Magnus-Type Integrator for Non-Autonomous Spdes 4599

DISCRETE AND CONTINUOUS doi:10.3934/dcds.2020194 DYNAMICAL SYSTEMS Volume 40, Number 8, August 2020 pp. 4597–4624

MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS

SPDES DRIVEN BY MULTIPLICATIVE NOISE

Antoine Tambue∗ Department of Computer science, Electrical engineering and Mathematical sciences Western Norway University of Applied Sciences, Inndalsveien 28, 5063 Bergen, Norway Center for Research in Computational and Applied Mechanics (CERECAM) and Department of Mathematics and Applied Mathematics University of Cape Town, 7701 Rondebosch, South Africa The African Institute for Mathematical Sciences(AIMS) of South Africa Jean Daniel Mukam FakultätfürMathematik, Technische UniversitätChemnitz 09126 Chemnitz, Germany

(Communicated by Lorenzo Zambotti)

Abstract. This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non- autonomous SPDE by the ﬁnite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square L2 norm. The result reveals how the convergence orders in both space and time depend on the regularity of the noise and the initial data. In particular, for multiplicative 2 2 trace class noise we achieve convergence order O h 1 + max(0, ln tm/h 1 +∆t 2 . Numerical simulations to illustrate our theoretical ﬁnding are provided.

1. Introduction. We consider the numerical approximations of the following semilinear parabolic non-autonomous SPDE driven by multiplicative noise dX = [A(t)X + F (t, X)]dt + B(t, X)dW (t), in Λ × (0,T ], (1) X(0) = X0, in Λ, in the Hilbert space L2(Λ), where Λ is a bounded domain of Rd, d = 1, 2, 3 and T ∈ (0, ∞). The family of unbounded linear operators A(t) are not necessarily self- A(t)s adjoint. Each A(t) is assumed to generate an analytic semigroup St(s) := e . The nonlinear functions F and B are respectively the drift and the diﬀusion parts. Precise assumptions on A(t), F and B to ensure the existence of the unique mild solution of (1) are given in the next section. The random initial data is denoted

2010 Mathematics Subject Classification. Primary: 65C30, 65J08, 65M60, 65M12, 65M15; Secondary: 65J15. Key words and phrases. Magnus-type integrator, stochastic partial differential equations, multiplicative noise, strong convergence, non-autonomous equations, finite element method. ∗ Corresponding author: Antoine Tambue, email: [email protected].

4597 4598 ANTOINE TAMBUE AND JEAN DANIEL MUKAM by X0. We denote by (Ω, F, P) a probability space with a filtration (Ft)t∈[0,T ] ⊂ F that fulfills the usual conditions (see [23, Definition 2.1.11]). The noise term W (t) is assumed to be a Q-Wiener process defined in the filtered probability space Ω, F, P, {Ft}t∈[0,T ] , where the covariance operator Q : H −→ H is assumed to be linear, self adjoint and positive definite. It is well known [23] that the noise can be represented as

∞ X √ W (t, x) = qiei(x)βi(t), (2) i=0 where (qi, ei)i∈N are the eigenvalues and eigenfunctions of the covariance operator Q, and (βi)i∈N are independent and identically distributed standard Brownian mo- tions. The deterministic counterpart of (1) finds applications in many fields such as quantum fields theory, electromagnetism, nuclear physics (see e.g. [3] and references therein). It is worth to mention that models based on SPDEs can offer a more realistic representation of the system than models based only on PDEs, due to uncertainty in the input data. In many situations it is very hard to exhibit explicit solutions of SPDEs. Numerical algorithms are therefore excellent tools to provide good approximations. Numerical approximations of (1) based on implicit, explicit Euler methods and exponential integrators with A(t) = A, where A is self-adjoint are thoroughly investigated in the literature, see e.g. [11, 14, 13, 28, 29, 17, 27] and the references therein. If we turn our attention to the case of time independent operator A(t) = A, with A not necessary self-adjoint, the list of references become remarkably short, see e.g., [16, 20]. To the best of our knowledge numerical approximations of (1) with time dependent linear operator A(t) are not yet investigated in the scientific literature, due to the complexity of the associated evolution operators U(t, s), 0 ≤ s ≤ t ≤ T . Our aim in this paper is to fill that gap and propose an explicit numerical scheme to approximate (1). We use the finite element method for spatial discretization and Magnus-type integrator for temporal discretization. Magnus-type integrator is based on a truncation of Magnus expansion, which was first proposed in [19] to represent the solution of non-autonomous homogeneous differential equation in the exponential form. Magnus expansion was further studied in [1,2,3]. The first numerical method based on Magnus expansion was proposed in [10] for deterministic time-dependent homogeneous Schrödingerequation. The study in [10] was extended in [5] for partial differential equation of the following form

0 u (t) = A(t)u(t) + b(t), 0 < t ≤ T, u(0) = u0. (3)

To build our novel scheme, we follow [5] and apply the Magnus-type integrator method to the semi-discrete problem (40) and obtain the fully discrete scheme (52), called stochastic Magnus-type integrators (SMTI). We investigate the strong convergence of the new fully discrete scheme toward the mild solution. Due to the complexity of the evolution operators U(t, s) and their corresponding semi discrete version Uh(t, s), novel technical estimates are provided to achieve convergence orders comparable of that of autonomous SPDEs [16, 14, 20]. The result indicates how the convergence orders in both space and time depend on the regularity of the initial data and the noise. In particular for multiplicative trace class noise, we achieve optimal convergence orders of O hβ + ∆tmin(β,1)/2, where β is the regularity parameter, deﬁned in Assumption 2.1. MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4599

The rest of this paper is organised as follows. Section2 provides the general setting, the fully discrete scheme and the main result. In Section3, we provide some preparatory results and we present the proof of the main result. Section4 provides some numerical experiments to sustain our theoretical result.

2. Mathematical setting, numerical scheme and main result.

2.1. Notations and main assumptions. Let (H, h., .iH , k.k) and (U, h., .iU , k.kU ) be two separable Hilbert spaces. We denote by L2(Ω,U) the Banach space of all equivalence classes of square-integrable U-valued random variables. Let L(U, H) be the space of bounded linear mappings from U to H endowed with the usual operator norm k.kL(U,H). By L2(U, H) := HS(U, H), we denote the space of Hilbert-Schmidt operators from U to H equipped with the norm ∞ X klk2 := klψ k2, l ∈ L (U, H), (4) L2(U,H) i 2 i=1 ∞ where (ψi)i=1 is an orthonormal basis of U. Note that this deﬁnition is independent of the orthonormal basis of U. For simplicity, we use the notations L(U, U) =: L(U). and L2(U, U) =: L2(U). For all l ∈ L(U, H) and l1 ∈ L2(U) we have ll1 ∈ L2(U, H) and

kll1kL2(U,H) ≤ klkL(U,H)kl1kL2(U). (5) 1 0 The space of Hilbert-Schmidt operators from Q 2 (H) to H is denoted by L2 := 1 1 0 L2(Q 2 (H),H) = HS(Q 2 (H),H). As usual, L2 is equipped with the norm 1 ∞ ! 2 1 X 1 2 0 klk 0 := klQ 2 k = klQ 2 e k , l ∈ L , (6) L2 HS i 2 i=1 ∞ where (ei)i=1 is an orthonormal basis of H. This deﬁnition is independent of the 0 orthonormal basis of H. For an L2- predictable stochastic process φ : [0,T ] × Λ −→ 0 L2 such that t Z 1 2 2 EkφQ kHSds < ∞, t ∈ [0,T ], (7) 0 the following relation called Itˆoisometry holds t 2 t t Z Z Z 1 2 2 2 E φdW (s) = EkφkL0 ds = EkφQ kHSds, t ∈ [0,T ], (8) 0 0 2 0 see e.g. [22, Step 2 in Section 2.3.2] or [23, Proposition 2.3.5]. In the sequel of this paper, we consider H = L2(Λ, R). To guarantee the existence of a unique mild solution of (1) and for the purpose of the convergence analysis, we make the following assumptions.

