ; and (4.7) is the approximationof u ( t ) . If N = h N γ + αγ i , then we have (cid:13)(cid:13) u ( t ) − u N ( t ) (cid:13)(cid:13) L ( D,U ) . N − γ + αd (cid:18) t H ε + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:19) . IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION Proof.
In the first place, using the triangle inequality, Corollary 3.2, Lemma 4.2and (4.7), we obtain (cid:13)(cid:13) u ( t ) − u N ( t ) (cid:13)(cid:13) L ( D,U ) . (cid:13)(cid:13) z ( t ) − z N ( t ) (cid:13)(cid:13) L ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t ∞ X i = N +1 λ − α i sin (cid:16) λ α i ( t m − s ) (cid:17) σ i φ i ( x )d β iH ( s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) . k z ( t ) − P N z ( t ) k L ( D,U ) + (cid:13)(cid:13) P N z ( t ) − z N ( t ) (cid:13)(cid:13) L ( D,U ) + t H ε N − γd . (cid:13)(cid:13) P N z ( t ) − z N ( t ) (cid:13)(cid:13) L ( D,U ) + N − γ + αd (cid:18) t H ε + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:19) . (4.11)Then we need to estimate the bound of (cid:13)(cid:13) P N z ( t ) − z N ( t ) (cid:13)(cid:13) L ( D,U ) . The definitionof projection operator P N implies that P N A − α sin (cid:0) A α t (cid:1) f = A − α sin (cid:0) A α t (cid:1) P N f. Thus, first performing P N on (3.12) and then doing subtraction with respect to (4.4)leads to (cid:13)(cid:13) P N z ( t ) − z N ( t ) (cid:13)(cid:13) L ( D,U ) = (cid:13)(cid:13)(cid:13)(cid:13)Z t A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) (cid:0) f N ( u ( s )) − f N (cid:0) u N ( s ) (cid:1)(cid:1) d s (cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) . Z t (cid:13)(cid:13) u ( s ) − u N ( s ) (cid:13)(cid:13) L ( D,U ) d s. Then the above estimates and the Gr¨onwall inequality imlpy (cid:13)(cid:13) u ( t ) − u N ( t ) (cid:13)(cid:13) L ( D,U ) . N − γ + αd (cid:18) t H ε + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:19) . (4.12)If N >
1, choosing ε = N ) , we have (cid:13)(cid:13) u ( t ) − u N ( t ) (cid:13)(cid:13) L ( D,U ) . N − α +4 ρ +2 α − d − ε d (cid:18) t H ε + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:19) . N − α +4 ρ − d d (cid:16) t H log( N ) + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:17) .
5. Fully discrete scheme.
In this section, we concern the time discretizationof (4.3). Meanwhile the error estimates of the fully discrete scheme are derived.Let z Nm and ¯ z Nm denote respectively the approximation of z N ( t m ) and ˙ z N ( t m ) withfixed time step size τ = TM and t m = mτ ( m = 0 , , , . . . , M ). Using the stochastictrigonometric method, we can get the full discrete scheme of (3.2) " z Nm +1 ¯ z Nm +1 = " cos (cid:0) A α τ (cid:1) A − α sin (cid:0) A α τ (cid:1) − A α sin (cid:0) A α τ (cid:1) cos (cid:0) A α τ (cid:1) z Nm ¯ z Nm (5.1) + τ " A − α sin (cid:0) A α τ (cid:1) f N (cid:0) u Nm (cid:1) cos (cid:0) A α τ (cid:1) f N (cid:0) u Nm (cid:1) , XING LIU AND WEIHUA DENG where u Nm = z Nm + P N Z t m A − α sin (cid:0) A α ( t m − s ) (cid:1) d B H ( s ) . By using recursion form of (5.1), we get " z Nm +1 ¯ z Nm +1 = " cos (cid:0) A α t m +1 (cid:1) A − α sin (cid:0) A α t m +1 (cid:1) − A α sin (cid:0) A α t m +1 (cid:1) cos (cid:0) A α t m +1 (cid:1) u N v N (5.2) + τ m X j =0 " A − α sin (cid:0) A α ( t m +1 − t j ) (cid:1) f N (cid:0) u Nj (cid:1) cos (cid:0) A α ( t m +1 − t j ) (cid:1) f N (cid:0) u Nj (cid:1) . Then we get the approximations of u ( t ) and v ( t ), that is, u Nm +1 = z Nm +1 + P N Z t m +1 A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) d B H ( s )(5.3)and v Nm +1 = ¯ z Nm +1 + P N Z t m +1 cos (cid:0) A α ( t m +1 − s ) (cid:1) d B H ( s ) . (5.4)As < H <
1, although R t m +1 sin (cid:0) A α ( t m +1 − s ) (cid:1) d B H ( s ) is a Gaussian process,it is difficult to accurately simulate this process; thus we give the approximation ofstochastic integral, that is P N m X j =0 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − t j ) (cid:1) d B H ( s ) . (5.5)Using Lemma 2.2, we obtain the error estimateE (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P N m X j =0 Z t j +1 t j A − α (cid:0) sin (cid:0) A α ( t m +1 − s ) (cid:1) − sin (cid:0) A α ( t m +1 − t j ) (cid:1)(cid:1) d B H ( s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . t Hm +1 τ min { γα , } N X i =0 λ min { γ,α }− α − ρi , which implies that this approximation (5.5) does not change the temporal convergencerate of scheme (5.1). The proof of this estimate is done in the same way as (5.11).For H = , the simulation of stochastic integral is easily implementable withoutapproximation. We now investigate the error estimates of the fully discrete scheme(5.1). The triangle inequality implies that (cid:13)(cid:13) u ( t m ) − u Nm (cid:13)(cid:13) L ( D,U ) . (cid:13)(cid:13) u ( t m ) − u N ( t m ) (cid:13)(cid:13) L ( D,U ) + (cid:13)(cid:13) u N ( t m ) − u Nm (cid:13)(cid:13) L ( D,U ) . Thus, we need to give the bound estimate of (cid:13)(cid:13) u N ( t m ) − u Nm (cid:13)(cid:13) L ( D,U ) . This boundestimate can be obtained by using the time H¨older regularity of u N ( t ), (4.7), and(5.3). Therefore combining the error estimate of the approximation (5.5), Corollary4.1, and Theorem 4.3 leads to the following results. IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION Proposition
Let u ( t m +1 ) and u Nm +1 be expressed by (3.5) and (5.8) , respec-tively. Suppose that Assumption 1-2 are satisfied. Suppose Corollary 4.1 and Theorem4.3 hold. Let N = h N γ + αγ i . Then (cid:13)(cid:13) u ( t m +1 ) − u Nm +1 (cid:13)(cid:13) L ( D,U ) . τ (cid:16) T H + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:17) + N − γ + αd (cid:18) T H ε + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:19) , γ > α and (cid:13)(cid:13) u ( t m +1 ) − u Nm +1 (cid:13)(cid:13) L ( D,U ) . τ γα (cid:18) T H ε + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:19) + N − γ + αd (cid:18) T H ε + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:19) , γ ≤ α. The proof of this proposition is similar to the one of Theorem 5.2 given in AppendixA. As γ > α , the derivative of u ( t ) is time H¨older continuous in the sense of mean-squared L p -norm, which means that the scheme (5.1) is not optimal to discretize (3.1)in time. Thus we can design a higher order scheme for the time discretization, as v ( t )belongs to L p ( D, U ). By modifying the scheme (5.1), we can get a better convergencerate than one of (5.1) in time. The modified scheme is as follows " z N ¯ z N = " cos (cid:0) A α τ (cid:1) A − α sin (cid:0) A α τ (cid:1) − A α sin (cid:0) A α τ (cid:1) cos (cid:0) A α τ (cid:1) z N ¯ z N (5.6) + " A − α (cid:0) − cos (cid:0) A α τ (cid:1)(cid:1) f N (cid:0) u N (cid:1) A − α sin (cid:0) A α τ (cid:1) f N (cid:0) u N (cid:1) and for m ≥ " z Nm +1 ¯ z Nm +1 = " cos (cid:0) A α τ (cid:1) A − α sin (cid:0) A α τ (cid:1) − A α sin (cid:0) A α τ (cid:1) cos (cid:0) A α τ (cid:1) z Nm ¯ z Nm (cid:21) (5.7) + " A − α (cid:0) − cos (cid:0) A α τ (cid:1)(cid:1) f N (cid:0) u Nm (cid:1) A − α sin (cid:0) A α τ (cid:1) f N (cid:0) u Nm (cid:1) + τA − α − A − α sin (cid:16) A α τ (cid:17) τ (cid:0) f N (cid:0) u Nm (cid:1) − f N (cid:0) u Nm − (cid:1)(cid:1) A − α − A − α cos (cid:16) A α τ (cid:17) τ (cid:0) f N (cid:0) u Nm (cid:1) − f N (cid:0) u Nm − (cid:1)(cid:1) . XING LIU AND WEIHUA DENG
Then a classic application of recursion gives u Nm +1 = cos (cid:0) A α t m +1 (cid:1) u N + A − α sin (cid:0) A α t m +1 (cid:1) v N (5.8) + m X j =0 A − α (cid:0) cos (cid:0) A α ( t m +1 − t j +1 ) (cid:1) − cos (cid:0) A α ( t m +1 − t j ) (cid:1)(cid:1) f N (cid:0) u Nj (cid:1) + m X j =1 A − α cos (cid:0) A α ( t m +1 − t j +1 ) (cid:1) (cid:0) f N (cid:0) u Nj (cid:1) − f N (cid:0) u Nj − (cid:1)(cid:1) − m X j =1 A − α R t j +1 t j cos (cid:0) A α ( t m +1 − s ) (cid:1) d sτ (cid:0) f N (cid:0) u Nj (cid:1) − f N (cid:0) u Nj − (cid:1)(cid:1) + P N Z t m +1 A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) d B H ( s )and v Nm +1 = − A α sin (cid:0) A α t m +1 (cid:1) u N + cos (cid:0) A α t m +1 (cid:1) v N (5.9) − m X j =0 A − α (cid:0) sin (cid:0) A α ( t m +1 − t j +1 ) (cid:1) − sin (cid:0) A α ( t m +1 − t j ) (cid:1)(cid:1) f N (cid:0) u Nj (cid:1) − m X j =1 A − α sin (cid:0) A α ( t m +1 − t j +1 ) (cid:1) (cid:0) f N (cid:0) u Nj (cid:1) − f N (cid:0) u Nj − (cid:1)(cid:1) + m X j =1 A − α R t j +1 t j sin (cid:0) A α ( t m +1 − s ) (cid:1) d sτ (cid:0) f N (cid:0) u Nj (cid:1) − f N (cid:0) u Nj − (cid:1)(cid:1) + P N Z t m +1 cos (cid:0) A α ( t m +1 − s ) (cid:1) d B H ( s ) . For < H < γ > α , if the time steps of (5.5) and (5.7) are the same, the desiredconvergence rate can not be got. Thus, a more precise approximation of stochasticintegral than (5.5) is expected. The inequality cos( t m +1 − r ) − cos( t m +1 − t j ) . | r − t j | θ (0 ≤ θ ≤
1) and the error equation R t j +1 t j R st j cos (cid:0) A α ( t m +1 − r ) (cid:1) d r d B H ( s ) of(5.5) imply that we can improve the accuracy of approximation for stochastic integralby the following scheme, that is,(5.10) P N m X j =0 Z t j +1 t j (cid:0) A − α sin (cid:0) A α ( t m +1 − t j ) (cid:1) − ( s − t j ) cos (cid:0) A α ( t m +1 − t j ) (cid:1)(cid:1) d B H ( s ) , which ensures the implementation of scheme (5.7) without loss of convergence rate;and Equation (5.10) is easy to simulate by using its explicit variance. Using Lemma2.2, the following error estimate of the approximation for stochastic integral is ob- IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P N m X j =0 Z t j +1 t j (cid:0) A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) − A − α sin (cid:0) A α ( t m +1 − t j ) (cid:1) + ( s − t j ) cos (cid:0) A α ( t m +1 − t j ) (cid:1)(cid:1) d B H ( s ) (cid:13)(cid:13) i =E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P N m X j =0 Z t j +1 t j Z st j (cid:0) cos (cid:0) A α ( t m +1 − t j ) (cid:1) − cos (cid:0) A α ( t m +1 − r ) (cid:1)(cid:1) d r d B H ( s ) (cid:13)(cid:13) i . N X i =0 λ − ρi m X j =0 Z t j +1 t j Z st j (cid:12)(cid:12)(cid:12) cos (cid:16) λ α i ( t m +1 − t j ) (cid:17) − cos (cid:16) λ α i ( t m +1 − r ) (cid:17)(cid:12)(cid:12)(cid:12) d r × m X k =0 Z t k +1 t k Z tt k (cid:12)(cid:12)(cid:12) cos (cid:16) λ α i ( t m +1 − t k ) (cid:17) − cos (cid:16) λ α i ( t m +1 − r ) (cid:17)(cid:12)(cid:12)(cid:12) d r | s − t | H − d s d t . τ min { γα , } N X i =0 λ min { γ − α,α }− ρi m X j =0 Z t j +1 t j m X k =0 Z t k +1 t k | s − t | H − d s d t = τ min { γα , } N X i =0 λ min { γ − α,α }− ρi Z t m +1 Z t m +1 | s − t | H − d s d t . t Hm +1 τ min { γα , } N X i =0 λ min { γ − α,α }− ρi , (5.11)the second inequality of which uses the factcos (cid:16) λ α i ( t m +1 − s ) (cid:17) − cos (cid:16) λ α i ( t m +1 − t ) (cid:17) . λ min { γ − α , α } i | s − t | min { γ − αα , } . We end this section by showing the error estimates of the fully discrete scheme(5.7) in L ( D, U ) norm.
Theorem
Let u ( t m +1 ) and u Nm +1 be expressed by (3.5) and (5.8) , respec-tively. Suppose that f ( u ) ∈ C ( R ) and f ′ ( u ) satisfies the Lipschitz condition, and theconditions of Corollary 4.1 and Theorem 4.3 are satisfied. Take N > and < τ < .If γ > α and N = h N γ + αγ i , then (i) for α < γ ≤ α , (cid:13)(cid:13) u ( t m +1 ) − u Nm +1 (cid:13)(cid:13) L ( D,U ) . τ α +2 ρ − d α (cid:16) T H | log( τ ) | + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:17) + N − ρ +2 α − d d (cid:16) T H log( N ) + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:17) ;8 XING LIU AND WEIHUA DENG (ii) for γ > α , (cid:13)(cid:13) u ( t m +1 ) − u Nm +1 (cid:13)(cid:13) L ( D,U ) . τ (cid:16) T H + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:17) + N − ρ +2 α − d d (cid:16) T H log( N ) + k u k L ( D, ˙ U γ + α ) + k v k L ( D, ˙ U γ ) (cid:17) . The detailed proof of Theorem 5.2 is given in Appendix A.Proposition 5.1 and Theorem 5.2 show that one can choose the appropriate tech-nique to solve (1.3), i.e., when α ≤ γ , use (5.1) to discretize (1.3), and if α > γ ,the scheme (5.7) can be chosen to obtain the approximation of u ( t ). In fact, when u ∈ L ( D, ˙ U γ ) and v ∈ L ( D, ˙ U γ − α ), the temporal rates of convergence still holdin Proposition 5.1 and Theorem 5.2.