Assumption 2.1. The initial data X0 :Ω −→ H is assumed to be measurable and β 2 2 satisﬁes X0 ∈ L Ω, D (−A(0)) , 0 ≤ β ≤ 2. Assumption 2.2. (i) As in [5,6,9] , we assume that D (A(t)) = D, 0 ≤ t ≤ T and the family of linear operators A(t): D ⊂ H −→ H to be uniformly 1 sectorial on 0 ≤ t ≤ T , i.e. there exist two constants c > 0 and θ ∈ 2 π, π such that −1 c (λI − A(t)) ≤ , λ ∈ Sθ, (9) L(L2(Λ)) |λ| 4600 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

iφ where Sθ := λ ∈ C : λ = ρe , ρ > 0, 0 ≤ |φ| ≤ θ . As in [9], by a standard scaling argument, we assume −A(t) to be invertible with bounded inverse. (ii) Similarly to [6,9,5, 22] , we require the following Lipschitz conditions: there exists a positive constant K1 such that −1 (A(t) − A(s)) (−A(0)) L(H) ≤ K1|t − s|, s, t ∈ [0,T ], (10) −1 (−A(0)) (A(t) − A(s)) L(D,H) ≤ K1|t − s|, s, t ∈ [0,T ]. (11) (iii) Since we are dealing with non smooth initial value, we follow [25] and assume that D ((−A(t))α) = D ((−A(0))α) , 0 ≤ t ≤ T, 0 ≤ α ≤ 1 (12) α and there exists a positive constant K2 such that for all u ∈ D((−A(0)) ) the following estimate holds uniformly for t ∈ [0,T ] −1 α α α K2 k(−A(0)) uk ≤ k(−A(t)) uk ≤ K2 k(−A(0)) uk . (13) Remark 1. As a consequence of Assumption 2.2 (i) and (iii), for all α ≥ 0 and δ ∈ [0, 1], there exists a constant C1 > 0 such that the following estimates hold uniformly for all t ∈ [0,T ]

α sA(t) −α (−A(t)) e ≤ C1s , s > 0, (14) L(H) −δ sA(t) δ (−A(t)) I − e ≤ C1s , s ≥ 0, (15) L(H) see e.g. [9, (2.1)]. Proposition 1. [21, Theorem 6.1, Chapter 5] Let ∆(T ) := {(t, s) : 0 ≤ s ≤ t ≤ T }. Under Assumption 2.2 there exists a unique evolution system [21, Deﬁnition 5.3, Chapter 5] U : ∆(T ) −→ L(H) such that

(i) There exists a positive constant K0 such that

kU(t, s)kL(H) ≤ K0, 0 ≤ s ≤ t ≤ T. (16) (ii) U(., s) ∈ C1(]s, T ]; L(H)), 0 ≤ s ≤ T , ∂U (t, s) = A(t)U(t, s), 0 ≤ s < t ≤ T, (17) ∂t K kA(t)U(t, s)k ≤ 0 , 0 ≤ s < t ≤ T. (18) L(H) t − s (iii) U(t, .)v ∈ C1([0, t[; H), 0 < t ≤ T , v ∈ D(A(0)) and ∂U (t, s)v = −U(t, s)A(s)v, 0 ≤ s ≤ t ≤ T, (19) ∂s −1 kA(t)U(t, s)A(s) kL(H) ≤ K0, 0 ≤ s ≤ t ≤ T. (20)

α − α − α Note that D (−A(t)) 2 = R (−A(t)) 2 = (−A(t)) 2 H ⊂ H, α ≥ 0, where for an operator G, R(G) stands for its range, see e.g., [24, Section 12.4.2]. We α α ˙ α 2 equip Ht := D (−A(t)) , α ∈ R with the norm kukα,t := k(−A(t)) 2 uk. Due α α to (12)-(13) and for ease of notation, we simply write H˙ and k.kα. Note that H˙ equipped with the inner product α α 2 2 ˙ α hu, viα := (−A(0)) u, (−A(0)) v H , u, v ∈ H (21) MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4601 is a separable R-Hilbert space, since H = L2(Λ, R) is a R-separable Hilbert space. See also [12, Section 2] or [7, Section 2]. We follow [25] and assume that the nonlinear operators F and B satisfy the following Lipschitz condition.

Assumption 2.3. The nonlinear operator F : [0,T ] × H −→ H is assumed to be β/2-H¨oldercontinuous with respect to the ﬁrst variable and Lipschitz continuous with respect to the second variable, i.e. there exists a positive constant K3 such that

β kF (s, 0)k ≤ K3, kF (t, u) − F (s, v)k ≤ K3 |t − s| 2 + ku − vk , (22) for all s, t ∈ [0,T ] and u, v ∈ H.

2 Assumption 2.4. We assume the diffusion function B : [0,T ] × H −→ L0 to be β/2-Höldercontinuous with respect to the first variable and Lipschitz continuous with respect to the second variable, i.e. there exists a positive constant K4 such that

β kB(s, 0)k 0 ≤ K , kB(t, u) − B(s, v)k 0 ≤ K |t − s| 2 + ku − vk , (23) L2 4 L2 4 for all s, t ∈ [0,T ] and u, v ∈ H.

The following theorem ensures the existence of a unique mild solution of (1).

Theorem 2.5. [25, Theorem 1.3] Let Assumptions 2.1, 2.2 (i)–(ii), 2.3 and 2.4 be fulﬁlled. Then there exists a unique predictable stochastic process 1 X : [0,T ]×Ω −→ H˙ γ (with γ ∈ min[0, min(1, β))), called mild solution of (1) and satisfying Z t Z t X(t) = U(t, 0)X0 + U(t, s)F (s, X(s))ds + U(t, s)B(s, X(s))dW (s), (24) 0 0

P-a.s. for all t ∈ [0,T ], where U(t, s) is the evolution system of Proposition1. Moreover, for all t ∈ [0,T ], X(t) ∈ L2(Ω, H˙ γ ) and there exists a positive constant K5 such that sup kX(t)kL2(Ω,H˙ γ ) ≤ K5 1 + kX0kL2(Ω,H˙ γ ) . (25) 0≤t≤T

Additionally, the solution process X(t), t ∈ [0,T ] is continuous with respect to 1 2 2 E[k.kγ ] . To achieve optimal convergence order in space for multiplicative noise when β ∈ [1, 2], we require the following further assumption, also used in [7, 12, 27, 16, 20, 14].

Assumption 2.6. We assume that there exists a positive constant c1 > 0, such β−1 1 β−1 that B(s, H˙ ) ⊂ HS Q 2 (H), H˙ and

β−1 β−1 (−A(0)) 2 B(s, v) ≤ c1 (1 + kvkβ−1) , v ∈ H˙ , s ∈ [0,T ], (26) 0 L2 where β comes from Assumption 2.1.

1up to modiﬁcations 4602 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

2.2. Fully discrete scheme and main result. In the rest of this work, for the sake of simplicity, we assume the family of linear operators A(t) in (1) to be of second order. To have a more precise description of A(t), let us introduce the following second order differential operator d d X ∂ ∂ X ∂ A(t) = q (t, x) − q (t, x) + q (t, x)I, (27) ∂x i,j ∂x j ∂x 0 i,j=1 i j j=1 j 1 where qi,j, qj and q0 are C functions in [0,T ] × Λ and qi,j satisfies the following ellipticity condition d X 2 d qi,j(t, x)ξiξj ≥ c1|ξ| , (t, x) ∈ [0,T ] × Λ, ξ ∈ R , (28) i,j=1 where c1 > 0 is a uniform constant. Furthermore, we assume that there exist c2 ≥ 0 and 0 < γ ≤ 1 such that the following Höldercontinuity holds γ |qi,j(t, x) − qi,j(s, x)| ≤ c2|t − s| , 0 ≤ t, s ≤ T, x ∈ Λ. As in [4, Chapter III], we introduce two spaces H and V , such that H ⊂ V , depending on the boundary conditions of −A(t). For Dirichlet boundary conditions, we take

H1(Λ) 1 ∞ V = H = H0 (Λ) = Cc (Λ) . (29) For Robin boundary condition, we take V = H1(Λ) and 2 H = {v ∈ H (Λ) : ∂v/∂vA + α0v = 0, on ∂Λ}, α0 ∈ R, (30) where ∂v/∂vA stands for the differentiation along the outer conormal vector vA, d pointing at n = (ni)i=1, given by q X ∂v ∂v/∂v = n (x)q (x) , x ∈ ∂Λ. (31) A i i,j ∂x i,j=1 j One can easily check that [4, Chapter III, (11.140)] the bilinear operator a(t), associated to −A(t) defined by a(t)(u, v) = h−A(t)u, viH , u ∈ D(A(t)), v ∈ V satisfies 2 a(t)(v, v) ≥ λ0kvk1, v ∈ V, t ∈ [0,T ], (32) where λ0 is a positive constant, independent of t. Note that a(t)(·, ·) is bounded in V × V ([4, Chapter III, (11.13)]), so the following operator A(t): V → V ∗ defined through a(t)(u, v) = h−A(t)u, vi u, v ∈ V, t ∈ [0,T ], is well defined, where V ∗ is the dual space of V and h·, ·i the duality pairing between V ∗ and V . Identifying H to its adjoint space H∗ by the Riesz representation theorem, we get the following continuous and dense inclusions ∗ V ⊂ H ⊂ V , and therefore hu, viH = hu, vi, u ∈ H, v ∈ V.