6. Numerical experiments.
In this section, we present numerical examplesto verify the theoretical results and the effect of the parameters α and ρ on theconvergence. All numerical errors are given in the sense of mean-squared L -norm.We solve (1.3) in the two-dimensional domain D = (0 , × (0 ,
1) by the proposedscheme (5.7) with x = ( x , x ), the smooth initial data u = sin( πx ) sin( πx )2 , and v = sin(4 πx ) sin(4 πx ). In D = (0 , × (0 , − ∆ are λ i,j = π ( i + j ) and φ i,j = 2 sin( iπx ) sin( jπx ) with i, j = 1 , , . . . , N . Unlessotherwise specified, we choose f ( u ( t )) = u ( t ). To calculate the convergence orders,the following formulas are used.convergence rate in space = ln (cid:18)(cid:13)(cid:13) u aNM − u NM (cid:13)(cid:13) L ( D,U ) / (cid:13)(cid:13)(cid:13) u NM − u N/aM (cid:13)(cid:13)(cid:13) L ( D,U ) (cid:19) ln a , convergence rate in time = ln (cid:18)(cid:13)(cid:13) u NaM − u NM (cid:13)(cid:13) L ( D,U ) / (cid:13)(cid:13)(cid:13) u NM − u NM/a (cid:13)(cid:13)(cid:13) L ( D,U ) (cid:19) ln a , where the constant a >
1. In numerical simulations, the errors (cid:13)(cid:13) u aNM − u NM (cid:13)(cid:13) L ( D,U ) are calculated by Monte Carlo method, i.e., (cid:13)(cid:13)(cid:13) u aN,MM − u N,MM (cid:13)(cid:13)(cid:13) L ( D,U ) = (cid:16) E h(cid:13)(cid:13) u aNM − u NM (cid:13)(cid:13) i(cid:17) ≈ K K X k =1 (cid:13)(cid:13) u aNM,k − u NM,k (cid:13)(cid:13) ! . We take K = 1000 as the number of the simulation trajectories. The symbol k represents the k -th trajectory. IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION Table 1
Spatial convergence rates with T = 0 . , M = 900 , H = 0 . and ρ = 1 . N α = 0 . α = 0 . α = 0 . N = N
256 2.610e-04 1.064e-04 4.810e-05576 1.608e-04 0.597 5.910e-05 0.725 2.365e-05 0.8751296 9.545e-05 0.643 3.220e-05 0.749 1.173e-05 0.865 N = h N γ + αγ i
256 1.051e-04 2.367e-05 5.912e-06576 4.787e-05 0.970 9.867e-06 1.079 2.073e-06 1.2921296 2.175e-05 0.973 4.055e-06 1.097 7.343e-07 1.280The spatial convergence rates of the scheme (5.7) is tested with the end time T = 0 . M = 900, which ensures the spatial error is the dominant one. In Table1, one can see that the spatial convergence rates tend to ρ + α − if N = N , and theconvergence rates are approximately equal to ρ +2 α − after postprocessing the sto-chastic integral (cid:16) N = h N γ + αγ i(cid:17) . And the convergence rates of the spectral Galerkinmethod are improved, as α increases. The numerical results verify the theoreticalones. −7 −6 −5 −4 −3 −2 M e rr o r α =0.2 α =0.4 α =0.6 α =0.8O( τ ) Fig. 1 . Temporal error convergence of the modified stochastic trigonometric method for thespace-time white noise ( H = 0 . . Next, we observe the behavior of the temporal convergence. We solve the problem(1.3) by using the scheme (5.7) with f ( u ( t )) = u ( t ), ρ = 2 . T = 0 .
6, and N = 400in Figures 1 and 2. The sufficiently big ρ and N guarantee that the dominant errorsarise from the temporal approximation. As H = , the simulation of the stochasticintegral R T sin ( λ i,j ( T − t )) d β ( t ) is easily implementable by using explicit variance ofthe stochastic integral, which is T − sin(2 λ i,j T )4 λ i,j . For H ∈ (cid:0) , (cid:1) , one can obtain theapproximation of the stochastic integral by using scheme (5.10). The simulation ofthe approximation is given in Appendix C. Figures 1 and 2 show that the temporal0 XING LIU AND WEIHUA DENG convergence rates have an order of 2 by using the proposed scheme, as γ > α , andthe convergence rates are independent of H . −7 −6 −5 −4 −3 −2 M e rr o r α =0.2 α =0.4 α =0.6 α =0.8O( τ ) Fig. 2 . Temporal error convergence of the modified stochastic trigonometric method for thespace-time fractional Gaussian noise ( H = 0 . . Table 2
Time convergence rates with N = 10 , T = 0 . , H = 0 . , and ρ = 0 . . M α = 0 . α = 0 . α = 0 . α < γ ≤ α , the convergence rates of the proposed scheme are close to α +2 ρ − α in time. For H = , from Table 2, one can see that the temporal convergence ratesreduce with the increase of α , for fixing ρ . As α = 0 .