So if we want to replace h·, ·i by the scalar product of h·, ·iH in H, we therefore need to have A(t)u ∈ H, for u ∈ V . So the domain of −A(t) is defined as D := D (−A(t)) = D (A(t)) = {u ∈ V,A(t)u ∈ H}. It is well known that [4, Chapter III, (11.11) & (11.110)] in the case of Dirichlet 1 2 boundary conditions D = H0 (Λ)∩H (Λ) and in the case of Robin boundary conditions D = H given by (30). We write the restriction of A(t): V −→ V ∗ to D (A(t)) MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4603 again A(t) which is therefore regarded as an operator of H (usually called the H realization of A(t)). The coercivity property (32) implies that −A(t) is a positive operator and its fractional powers are well defined ([15,4]). The following equivalence of norms holds [4, 15] α α α kvkHα(Λ) ≡ k((−A(t)) 2 vk := kvkα, v ∈ D((−A(t)) 2 ) ∩ H (Λ), (33) for any α ∈ [0, 2]. It is well known that A(t) is sectorial and generates an analytic A(s)t semigroup Ss(t) =: e , see e.g., [4, Chapter III] or [9, (2.1)]. The family of linear operators A(t) satisfies Assumption 2.2, see e.g. [21, Section 7.6]. It is well known that the family of operators {A(t)}0≤t≤T generate a two parameters evolution system {U(t, s)}0≤s≤t≤T , see e.g. [4, Page 832]. In the rest of this work the noise is considered to be trace-class. In the abstract 1 form (1), the nonlinear functions F : H −→ H and B : H −→ HS(Q 2 (H),H) are defined by (F (t, v))(x) = f(x, t, v(x)), (B(t, v)u)(x) = b(x, t, v(x)).u(x), (34)

1 for all x ∈ Λ, v ∈ H and u ∈ Q 2 (H), where f :Λ × R −→ R and b :Λ × R −→ R are continuously differentiable functions with globally bounded derivatives. Let us now turn our attention to the space discretization of the problem (1). We start by splitting the domain Λ in finite triangles. Let Th be the triangulation with maximal length h satisfying the usual regularity assumptions, and Vh ⊂ V be the space of continuous functions that are piecewise linear over the triangulation Th. 2 We consider the projection Ph from H = L (Λ) to Vh defined for every u ∈ H by

hPhu, χiH = hu, χiH , u ∈ H, χ ∈ Vh. (35)

For all t ∈ [0,T ], the discrete operator Ah(t): Vh −→ Vh is deﬁned by 1/2 ∗1/2 h−Ah(t)φ, χiH = h(−A(t)) φ, (−A(t)) χiH = a(t)(φ, χ), φ, χ ∈ Vh. (36) Note that (−A(t))∗1/2 is the adjoint of (−A(t))1/2. As in [15, (2.9)] or [4,8], it 1 follows that there exist constants C2 > 0 and θ ∈ ( 2 π, π) such that C k(λI − A (t))−1k ≤ 2 , λ ∈ S (37) h L(H) |λ| θ holds uniformly for h > 0 and t ∈ [0,T ]. It also holds that for any t ∈ [0,T ], h sAh(t) Ah(t) generates an analytic semigroup St (s) =: e , s ∈ [0,T ] and the smooth properties (14) and (15) hold for Ah uniformly for h > 0 and t ∈ [0,T ], i.e. for all α ≥ 0 and δ ∈ [0, 1], there exists a positive constant C3 such that the following estimates hold uniformly for h > 0 and t ∈ [0,T ], see e.g. [4,8]

α sAh(t) −α (−Ah(t)) e ≤ C3s , s > 0, (38) L(H) −δ sAh(t) δ (−Ah(t)) I − e ≤ C3s , s ≥ 0. (39) L(H) h The semi-discrete version of (1) consists of ﬁnding X (t) ∈ Vh, t ∈ [0,T ] such that h X (0) := PhX0 and h h h h dX (t) = [Ah(t)X (t) + PhF (t, X (t))]dt + PhB(t, X (t))dW (t), t ∈ (0,T ]. (40) Let us consider the following linear non-autonomous diﬀerential equations 0 y (t) = A(t)y(t), t ≥ t0, with y(t0) given. (41) 4604 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

• It is well known that if A(t) is a one dimensional matrix, the solution of (41) is given by Z t y(t) = exp A(s)ds y(t0), t ≥ t0. (42) t0

• If A(t) is an n × n matrix with n ∈ N and commute, i.e. A(t1)A(t2) = A(t2)A(t1), t1, t2 ≥ t0, then the solution of (41) is again given by (42). • It was shown by Magnus [19, Theorem III] that in the general case where the matrices A(t), t ∈ [0,T ] are time dependents and does not commute, the solution of (41) can be given in the following exponential form

Θ(t−t0) y(t) = e y(t0), t ≥ t0, (43) where Θ(t) called Magnus expansion is given by the following series [19, (3.28)] Z t 1 Z t Z τ Θ(t) = A(τ)dτ + A(τ), A(σ)dσ dτ t0 2 t0 t0 1 Z t Z τ Z σ + A(µ)dµ, A(σ) dσ, A(τ) dτ 4 t0 t0 t0 1 Z t Z τ Z τ + A(σ)dσ, A(µ)dµ, A(τ) dτ + ··· . (44) 12 t0 t0 t0 Here the Lie-product [u, v] of u and v is given by [u, v] = uv − vu. For deterministic problems, numerical methods based on this expansion received some attentions since one decade, see e.g. [3,5, 10, 18]. For the time-dependent Schrödingerequation [5], the Magnus expansion (44) was truncated after the first term and the integral was approximated by the mid-point rule. This mid-point rule approximation of Θ(t) was also used in [10] to obtain a second-order Magnus type integrator for non-autonomous deterministic parabolic partial differential equation (PDE). Note that the convergence analysis in [5, 10] was only done in time. Throughout this paper, we take tm = m∆t ∈ [0,T ], where T = M∆t for m, M ∈ N, m ≤ M. To build our numerical method, we follow the case of linear problem and write the solution of our semi-discrete problem (40) at time tm+1 as follows

Z tm+1 h Θh(∆t) h Θh(tm+1−s) h X (tm+1) = e X (tm) + e PhF (s, X (s))ds tm Z tm+1 Θh(tm+1−s) h + e PhB(s, X (s))dW (s), (45) tm h where Θh is given by (44) with A(t) replaced by Ah(t). Let Xm be the numerical h approximation of X (tm). To approximate the ﬁrst term of (45), we truncate (44) after the ﬁrst term and use the rectangle rule with left point rule to approximate the remaining integral. This yields

t R m+1 A (s)ds Θh(∆t) h tm h h ∆tAh(tm) h e X (tm) ≈ e Xm ≈ e Xm. (46) The second integral in (45) is approximated as follows

Z tm+1 Z tm+1 Θh(tm+1−s) h Θh(tm+1−s) h e PhF (s, X (s))ds ≈ e PhF (tm,Xm)ds. (47) tm tm

To approximate Θh(tm+1 − s), we truncate (44) after the ﬁrst term and this yields MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4605

Z tm+1 Z tm+1 Θh(tm+1 − s) ≈ Ah(s)ds ≈ Ah(tm)ds = Ah(tm)(tm+1 − s). (48) s s Substituting (48) in (47) yields the following approximation

Z tm+1 Z tm+1 Θh(tm+1−s) h Ah(tm)(tm+1−s) h e PhF (s, X (s))ds ≈ e PhF (tm,Xm)ds. (49) tm tm The last term in (45) is approximated as follows

Z tm+1 Θh(tm+1−s) h e PhB(s, X (s))dW (s) tm Z tm+1 Θh(tm+1−tm) h ≈ e PhB(tm,Xm)dW (s) tm Z tm+1 Ah(tm)∆t h ≈ e PhB(tm,Xm)dW (s). (50) tm Substituting (50), (49) and (46) in (45) yields the following fully discrete scheme for (1), called stochastic Magnus-type integrators (SMTI)

Z tm+1 h ∆tAh(tm) h (tm+1−s)Ah(tm) h Xm+1 = e Xm + e PhF tm,Xm ds tm Z tm+1 ∆tAh(tm) h + e PhB tm,Xm dW (s). (51) tm Note that the numerical scheme (51) can be written as follows

h ∆tAh,m h h Xm+1 = e Xm + ∆tϕ1(∆tAh,m)PhF tm,Xm

∆tAh,m h + e PhB tm,Xm ∆Wm, m = 0, ··· ,M, (52) h X0 = PhX0, Ah,m := Ah (tm), where the linear operator ϕ1(∆tAh,m) is given by 1 Z ∆t (∆t−s)Ah,m ϕ1(∆tAh,m) := e ds, (53) ∆t 0 and for any M ∈ N, ∆t = T/M, tm = m∆t, m = 0, 1, ··· ,M and

∆Wm := W ((m + 1)∆t) − W (m∆t). (54) We also note that an equivalent formulation of the numerical scheme (52), easy for simulation is given by h Xm+1 h h = Xm + PhB tm,Xm ∆Wm

h n h h o h i + ∆tϕ1(∆tAh,m) Ah,m Xm + PhB tm,Xm ∆Wm + PhF tm,Xm . (55) With the numerical method in hand, we can now state its strong convergence result toward the mild solution, which is in fact our main result. In the rest of this paper C is a generic constant independent of h, m, M and ∆t that may change from one place to another. Theorem 2.7. [Main result] Let Assumptions 2.1, 2.2, 2.3 and 2.4 be fulﬁlled. (i) If 0 ≤ β < 1, then the following error estimate holds

1 β h 2 2 β 2 EkX(tm) − Xmk ≤ C h + ∆t . (56) 4606 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

(ii) If 1 ≤ β < 2 and moreover if Assumption 2.6 is satisﬁed, then the following error estimate holds 1 1 h 2 2 β 2 EkX(tm) − Xmk ≤ C h + ∆t . (57) (iii) If β = 2 and if Assumption 2.6 is fulﬁlled, then the following error estimate holds 1 h 1 i h 2 2 2 2 2 EkX(tm) − Xmk ≤ C h 1 + max(0, ln(tm/h ) + ∆t . (58)

3. Proof of the main result. The proof of the main result needs some preparatory results. 3.1. Preparatory results. The following lemmas will be useful in our convergence proof. Lemma 3.1. [26,4] Let Assumption 2.2 be fulﬁlled. Then for any γ ∈ [0, 1], the following estimates hold uniformly in h > 0 and t ∈ [0,T ] −1 −γ −γ −γ K k(−(Ah(0)) vk ≤ k((−Ah(t)) vk ≤ Kk((−Ah(0)) vk, v ∈ Vh, (59) −1 γ γ γ K k(−(Ah(0)) vk ≤ k((−Ah(t)) vk ≤ Kk((Ah(0)) vk, v ∈ Vh ∩ D, (60) where K is a positive constant independent of t and h. Remark 2. From [4, Chapter III] it is well known that there exists a unique evolution system Uh : ∆(T ) −→ L(H), satisfying [21, (6.3), Page 149] Z t h h h Uh(t, s) = Ss (t − s) + Sτ (t − τ)R (τ, s)dτ, (61) s ∞ h Ah(s)t h P h h where Ss (t) := e , R (t, s) := Rm(t, s), with Rm(t, s) satisfying the fol- m=1 lowing recurrence relation [21, (6.22), Page 153] Z t h h h Rm+1 = R1 (t, s)Rm(τ, s)dτ, m ≥ 1 (62) s h h and R1 (t, s) := (Ah(s) − Ah(t))Ss (t − s). Note also that from [21, (6.6), Chapter 5, Page 150], the following identity holds Z t h h h h R (t, s) = R1 (t, s) + R1 (t, τ)R (τ, s)dτ. (63) s The mild solution of (40) is therefore given by Z t h h X (t) = Uh(t, 0)PhX0 + Uh(t, s)PhF s, X (s) ds 0 Z t h + Uh(t, s)PhB s, X (s) dW (s). (64) 0

Lemma 3.2. Under Assumption 2.2, the evolution system Uh : ∆(T ) −→ H satis- ﬁes the following 1 (i) Uh(., s) ∈ C (]s, T ]; L(H)), 0 ≤ s ≤ T and ∂U h (t, s) = A (t)U (t, s), 0 ≤ s ≤ t ≤ T, (65) ∂t h h C kA (t)U (t, s)k ≤ , 0 ≤ s < t ≤ T. (66) h h L(H) t − s MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4607

1 (ii) Uh(t, .)u ∈ C ([0, t[; H), 0 < t ≤ T , u ∈ D(Ah(0)) and ∂U h (t, s)u = −U (t, s)A (s)u, 0 ≤ s ≤ t ≤ T, (67) ∂s h h −1 kAh(t)Uh(t, s)Ah(s) kL(H) ≤ C, 0 ≤ s ≤ t ≤ T. (68) Proof. The proof is similar to that of [21, Theorem 6.1, Chapter 5] using (39), (38). See also [4, Chapter III]. Lemma 3.3. [26] or [4, Chapter III]. Let Assumption 2.2 be fulfilled. (i) The following estimates hold C kRh(t, s)k ≤ C, kRh (t, s)k ≤ (t − s)m−1, m ≥ 1, (69) 1 L(H) m L(H) m! h kR (t, s)kL(H) ≤ C, kUh(t, s)kL(H) ≤ C, 0 ≤ s ≤ t ≤ T. (70) (ii) For any 0 ≤ α ≤ 1, 0 ≤ γ ≤ 1 and 0 ≤ s ≤ t ≤ T , the following estimates hold α −α k(−Ah(r)) Uh(t, s)kL(H) ≤ C(t − s) , r ∈ [0,T ], (71) α −α kUh(t, s)(−Ah(r)) kL(H) ≤ C(t − s) , r ∈ [0,T ], (72) α −γ γ−α k(−Ah(r)) Uh(t, s)(−Ah(s)) kL(H) ≤ C(t − s) , r ∈ [0,T ]. (73) (iii) For any 0 ≤ s ≤ t ≤ T the following useful estimates hold −γ γ k (Uh(t, s) − I)(−Ah(s)) kL(H) ≤ C(t − s) , 0 ≤ γ ≤ 1, (74) −γ γ k −Ah(r)) (Uh(t, s) − I kL(H) ≤ C(t − s) , 0 ≤ γ ≤ 1. (75) The following space and time regularity of the semi-discrete problem (40) will be useful in our convergence analysis. Lemma 3.4. Let Assumptions 2.1, 2.2 (i)-(ii), 2.3 and 2.4 be fulfilled with the corresponding 0 ≤ β < 1. Then for all γ ∈ [0, β] the following estimates hold γ h k(−Ah(r)) 2 X (t)kL2(Ω,H) ≤ C, 0 ≤ r, t ≤ T, (76) h h β kX (t2) − X (t1)kL2(Ω,H) ≤ C(t2 − t1) 2 , 0 ≤ t1 ≤ t2 ≤ T. (77) Moreover if Assumption 2.6 is fulfilled, then (76) and (77) hold for β = 1.

h 2 Proof. We ﬁrst show that sup kX (t)kL2(Ω,H) ≤ C. Taking the norm in both t∈[0,T ] 2 2 2 2 sides of (64) and using the inequality (a + b + c) ≤ 3a + 3b + 3c , a, b, c ∈ R+ yields h 2 2 kX (t)kL2(Ω,H) ≤ 3kUh(t, 0)PhX0kL2(Ω,H) Z t 2 h + 3 Uh(t, s)PhF s, X (s) ds ds 0 L2(Ω,H) Z t 2 h + 3 Uh(t, s)PhB s, X (s) dW (s) 0 L2(Ω,H)

:= I0 + I1 + I2. (78)

Using Lemma 3.3 (i) and the uniform boundedness of Ph, it holds that 2 I0 ≤ 3kX0kL2(Ω,H) ≤ C. (79) 4608 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

Using again Lemma 3.3 (i), Assumption 2.3 and the uniform boundedness of Ph, it holds that

Z t 2 h I1 ≤ 3 kUh(t, s)PhF s, X (s) kL2(Ω,H) 0 Z t 2 h ≤ C C + kX (s)kL2(Ω,H) ds . 0 Using H¨olderinequality yields Z t h 2 I1 ≤ C + C kX (s)kL2(Ω,H)ds. (80) 0 Applying the itˆo-isometry, using Lemma 3.3 (i) and Assumption 2.4, it holds that

Z t Z t h 2 h 2 I2 = 3 EkUh(t, s)PhB s, X (s) kL0 ds ≤ C + C kX (t)kL2(Ω,H)ds. (81) 0 2 0 Substituting (81), (80) and (79) in (78) yields

Z t h 2 h 2 kX (t)kL2(Ω,H) ≤ C + C kX (s)kL2(Ω,H)ds. (82) 0 Applying the continuous Gronwall lemma to (82) yields

h 2 kX (t)kL2(Ω,H) ≤ C, t ∈ [0,T ]. (83)

γ Let us now prove (76). Pre-multiplying (64) by (−Ah(r)) 2 , taking the norm in both sides and using triangle inequality yields

γ h (−Ah(r)) 2 X (t) L2(Ω,H) γ ≤ (−Ah(r)) 2 Uh(t, 0)PhX0 L2(Ω,H) Z t γ h 2 + (−Ah(r)) Uh(t, s)PhF s, X (s) ds 0 L(Ω,H) Z t γ h + (−Ah(r)) 2 Uh(t, s)PhB s, X (s) dW (s) 0 L2(Ω,H)