5, 0 .
7, and 0 .
9, the theoreticalconvergence rates are approximately 1 . . . H =0 .
6, Table 3 demonstrates that the time convergence rates increase with the increaseof ρ , for fixing α . The temporal theoretical convergence rates are approximately 1 . . . ρ = 0 .
75, 0 .
80, and 0 .
85, respectively. Tables 2 and 3 show thatthe numerical results confirm the error estimates in Theorem 5.2.
Table 3
Time convergence rates with N = 10 , T = 0 . , H = 0 . , and α = 0 . . M ρ = 0 .
75 Rate ρ = 0 . ρ = 0 .
85 Rate4 4.568e-03 3.543e-03 2.907e-038 1.559e-03 1.551 1.156e-03 1.616 8.956e-04 1.69916 5.330e-04 1.548 3.733e-04 1.630 2.704e-04 1.728
IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION
7. Conclusion.
This paper discusses the numerical schemes and their erroranalyses for the equation describing the wave propagation with attenuation and pos-sible external disturbance. Two kinds of external noises (white noise and fractionalGaussian noise) are considered. The regularity results of the mild solution of theequation are obtained. The spectral Galerkin method is used for space approximationand the stochastic trigonometric method for time approximation. The detailed erroranalyses are performed. The techniques of equivalent transformation and postprocess-ing the stochastic integral improve the convergence rate in space from (2 ρ + α − d ) /d to (2 ρ + 2 α − d ) /d . For the temporal approximation, by modifying the stochastictrigonometric method, when γ > α , the superlinear convergence is obtained. Theconvergence rates of the designed schemes are independent of hurst index H . Theextensive numerical experiments confirm the theoretical results. Appendix A. Proof of Theorem 5.2.
Proof.
First, combining (4.7), (5.8), (5.11), the assumptions, and Corollary 4.1leads to (cid:13)(cid:13) u N ( t m +1 ) − u Nm +1 (cid:13)(cid:13) L ( D,U ) (A.1) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) (cid:2) f N (cid:0) u N ( s ) (cid:1) − f N (cid:0) u N ( t j ) (cid:1) − Z st j f ′ N (cid:0) u N ( t j ) (cid:1) v N ( t j )d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 A − α τ cos (cid:0) A α ( t m +1 − t j +1 ) (cid:1) − R t j +1 t j cos (cid:0) A α ( t m +1 − s ) (cid:1) d sτ × (cid:2) τ f ′ N (cid:0) u N ( t j ) (cid:1) v N ( t j ) − f N (cid:0) u Nj (cid:1) + f N (cid:0) u Nj − (cid:1)(cid:3)(cid:13)(cid:13) L ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) (cid:2) f N (cid:0) u N ( t j ) (cid:1) − f N (cid:0) u Nj (cid:1)(cid:3) d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) (cid:2) f N (cid:0) u N ( s ) (cid:1) − f N (cid:0) u N (cid:1)(cid:3) d s (cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) + τ min { γα , } N X i =0 λ min { γ − α,α }− ρi ! . J + J + m X j =0 τ (cid:13)(cid:13) u N ( t j ) − u Nj (cid:13)(cid:13) L ( D,U ) + τ + τ min { γα , } N X i =0 λ min { γ − α,α }− ρi ! . XING LIU AND WEIHUA DENG
When < H <
1, let θ = min (cid:8) γ − αα , (cid:9) . We have J . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) × "Z st j f ′ N (cid:0) u N ( r ) (cid:1) v N ( r )d r − Z st j f ′ N (cid:0) u N ( t j ) (cid:1) v N ( t j )d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) × Z st j (cid:0) f ′ N (cid:0) u N ( r ) (cid:1) − f ′ N (cid:0) u N ( t j ) (cid:1)(cid:1) v N ( r )d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) × Z st j f ′ N (cid:0) u N ( t j ) (cid:1) (cid:0) cos (cid:0) A α ( r − t j ) (cid:1) − I (cid:1) v N ( t j )d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) × Z st j f ′ N (cid:0) u N ( t j ) (cid:1) (cid:0) v N ( r ) − cos (cid:0) A α ( r − t j ) (cid:1) v N ( t j ) (cid:1) d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) . m X j =1 Z t j +1 t j Z st j (cid:13)(cid:13)(cid:0) u N ( r ) − u N ( t j ) (cid:1) × v N ( r ) (cid:13)(cid:13) L ( D,U ) d r d s + m X j =1 τ θ Z t j +1 t j Z st j (cid:13)(cid:13)(cid:13) A θα v N ( t j ) (cid:13)(cid:13)(cid:13) L ( D,U ) d r d s + II . m X j =1 Z t j +1 t j Z st j Z rt j (cid:13)(cid:13) v N ( t ) v N ( r ) (cid:13)(cid:13) L ( D,U ) d t d r d s + m X j =1 τ θ (cid:13)(cid:13)(cid:13) A θα v N ( t j ) (cid:13)(cid:13)(cid:13) L ( D,U ) + II . m X j =1 Z t j +1 t j Z st j Z rt j (cid:13)(cid:13)(cid:13)(cid:12)(cid:12) v N ( t ) (cid:12)(cid:12) + (cid:12)(cid:12) v N ( r ) (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) L ( D,U ) d t d r d s + m X j =1 τ θ (cid:13)(cid:13)(cid:13) A θα v N ( t j ) (cid:13)(cid:13)(cid:13) L ( D,U ) + II . τ (cid:16) k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) + m X j =1 τ θ (cid:13)(cid:13)(cid:13) A θα v N ( t j ) (cid:13)(cid:13)(cid:13) L ( D,U ) + II.