:= II0 + II1 + II2. (84)

− γ γ Inserting (−Ah(0)) 2 (−Ah(0)) 2 , using Lemmas 3.3 (ii) 3.1, it holds that

γ − γ γ II0 ≤ k(−Ah(r)) 2 Uh(t, 0)(−Ah(0)) 2 kL(H)k(−Ah(0)) 2 X0k ≤ C. (85)

Using Lemmas 3.1, 3.3 (ii), Assumption 2.3 and (83) yields

Z t γ h II ≤ C (−A (s)) 2 U (t, s) sup F s, X (s) ds 1 h h L2(Ω,H) 0 L(H) t∈[0,T ] Z t h − γ ≤ C sup 1 + kX (s)kL2(Ω,H) (t − s) 2 ds ≤ C. (86) s∈[0,T ] 0 MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4609

Applying the Itˆo-isometry, using Lemmas 3.1, 3.3 (ii), Assumption 2.4 and (83) yields

Z t 2 2 γ h II = (−Ah(0)) 2 Uh(t, s)PhB s, X (s) ds 2 E 0 0 L2 Z t h 2 −γ ≤ C sup 1 + kX (s)kL2(Ω,H) (t − s) ds ≤ C. (87) s∈[0,T ] 0 Substituting (87), (86) and (85) in (84) completes the proof of (76). The proof of (77) follows from (64). In fact from (64), we have

h h kX (t2) − X (t1)kL2(Ω,H)

≤ k(Uh(t2, 0) − Uh(t1, 0)) PhX0kL2(Ω,H) Z t1 h + (Uh(t2, s) − Uh(t1, s)) PhF s, X (s) L2(Ω,H) ds 0 Z t2 h + Uh(t2, s)PhF s, X (s) L2(Ω,H) ds t1 Z t1 h + Uh(t2, s) − Uh(t1, s))PhB s, X (s) dW (s) 0 L2(Ω,H) Z t2 h + Uh(t2, s)PhB s, X (s) dW (s) t1 L2(Ω,H)

:= III0 + III1 + III2 + III3 + III4. (88)

Inserting an appropriate power of −Ah(t1), using Lemmas 3.3 (ii)-(iii) and [20, Lemma 1] yields

III0 = k(Uh(t2, t1) − I)Uh(t1, 0)PhX0kL2(Ω,H) − β ≤ (Uh(t2, t1) − I)(−Ah(t1)) 2 L(H) β − β β × (−Ah(t1)) 2 Uh(t1, 0)(−Ah(t1)) 2 (−Ah(t1)) 2 PhX0 L(H) L2(Ω,H) β ≤ C(t2 − t1) 2 . (89) Using Assumption 2.4,(76), Lemma 3.3 (ii) and (iii) yields

Z t1 h III1 ≤ k(Uh(t2, t1) − I)Uh(t1, s)kL(H) PhF s, X (s) L2(Ω,H) ds 0 Z t1 − β β 2 2 ≤ C (Uh(t2, t1) − I)(−Ah(t1)) (−Ah(t1)) Uh(t1, s) ds 0 L(H) L(H) Z t1 β − β ≤ C (t2 − t1) 2 (t1 − s) 2 ds 0 β ≤ C(t2 − t1) 2 . (90) Using Lemma 3.3 (i) and Assumption 2.3, it holds that

Z t2 h III2 ≤ C sup F s, X (s) L2(Ω,H) ds ≤ C(t2 − t1). (91) t1 s∈[0,T ] 4610 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

Using the Itˆo-isometry, Assumption 2.6,(76), Lemma 3.3 (ii)-(iii) and following the same lines as the estimate of III1 yields 2 β III3 ≤ C(t2 − t1) . (92)

Using the Itˆo-isometryand following the same lines as that of III2 yields 2 III4 ≤ C(t2 − t1). (93) Substituting (93), (92), (91), (90) and (89) in (88) completes the proof of (77). Let us consider the following deterministic problem: ﬁnd u ∈ V such that u0 = A(t)u, u(τ) = v, τ ≥ 0, t ∈ (τ, T ]. (94)

The corresponding semi-discrete problem in space is: ﬁnd uh ∈ Vh such that 0 uh(t) = Ah(t)uh, uh(τ) = Phv, τ ≥ 0, t ∈ (τ, T ]. (95) Let us deﬁne the operator

Th(t, τ) := U(t, τ) − Uh(t, τ)Ph, (96) so that u(t) − uh(t) = Th(t, τ)v. The following lemma will be useful in our convergence analysis. Lemma 3.5. [26] Let r ∈ [0, 2] and 0 ≤ γ ≤ r. Let Assumption 2.2 be fulﬁlled. Then the following error estimate holds for the semi-discrete approximation (95)

r − (r−γ) γ ku(t) − uh(t)k = kTh(t, τ)vk ≤ Ch (t − τ) 2 kvkγ , v ∈ H˙ . (97) Proposition 2. [Space error] Let Assumptions 2.1, 2.2, 2.3 and 2.4 be fulfilled. Let X(t) and Xh(t) be the mild solution of (1) and (40) respectively. (i) If 0 ≤ β < 1, then the following error estimate holds h β kX(t) − X (t)kL2(Ω,H) ≤ Ch , 0 ≤ t ≤ T. (98) (ii) If 1 ≤ β < 2 and furthermore if Assumption 2.6 is fulfilled, then the following error estimate holds h β kX(t) − X (t)kL2(Ω,H) ≤ Ch , 0 ≤ t ≤ T, (99) (iii) If β = 2 and furthermore if Assumption 2.6 is fulfilled, then the following error estimate holds h 2 2 kX(t) − X (t)kL2(Ω,H) ≤ Ch 1 + max 0, ln(t/h ) , 0 < t ≤ T. (100) Proof. Subtracting (64) form (24), taking the L2 norm and using triangle inequality yields h kX(t) − X (t)kL2(Ω,H)

≤ kU(t, 0)X0 − Uh(t, 0)PhX0kL2(Ω,H) Z t h + U(t, s)F (s, X(s)) − Uh(t, s)PhF s, X (s) ds 0 L2(Ω,H) Z t h + U(t, s)B (s, X(s)) − Uh(t, s)PhB s, X (s) dW (s) 0 L2(Ω,H)

=: IV0 + IV1 + IV2. (101) Using Lemma 3.5 with r = γ = β yields β β IV0 ≤ Ch kX0kL2(Ω,H˙ β ) ≤ Ch . (102) MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4611

Using Lemma 3.5 with r = β, γ = 0, Assumption 2.3, Lemmas 3.4 and 3.3 yields Z t h IV1 ≤ U(t, s)F (s, X(s)) − U(t, s)F s, X (s) L2(Ω,H) ds 0 Z t h h + U(t, s)F s, X (s) − Uh(t, s)PhF s, X (s) L2(Ω,H) ds 0 t t Z Z β h β − 2 ≤ C X(s) − X (s) L2(Ω,H) ds + Ch (t − s) ds 0 0 Z t β h ≤ Ch + C X(s) − X (s) L2(Ω,H) ds. (103) 0 β−1 Using the Itˆo-isometryproperty, Lemma 3.4, Lemma 3.5 with r = β and γ = 2 , Assumption 2.6 yields Z t 2 h 2 IV2 = U(t, s)B (s, X(s)) − Uh(t, s)PhB s, X (s) L0 ds 0 2 Z t h 2 ≤ U(t, s)B (s, X(s)) − U(t, s)B s, X (s) L0 ds 0 2 Z t h h 2 + U(t, s)B s, X (s) − Uh(t, s)PhB s, X (s) L0 ds 0 2 Z t Z t h 2 2β −1+β ≤ C X(s) − X (s) L2(Ω,H) ds + Ch (t − s) ds 0 0 Z t 2β h 2 ≤ Ch + C X(s) − X (s) L2(Ω,H) ds. (104) 0 Substituting (104), (103) and (102) in (101) yields Z t h 2 2β h 2 X(t) − X (t) L2(Ω,H) ≤ Ch + C X(s) − X (s) L2(Ω,H) ds. (105) 0 Applying the continuous Gronwall lemma to (105) yields h β X(t) − X (t) L2(Ω,H) ≤ Ch . (106)

For non commutative operators Hj on a Banach space, we introduce the following notation for the composition of operators k Y H H ··· H if k ≥ l, H = k k−1 l (107) j I if k < l. j=l The following lemma will be useful in our convergence proof. Lemma 3.6. [26] Let Assumption 2.2 be fulﬁlled. Then the following estimate holds   m Y ∆tAh,j γ −γ  e  (−Ah,l) ≤ Ct , (108) m−l j=l L(H)   m γ1 Y ∆tAh,j −γ2 γ2−γ1 (−Ah,k)  e  (−Ah,l) ≤ Ct , (109) m−l j=l L(H) 4612 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

0 ≤ γ1 ≤ 1, 0 ≤ γ < 1, 0 < γ2 ≤ 1, for any 0 ≤ l < m, where C is a positive constant independent of m, l, h and ∆t. Lemma 3.7. (i) For all α ≥ 0, the following estimate holds

h α −α R (t, s)(−Ah(s)) L(H) ≤ C(t − s) , t, s ∈ [0,T ]. (110) (ii) For all α ∈ [0, 1], the following estimate holds

∆tAh,j−1 −α 1+α Uh(tj, tj−1) − e (−Ah,j−1) ≤ C∆t . (111) L(H) (iii) For all α ∈ [0, 1), the following estimate holds

∆tAh,j−1 α 1−α Uh(tj, tj−1) − e (−Ah,j−1) L(H) ≤ C∆t . (112) (iv) For all α ∈ [0, 1], the following estimate holds

−α ∆tAh,j−1 1+α (−Ah,j−1) Uh(tj, tj−1) − e L(H) ≤ C∆t . (113) Proof. From the integral equation (63), we have

h α Ah(s)(t−s) α R (t, s)(−Ah(s)) = e (−Ah(s)) Z t h h α + R1 (t, τ)R (τ, s)(−Ah(s)) dτ. (114) s Taking the norm in both sides of (114), using (39) and Lemma 3.3 yields

h α Ah(s)(t−s) α R (t, s)(−Ah(s)) ≤ e (−Ah(s)) (115) L(H) L(H) Z t h h α + kR1 (τ, s)kL(H) R (τ, s)(−Ah(s)) L(H) dτ s Z t −α h α ≤ C(t − s) + C R (τ, s)(−Ah(s)) L(H) dτ. s Applying the continuous Gronwall lemma to (115) yields

h α −α R (t, s)(−Ah(s)) L(H) ≤ C(t − s) . (116) This completes the proof of (i). From (61) and (63), we have

∆tAh,j−1 Uh(tj, tj−1) − e Z tj (tj −τ)Ah(τ) = e Rh(τ, tj−1)dτ tj−1 Z tj (tj −τ)Ah(τ) h = e R1 (τ, tj−1)dτ tj−1 " # Z tj Z τ (tj −τ)Ah(τ) h h + e R1 (τ, s)R (s, tj−1)ds dτ tj−1 tj−1

Z tj (tj −τ)Ah(τ) Ah,j−1(τ−tj−1) = e (Ah(τ) − Ah(tj−1)) e dτ tj−1 " # Z tj Z τ (tj −τ)Ah(τ) h h + e R1 (τ, s)R (s, tj−1)ds dτ. (117) tj−1 tj−1 MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4613

Therefore, from (117), for all α ∈ [0, 1], using (39) and Lemma 3.3, it holds that

∆tAh,j−1 −α Uh(tj, tj−1) − e (−Ah,j−1) L(H) Z tj −1 (tj −τ)Ah(τ) ≤ e (Ah(τ) − Ah(tj−1)) (−Ah,j−1) tj−1

Ah,j−1(τ−tj−1) 1−α .e (−Ah,j−1) dτ L(H) " # Z tj Z τ (tj −τ)Ah(τ) h h + e kR1 (τ, s)R (s, tj−1)kL(H)ds dτ L(H) tj−1 tj−1 Z tj (tj −τ)Ah(τ) −1 ≤ e (Ah(τ) − Ah(tj−1)) (−Ah,j−1) L(H) L(H) tj−1 Z tj Z τ Ah,j−1(τ−tj−1) 1−α × e (−Ah,j−1) dτ + C dsdτ L(H) tj−1 tj−1 Z tj α 2 1+α ≤ C (τ − tj−1) dτ + C∆t ≤ C∆t . (118) tj−1 This completes the proof of (ii). The proof of (iii) and (iv) are similar to that of (ii) using (i). The following lemma can be found in [15]

Lemma 3.8. For all α1, α2 > 0 and α ∈ [0, 1), there exist two positive constants

Cα1,α2 and Cα,α2 such that m X −1+α1 −1+α2 −1+α1+α2 ∆t tm−j+1tj ≤ Cα1,α2 tm , (119) j=1 m X −α −1+α2 −α+α2 ∆t tm−j+1tj ≤ Cα,α2 tm . (120) j=1 Proof. The proof of (119) follows from the comparison with the integral Z t (t − s)−1+α1 s−1+α2 ds. (121) 0 The proof of (120) is a consequence of (119). The following lemma is fundamental in our convergence analysis. Lemma 3.9. Let Assumption 2.2 be fulﬁlled. Then for all 1 ≤ i ≤ m ≤ M. (i) The following estimate holds     m m−1 Y Y ∆tAh,j 1−  Uh(tj, tj−1) −  e  ≤ C∆t , (122)

j=i j=i−1 L(H) where > 0 is a positive number small enough. (ii) The following estimate also holds     m m−1 Y Y ∆tAh,j −  Uh(tj, tj−1) −  e  (−Ah,i−1) ≤ C∆t. (123)

j=i j=i−1 L(H) 4614 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

Proof. First of all note that

 m   m−1  Y Y ∆tAh,j  Uh(tj, tj−1) −  e  j=i j=i−1

 m   m  Y Y ∆tAh,j−1 =  Uh(tj, tj−1) −  e  . (124) j=i j=i Using the telescopic sum, (124) can be written as follows

 m   m  Y Y ∆tAh,j−1  Uh(tj, tj−1) −  e  j=i j=i

m−i+1  m  X Y ∆tAh,i+k−2 =  Uh(tj, tj−1) Uh (ti+k−1, ti+k−2) − e k=1 j=i+k

i+k−2  Y ∆tA .  e h,j−1  . (125) j=i Writing down explicitly the ﬁrst term of (125) gives

 m   m  Y Y ∆tAh,j−1  Uh(tj, tj−1) −  e  j=i j=i

 m  Y ∆tAh,i−1 =  Uh(tj, tj−1) Uh(ti, ti−1) − e j=i+1

m−i+1  m  X Y ∆tAh,i+k−2 +  Uh(tj, tj−1) Uh (ti+k−1, ti+k−2) − e k=2 j=i+k

i+k−2  Y ∆tA .  e h,j−1  . (126) j=i Taking the norm in both sides of (126), using Lemma 3.3, Lemma 3.7 (ii) and Lemma 3.6 yields     m m Y Y ∆tAh,j−1  Uh(tj, tj−1) −  e 

j=i j=i L(H)

∆tAh,i−1 ≤ kUh(tm−i+1, ti)kL(H) Uh(ti, ti−1) − e L(H) m−i+1 X + kUh(tm, ti+k−1)kL(H) k=2

∆tAh,i+k−2 −1+ × Uh(ti+k−1, ti+k−2) − e (−Ah,i+k−2) L(H)   i+k−2 1− Y ∆tAh,j−1 × (−Ah,i+k−2)  e 

j=i L(H) MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4615

m−i+1 X 2− −1+ 1− ≤ C∆t + C ∆t tk−1 ≤ C∆t . k=2 (127) This completes the proof of (i). The proof of (ii) is similar to that of (i) using (109) and Lemma 3.8. With the above preparatory results in hand, we can now prove our main result.

3.2. Proof of Theorem 2.7. Using triangle inequality, we split the fully discrete error in two parts as follows. h h h h kX(tm) − XmkL2(Ω,H) ≤ kX(tm) − X (tm)kL2(Ω,H) + kX (tm) − XmkL2(Ω,H) =: V + VI. (128) The space error V is estimated in Lemma 3.5. It remains to estimate the time error VI. Note that the mild solution of (40) can be written as follows.