The condition γ > α implies that ρ > d . Then combining the fact that { β iH ( t ) } i ∈ N IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION II = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) Z st j f ′ N (cid:0) u N ( t j ) (cid:1) × " − A α sin (cid:0) A α ( r − t j ) (cid:1) u N ( t j ) + Z rt j cos (cid:0) A α ( r − y ) (cid:1) f N (cid:0) u N ( y ) (cid:1) d y + Z rt j N X i =1 cos (cid:16) λ α i ( r − y ) (cid:17) σ i φ i ( x )d β iH ( y ) d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) . τ θ m X j =1 (cid:13)(cid:13)(cid:13) A α ( θ +1)2 u N ( t j ) (cid:13)(cid:13)(cid:13) L ( D,U ) + τ (cid:16) T H + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) + N X i =1 λ − ρi E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) Z st j f ′ N (cid:0) u N ( t j ) (cid:1) × Z rt j cos (cid:16) λ α i ( r − y ) (cid:17) φ i ( x )d β iH ( y )d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . N X i =1 λ − ρi E Z D m X j =1 m X k =1 Z t j +1 t j Z st j Z t k +1 k j Z tk j Z rt j Z r t k A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) × f ′ N (cid:0) u N ( t j ) (cid:1) cos (cid:16) λ α i ( r − y ) (cid:17) φ i ( x ) A − α sin (cid:0) A α ( t m +1 − t ) (cid:1) f ′ N (cid:0) u N ( t k ) (cid:1) × cos (cid:16) λ α i ( r − y ) (cid:17) φ i ( x ) | y − y | H − d y d y d r d s d r d t d x i(cid:17) + τ θ m X j =1 (cid:13)(cid:13)(cid:13) A α ( θ +1)2 u N ( t j ) (cid:13)(cid:13)(cid:13) L ( D,U ) . τ m X j =1 m X k =1 E (cid:2)(cid:13)(cid:13) f ′ N (cid:0) u N ( t j ) (cid:1) φ i ( x ) (cid:13)(cid:13) (cid:13)(cid:13) f ′ N (cid:0) u N ( t k ) (cid:1) φ i ( x ) (cid:13)(cid:13)(cid:3) × Z t j +1 t j Z t k +1 t k | y − y | H − d y d y ! + τ θ m X j =1 (cid:13)(cid:13)(cid:13) A α ( θ +1)2 u N ( t j ) (cid:13)(cid:13)(cid:13) L ( D,U ) . Combining the above estimates and Corollary 4.1 leads to J . τ γα (cid:18) T H ε + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) , α < γ ≤ α and J . τ (cid:16) T H + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) , γ > α. XING LIU AND WEIHUA DENG
Similar to J , one gets J . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 A − α R t j +1 t j R t j +1 s sin (cid:0) A α ( t m +1 − r ) (cid:1) d r d sτ (cid:2) f N (cid:0) u N ( t j − ) (cid:1) − f N (cid:0) u N ( t j ) (cid:1) + τ f ′ N (cid:0) u N ( t j ) (cid:1) v N ( t j ) (cid:3)(cid:13)(cid:13) L ( D,U ) + τ m X j =1 (cid:13)(cid:13)(cid:2) f N (cid:0) u N ( t j ) (cid:1) − f N (cid:0) u N ( t j − ) (cid:1) − f N (cid:0) u Nj (cid:1) + f N (cid:0) u Nj − (cid:1)(cid:3)(cid:13)(cid:13) L ( D,U ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 A − α R t j +1 t j R t j +1 s sin (cid:0) A α ( t m +1 − r ) (cid:1) d r d sτ × "Z t j t j − f ′ N (cid:0) u N ( t j ) (cid:1) v N ( t j )d r − Z t j t j − f ′ N (cid:0) u N ( r ) (cid:1) v N ( t j )d r L ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 A − α R t j +1 t j R t j +1 s sin (cid:0) A α ( t m +1 − r ) (cid:1) d r d sτ × "Z t j t j − f ′ N (cid:0) u N ( r ) (cid:1) (cid:0) v N ( t j ) − cos (cid:0) A α ( t j − r ) (cid:1) v N ( r ) (cid:1) d r L ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 A − α R t j +1 t j R t j +1 s sin (cid:0) A α ( t m +1 − r ) (cid:1) d r d sτ × "Z t j t j − f ′ N (cid:0) u N ( r ) (cid:1) (cid:0) cos (cid:0) A α ( t j − r ) (cid:1) − I (cid:1) v N ( r )d r L ( D,U ) + τ m X j =0 (cid:13)(cid:13) u N ( t j ) − u Nj (cid:13)(cid:13) L ( D,U ) . For α < γ ≤ α , we have J . τ γα (cid:18) T H ε + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) + τ m X j =0 (cid:13)(cid:13) u N ( t j ) − u Nj (cid:13)(cid:13) L ( D,U ) . When γ > α , we also have J . τ (cid:16) T H + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) + τ m X j =0 (cid:13)(cid:13) u N ( t j ) − u Nj (cid:13)(cid:13) L ( D,U ) . IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION J , and J leads to (cid:13)(cid:13) u N ( t m +1 ) − u Nm +1 (cid:13)(cid:13) L ( D,U ) . τ γα (cid:18) T H ε + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) , α < γ ≤ α and (cid:13)(cid:13) u N ( t m +1 ) − u Nm +1 (cid:13)(cid:13) L ( D,U ) . τ (cid:16) T H + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) , γ > α. When H = , using the same steps in (A.1), we get (cid:13)(cid:13) u N ( t m +1 ) − u Nm +1 (cid:13)(cid:13) L ( D,U ) . I + I + m X j =0 τ (cid:13)(cid:13) u N ( t j ) − u Nj (cid:13)(cid:13) L ( D,U ) + τ . For α < γ ≤ α , using Corollary 4.1 and the assumptions of f , we obtain I . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) (A.2) × Z st j (cid:0) f ′ N (cid:0) u N ( r ) (cid:1) − f ′ N (cid:0) u N ( t j ) (cid:1)(cid:1) v N ( r )d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) × Z st j f ′ N (cid:0) u N ( t j ) (cid:1) (cid:0) cos (cid:0) A α ( r − t j ) (cid:1) − I (cid:1) v N ( t j )d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) × Z st j f ′ N (cid:0) u N ( t j ) (cid:1) (cid:0) v N ( r ) − cos (cid:0) A α ( r − t j ) (cid:1) v N ( t j ) (cid:1) d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) . m X j =1 Z t j +1 t j Z st j (cid:13)(cid:13)(cid:0) u N ( r ) − u N ( t j ) (cid:1) × v N ( r ) (cid:13)(cid:13) L ( D,U ) d r d s + m X j =1 τ γ − αα Z t j +1 t j Z st j (cid:13)(cid:13)(cid:13) A γ − α v N ( t j ) (cid:13)(cid:13)(cid:13) L ( D,U ) d r d s + III . τ γα (cid:18) T H ε + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) + III. XING LIU AND WEIHUA DENG
Combining the fact that { β iH ( t ) } i ∈ N are mutually independent and Equation (4.10),we have III = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 Z t j +1 t j A − α sin (cid:0) A α ( t m +1 − s ) (cid:1) Z st j f ′ N (cid:0) u N ( t j ) (cid:1) × " − A α sin (cid:0) A α ( r − t j ) (cid:1) u N ( t j ) + Z rt j cos (cid:0) A α ( r − y ) (cid:1) f N (cid:0) u N ( y ) (cid:1) d y + Z rt j N X i =1 cos (cid:16) λ α i ( r − y ) (cid:17) σ i φ i ( x )d β i ( y ) d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D,U ) . τ γα m X j =1 (cid:13)(cid:13)(cid:13) A γ u N ( t j ) (cid:13)(cid:13)(cid:13) L ( D,U ) + τ (cid:16) T H + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) + m X j =1 E "(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t j +1 t j N X i =1 λ − α i sin (cid:16) λ α i ( t m +1 − s ) (cid:17) Z st j f ′ N (cid:0) u N ( t j ) (cid:1) × Z rt j cos (cid:16) λ α i ( r − y ) (cid:17) σ i φ i ( x )d β i ( y )d r d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . τ γα (cid:18) T H ε + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:19) . In first inequality, we employ the fact that Brownian motion is a process with inde-pendent increment, that isE "Z t j +1 t j Z st j f ′ N (cid:0) u N ( t j ) (cid:1) Z rt j cos (cid:16) λ α i ( r − y ) (cid:17) d β i ( y )d r d s × Z t k +1 t k Z st k f ′ N (cid:0) u N ( t k ) (cid:1) Z rt k cos (cid:16) λ α i ( r − y ) (cid:17) d β i ( y )d r d s (cid:21) = 0 , j = k. Then I . τ γα (cid:18) T H ε + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:19) . Similar to J and I , we have I . τ γα (cid:18) T H ε + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) + τ m X j =0 (cid:13)(cid:13) u N ( t j ) − u Nj (cid:13)(cid:13) L ( D,U ) . If γ > α , then I . τ (cid:16) T H + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) IGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION I . τ (cid:16) T H + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) + τ m X j =0 (cid:13)(cid:13) u N ( t j ) − u Nj (cid:13)(cid:13) L ( D,U ) . Using the above estimates and the discrete Gr¨onwall inequality, we obtain (cid:13)(cid:13) u N ( t m +1 ) − u Nm +1 (cid:13)(cid:13) L ( D,U ) (A.3) . τ γα (cid:18) T H ε + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) , α < γ ≤ α and (cid:13)(cid:13) u N ( t m +1 ) − u Nm +1 (cid:13)(cid:13) L ( D,U ) (A.4) . τ (cid:16) T H + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) + k u k L ( D, ˙ U γ ) + k v k L ( D, ˙ U γ − α ) (cid:17) , γ > α. Take 0 < τ < ε = | log( τ ) | . Combining above estimates and Theorem 4.3, weobtain the desired results. Appendix B. Definitions of the cosine and sine operators.