Z tm h h h X (tm) = Uh(tm, tm−1)X (tm−1) + Uh(tm, s)PhF s, X (s) ds tm−1 Z tm h + Uh(tm, s)PhB s, X (s) dW (s). (129) tm−1 Iterating the mild solution (129) yields h X (tm)  m  Z tm Y h =  Uh(tj, tj−1) PhX0 + Uh(tm, s)PhF s, X (s) ds (130) j=1 tm−1

Z tm h + Uh(tm, s)PhB s, X (s) dW (s) tm−1 m−1  m  Z tm−k X Y h +  Uh(tj, tj−1) Uh(tm−k, s)PhF s, X (s) ds k=1 tm−k−1 j=m−k+1

m−1  m  Z tm−k X Y h +  Uh(tj, tj−1) Uh(tm−k, s)PhB s, X (s) dW (s). k=1 tm−k−1 j=m−k+1

h Iterating the numerical scheme (51) by substituting Xj , j = m − 1, ··· , 1 only in the ﬁrst term of (51) by their expressions yields h Xm m−1  Z tm Y ∆tAh,j h (tm−s)Ah,m−1 h =  e  X0 + e PhF tm−1,Xm−1 ds (131) j=0 tm−1

Z tm ∆tAh,m−1 h + e PhB tm−1,Xm−1 dW (s) tm−1 m−1  m−1  Z tm−k X Y ∆tAh,j (tm−k−s)Ah,m−k−1 h +  e  e PhF tm−k−1,Xm−k−1 ds k=1 tm−k−1 j=m−k 4616 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

m−1  m−1  Z tm−k X Y ∆tAh,j ∆tAh,m−k−1 h +  e  e PhB tm−k−1,Xm−k−1 dW (s). k=1 tm−k−1 j=m−k

Substracting (131) from (130) yields

h h X (tm) − Xm  m  m−1  Y Y ∆tAh,j =  Uh(tj, tj−1) PhX0 −  e  PhX0 j=1 j=0 Z tm h i h (tm−s)Ah,m−1 h + Uh(tm, s)PhF s, X (s) − e PhF tm−1,Xm−1 ds tm−1 Z tm h ∆tAh,m−1 h + Uh(tm, s)PhB s, X (s) − e PhB tm−1,Xm−1 dW (s) tm−1 m−1  m  Z tm−k X Y h +  Uh(tj, tj−1) Uh(tm−k, s)PhF s, X (s) ds k=1 tm−k−1 j=m−k+1

m−1  m−1  Z tm−k X Y ∆tAh,j (tm−k−s)Ah,m−k−1 h −  e  e PhF tm−k−1,Xm−k−1 ds k=1 tm−k−1 j=m−k

m−1  m  Z tm−k X Y h +  Uh(tj, tj−1) Uh(tm−k, s)PhB s, X (s) dW (s) k=1 tm−k−1 j=m−k+1

m−1  m−1  Z tm−k X Y ∆tAh,j ∆tAh,m−k−1 h −  e  e PhB tm−k−1,Xm−k−1 dW (s) k=1 tm−k−1 j=m−k

=: VI1 + VI2 + VI3 + VI4 + VI5. (132)

Taking the norm in both sides of (132) yields

5 h h 2 X 2 kX (tm) − XmkL2(Ω,H) ≤ 25 kVIikL2(Ω,H). (133) i=1

In what follows, we estimate separately kVIikL2(Ω,H), i = 1, ··· , 5.

3.2.1. Estimate of VI1, VI2 and VI3. Using Lemma 3.9, it holds that     m m−1 Y Y ∆tAh,j kVI1kL2(Ω,H) ≤  Uh(tj, tj−1) −  e  kX0kL2(Ω,H)

j=1 j=0 L(H) ≤ C∆t1−. (134)

Using triangle inequality yields

kVI2kL2(Ω,H) Z tm h ≤ Uh(tm, s)PhF s, X (s) ds L2(Ω,H) tm−1 MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4617

Z tm h i (tm−s)Ah,m−1 h h + e PhF tm−1,Xm−1 − PhF tm−1,X (tm−1) ds L2(Ω,H) tm−1 Z tm (tm−s)Ah,m−1 h + e PhF tm−1,X (tm−1) ds. L2(Ω,H) tm−1

Using (38), Lemma 3.3, Assumption 2.3 and Theorem 2.5, it holds that

Z tm Z tm Z tm h h kVI2kL2(Ω,H) ≤ C ds + C kX (tm−1) − Xm−1kL2(Ω,H)ds + C ds tm−1 tm−1 tm−1 h h ≤ C∆t + C∆tkX (tm−1) − Xm−1kL2(Ω,H). (135)

Applying the Itˆo-isometry, using Assumption 2.4,(38), Theorem 2.5 and Lemma 3.3 yields

2 kVI3kL2(Ω,H) Z tm h 2 ≤ 9 E Uh(tm, s)PhB s, X (s) 0 ds L2 tm−1 t Z m 2 ∆tAh,m−1 h h + 9 E e PhB tm−1,Xm−1 − PhB tm−1,X (tm−1) 0 ds L2 tm−1 t Z m 2 ∆tAh,m−1 h + 9 E e PhF tm−1,X (tm−1) 0 ds L2 tm−1 Z tm Z tm Z tm h h 2 ≤ C ds + C kX (tm−1) − Xm−1kL2(Ω,H)ds + C ds tm−1 tm−1 tm−1 h h 2 ≤ C∆t + C∆tkX (tm−1) − Xm−1kL2(Ω,H). (136)

3.2.2. Estimate of VI4. To estimate VI4, we split it in ﬁve terms as follows.

m−1  m  X Z tm−k Y VI4 =  Uh(tj, tj−1) Uh(tm−k, s) k=1 tm−k−1 j=m−k+1 h h PhF s, X (s) − PhF tm−k−1,X (tm−k−1) ds

m−1  m  X Z tm−k Y +  Uh(tj, tj−1) k=1 tm−k−1 j=m−k+1 h [Uh(tm−k, s) − Uh(tm−k, tm−k−1)] PhF tm−k−1,X (tm−k−1) ds

m−1  m   m−1  Z tm−k X Y Y ∆tAh,j +  Uh(tj, tj−1) −  e  k=1 tm−k−1 j=m−k j=m−k−1 h PhF tm−k−1,X (tm−k−1) ds

m−1  m−1  Z tm−k X Y ∆tA ∆tA (t −s)A +  e h,j  e h,m−k−1 − e m−k h,m−k−1 k=1 tm−k−1 j=m−k h PhF tm−k−1,X (tm−k−1) ds 4618 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

m−1  m−1  Z tm−k X Y ∆tA (t −s)A +  e h,j  e m−k h,m−k−1 k=1 tm−k−1 j=m−k h h PhF tm−k−1,X (tm−k−1) − PhF tm−k−1,Xm−k−1 ds

=: VI41 + VI42 + VI43 + VI44 + VI45. (137)

Using triangle inequality and Assumption 2.3 yields

kVI41kL2(Ω,H) m−1 Z tm−k X h h ≤ C PhF s, X (s) − PhF tm−k−1,X (tm−k−1) L2(Ω,H) ds k=1 tm−k−1 m−1 Z tm−k X β ≤ C (s − tm−k−1) 2 ds k=1 tm−k−1 m−1 Z tm−k X h h + C kX (s) − X (tm−k−1)kL2(Ω,H)ds. k=1 tm−k−1

Using Lemma 3.3 yields

m−1 β Z tm−k min(β,1) min(β,1) 2 X 2 2 kVI41kL2(Ω,H) ≤ C∆t + C (s − tm−k−1) ds ≤ C∆t . (138) k=1 tm−k−1

Using Triangle inequality and inserting an appropriate power of Ah,m−1 yields

kVI42kL2(Ω,H) m−1 X Z tm−k ≤ C kUh(tm, tm−k)Uh(tm−k, s)(I − Uh(s, tm−k−1)kL(H) k=1 tm−k−1 h × PhF tm−k−1,X (tm−k−1) L2(Ω,H) ds m−1 Z tm−k X 1− ≤ C Uh(tm, tm−k)(−Ah,m−k) L(H) k=1 tm−k−1 −1+ 1− × (−Ah,m−k) Uh(tm−k, s)(−Ah,m−k) L(H) −1+ × (−Ah,m−k) (I − Uh(s, tm−k−1)) L(H) ds.

Using Lemma 3.3, Assumption 2.3 and Theorem 2.5 yields

m−1 Z tm−k X −1+ 1− kVI42kL2(Ω,H) ≤ C (tm − tm−k) (s − tm−k−1) ds k=1 tm−k−1 m−1 Z tm−k 1− X −1+ ≤ C∆t tk ds k=1 tm−k−1 m−1 1− X −1+ 1− ≤ C∆t ∆ttk ≤ C∆t . (139) k=1 MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4619

Using triangle inequality and inserting appropriate power of Ah,m−1 yields

m−1  m−1  Z tm−k X Y ∆tAh,j kVI43kL2(Ω,H) ≤  e 

k=1 tm−k−1 j=m−k e(s−tm−k−1)Ah,m−k−1 − I e(tm−k−s)Ah,m−k−1 L(H) h × PhF tm−k−1,X (tm−k−1) L2(Ω,H) ds m−1  m−1  Z tm−k X Y ∆tAh,j 1− ≤ C  e  (−Ah,m−k−1)

k=1 tm−k−1 j=m−k L(H) −1+ (s−tm−k−1)Ah,m−k−1 × (−Ah,m−k−1) e − I L(H)

× e(tm−k−s)Ah,m−k−1 ds. L(H)