In term of the eigenpairs { ( λ i , φ i ) } ∞ i =1 , the cosine and sine operators can be ex-pressed as sin ( A α t ) u ( t ) = ∞ X i =1 sin ( λ αi t ) h u ( t ) , φ i ( x ) i φ i ( x )= ∞ X i =1 ∞ X j =1 ( − j − ( λ αi t ) j − (2 j − h u ( t ) , φ i ( x ) i φ i ( x )and cos ( A α t ) u ( t ) = ∞ X i =1 cos ( λ αi t ) h u ( t ) , φ i ( x ) i φ i ( x )= ∞ X i =1 ∞ X j =0 ( − j ( λ αi t ) j (2 j )! h u ( t ) , φ i ( x ) i φ i ( x ) . Appendix C. Simulation of stochastic integral for fBm.
Suppose 0 ≤ t ≤ · · · ≤ t m ≤ · · · ≤ t M = T ( m = 1 , , . . . , M −
1) and the fixedsizes of the mesh τ = t m +1 − t m . Let’s consider the following vector Z = Z t s d β H ( s ) , Z t t ( s − t )d β H ( s ) , . . . , Z t M t M − ( s − t M − )d β H ( s ) ! . The stochastic integral R t m +1 t m ( s − t m )d β H ( s ) is a Gaussian process with mean 0. TheCholesky method can be applied to stationary and non-stationary Gaussian processes.8 XING LIU AND WEIHUA DENG
Thus, we use the Ckolesky method to simulate (5.10). The probability distributionof the vector Z is normal with mean 0 and the covariance matrix Σ. Let Σ i,j be theelement of row i , column j of matrix Σ. ThenΣ i,j = E "Z t j +1 t j ( s − t j )d β H ( s ) Z t k +1 t k ( t − t k )d β H ( t ) . By using Lemma 2.2, for j > k , we haveE "Z t j +1 t j ( s − t j )d B H ( s ) Z t k +1 t k ( t − t k )d B H ( t ) = H (2 H − Z t j +1 t j Z t k +1 t k ( s − t j )( t − t k )( s − t ) H − d t d s = − τ t j +1 − t k +1 ) H + τ H + 1) (cid:0) ( t j +1 − t k ) H +1 − ( t j − t k +1 ) H +1 (cid:1) − H + 1)(2 H + 2) (cid:0) ( t j +1 − t k ) H +2 − t j − t k ) H +2 + ( t j − t k +1 ) H +2 (cid:1) = − τ H j − k ) H + τ H H + 1) (cid:0) ( j + 1 − k ) H +1 − ( j − k − H +1 (cid:1) − τ H H + 1)(2 H + 2) (cid:0) ( j + 1 − k ) H +2 − j − k ) H +2 + ( j − k − H +2 (cid:1) . When j = k , E "Z t j +1 t j ( s − t j )d B H ( s ) Z t j +1 t j ( t − t j )d B H ( t ) = τ H +2 H + 2 . When Σ is a symmetric positive matrix, the covariance matrix Σ can be written as L ( M ) L ( M ) ′ , where the matrix L ( M ) is lower triangular matrix and the matrix L ( M ) ′ is the transpose of L ( M ). Let V = ( V , V , . . . , V M ). The elements of the vector V are a sequence of independent and identically distributed standard normal randomvariables. Since Z = L ( M ) V , then Z can be simulated. Let l i,j be the element of row i , column j of matrix L ( M ). That is,Σ i,j = j X k =1 l i,k l j,k , j ≤ i. As i = j = 1, we have l , = Σ , . The l i,j satisfies l i +1 , = Σ i +1 , l , ,l i +1 ,i +1 = Σ i +1 ,i +1 − i X k =1 l i +1 ,k ,l i +1 ,j = 1 l j,j Σ i +1 ,j − j − X k =1 l i +1 ,k l j,k ! , < j ≤ i. REFERENCESIGHER ORDER APPROXIMATION FOR STOCHASTIC WAVE EQUATION [1] E. Al`os, O. Mazet, and D. Nualart , Stochastic calculus with respect to Gaussian processes ,Ann. Probab., 29 (2001), pp. 766-801.[2]
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