Using Lemma 3.6, Assumption 2.3, Theorem 2.5,(38) and (39) yields

m−1 Z tm−k X −1+ 1− kVI43kL2(Ω,H) ≤ C tk (s − tm−k−1) ds k=1 tm−k−1 m−1 Z tm−k 1− X −1+ 1− ≤ C∆t tk ∆tds ≤ C∆t . (140) k=1 tm−k−1

Using triangle inequality and inserting an appropriate power of Ah,m−1 yields

m−1  m−1  Z tm−k X Y ∆tAh,j kVI44kL2(Ω,H) ≤  e 

k=1 tm−k−1 j=m−k I − e(s−tm−k−1)Ah,m−k−1 e(tm−k−s)Ah,m−k−1 L(H) h × PhF tm−k−1,X (tm−k−1) L2(Ω,H) ds m−1  m−1  Z tm−k X Y ∆tAh,j 1− ≤ C  e  (−Ah,m−k)

k=1 tm−k−1 j=m−k L(H) −1+ (s−tm−k−1)Ah,m−k−1 × (−Ah,m−k) I − e L(H)

× e(tm−k−s)Ah,m−k−1 ds. L(H)

Using Lemma 3.6,(38), (39), Assumption 2.3 and Lemma 3.3 yields

m−1 Z tm−k X −1+ 1− kVI44kL2(Ω,H) ≤ C tk (s − tm−k−1) ds k=1 tm−k−1 m−1 Z tm−k 1− X −1+ 1− ≤ C∆t tk ds ≤ C∆t . (141) k=1 tm−k−1 4620 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

Using Lemma 3.6 and Assumption 2.3 yields m−1 Z tm−k X h h kVI45kL2(Ω,H) ≤ C kX (tm−k−1) − Xm−k−1kL2(Ω,H)ds k=1 tm−k−1 m−1 X h h ≤ C∆t kX (tk) − Xk kL2(Ω,H). (142) k=0 Substituting (142), (141), (140), (139) and (138) in (137) yields m−1 min(β,1)/2 X h h kVI4kL2(Ω,H) ≤ C∆t + C∆t kX (tk) − Xk kL2(Ω,H). (143) k=0

3.2.3. Estimate of VI5. To estimate VI5, we split it in four terms as follows

m−1  m  X Z tm−k Y VI5 =  Uh(tj, tj−1) Uh(tm−k, s) k=1 tm−k−1 j=m−k+1 h h PhB s, X (s) − PhB tm−k−1,X (tm−k−1) dW (s)

m−1  m  X Z tm−k Y +  Uh(tj, tj−1) k=1 tm−k−1 j=m−k+1 h [Uh(tm−k, s) − Uh(tm−k, tm−k−1)] PhB tm−k−1,X (tm−k−1) dW (s)

m−1  m   m−1  Z tm−k X Y Y ∆tAh,j +  Uh(tj, tj−1) −  e  k=1 tm−k−1 j=m−k j=m−k−1 h PhB tm−k−1,X (tm−k−1) dW (s)

m−1  m−1  Z tm−k X Y ∆tA +  e h,j  k=1 tm−k−1 j=m−k−1 h h PhB tm−k−1,X (tm−k−1) − PhB tm−k−1,Xm−k−1 dW (s)

=: VI51 + VI52 + VI53 + VI54. (144) Using the Itˆo-isometryproperty, Lemma 3.3, Assumption 2.4 and Lemma 3.4 yields m−1 Z tm−k 2 X h kVI51kL2(Ω,H) = E Uh(tm, s) PhB s, X (s) k=1 tm−k−1 h 2 −PhB tm−k−1,X (tm−k−1) 0 ds L2 m−1 Z tm−k X β ≤ C (s − tm−k−1) ds k=1 tm−k−1 m−1 Z tm−k X h h 2 + C X (s) − X (tm−k−1) L2(Ω,H) ds k=1 tm−k−1 m−1 Z tm−k β X min(β,1) ≤ C∆t + C (s − tm−k−1) ds k=1 tm−k−1 ≤ C∆tmin(β,1). (145) MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4621

Applying the Itˆo-isometry, using Lemma 3.3, Assumption 2.4 and Lemma 3.4 yields

m−1 Z tm−k 2 X kVI52kL2(Ω,H) = E kUh(tm, tm−k)Uh(tm−k, s) k=1 tm−k−1 h 2 (I − Uh(s, tm−k−1)) PhB tm−k−1,X (tm−k−1) 0 ds L2 m−1 t Z m−k 1− 2 X 2 ≤ C Uh(tm, tm−k)(−Ah,m−k) L(H) k=1 tm−k−1 −1+ 1− 2 2 2 ×k(−Ah,m−k) Uh(tm−k, s)(−Ah,m−k) kL(H) −1+ 2 2 × (−Ah,m−k) (I − Uh(s, tm−k−1)) ds L(H) m−1 Z tm−k X −1+ 1− ≤ C tk (s − tm−k−1) ds k=1 tm−k−1 m−1 Z tm−k 1− X −1+ 1− ≤ C∆t tk ds ≤ C∆t . (146) k=1 tm−k−1 Applying the Itˆo-isometry, Lemma 3.9, Assumption 2.4 and Lemma 3.4 yields

m−1  m   m−1  Z tm−k 2 X Y Y ∆tAh,j kVI53kL2(Ω,H) = E  Uh(tj , tj−1) −  e 

k=1 tm−k−1 j=m−k j=m−k−1 2 h .PhB tm−k−1,X (tm−k−1) ds 0 L2 m−1 Z tm−k X 1− 1− ≤ C ∆t ds ≤ C∆t . (147) k=1 tm−k−1 Applying the Itˆo-isometry, Lemma 3.6 and Assumption 2.4 yields

m−1  m−1  Z tm−k 2 X Y ∆tAh,j kVI54kL2(Ω,H) = E  e 

k=1 tm−k−1 j=m−k−1 2 h h h i . PhB tm−k−1,X (tm−k−1) − PhB tm−k−1,Xm−k−1 ds 0 L2 m−1 Z tm−k 2 X h h ≤ C X (tm−k−1) − Xm−k−1 ds L2(Ω,H) k=1 tm−k−1 m−1 X h h 2 ≤ C∆t kX (tk) − Xk kL2(Ω,H). (148) k=0 Substituting (148), (147), (146) and (145) in (144) yields

m−1 2 min(β,1) X h h 2 kVI5kL2(Ω,H) ≤ C∆t + C∆t kX (tk) − Xk kL2(Ω,H). (149) k=0 Substituting (149), (143), (136), (135) and (134) in (132) yields

m−1 h h 2 min(β,1−) X h h 2 kX (tm) − XmkL2(Ω,H) ≤ C∆t + C∆t kX (tk) − Xk kL2(Ω,H). (150) k=0 4622 ANTOINE TAMBUE AND JEAN DANIEL MUKAM

Applying the discrete Gronwall lemma to (150) yields h h min(β,1−)/2 kX (tm) − XmkL2(Ω,H) ≤ C∆t . (151) Note that to achieve optimal convergence 1/2 when β ≥ 1, we only need to re- estimate kVI52kL2(Ω,H) and kVI53kL2(Ω,H) by using Assumption 2.6 and Lemma 3.9 (ii). This is straightforward. The proof of Theorem 2.7 is therefore completed.

4. Numerical experiments. We consider the following stochastic dominated ad- vection diffusion reaction equation with constant diagonal diffusion tensor e−tX dX = (1 + e−t)(∆X − ∇ · (qX)) − dt + XdW, X(0) = 0, (152) |X| + 1 with mixed Neumann-Dirichlet boundary conditions on Λ = [0,L1] × [0,L2]. The Dirichlet boundary condition is X = 1 at Γ = {(x, y): x = 0} and we use the homogeneous Neumann boundary conditions elsewhere. The eigenfunctions (1) (2) {ei,j} = {ei ⊗ ej }i,j≥0 of the covariance operator Q are the same as for the Laplace operator −∆ with homogeneous boundary condition, given by r r (l) 1 (l) 2 iπ e0 (x) = , ei (x) = cos x , i ∈ N, Ll Ll Ll where l ∈ {1, 2} , x ∈ Λ. We assume that the noise can be represented as X p W (x, t) = λi,jei,j(x)βi,j(t), (153) 2 (i,j)∈N where βi,j(t) are independent and identically distributed standard Brownian mo- 2 tions, λi,j,(i, j) ∈ N are the eigenvalues of Q, with 2 2−(β+δ) λi,j = i + j , β > 0, (154) in the representation (153) for some small δ > 0. To obtain trace class noise, it is enough to have β +δ > 1. In our simulations, we take β ∈ {1.5, 2} and δ = 0.001. In (34), we take b(x, u) = u, x ∈ Λ and u ∈ R. Therefore, from [12, Section 4] it follows that the operators B defined by (34) fulfills Assumption 2.4 and Assumption 2.6. e−tv The function F is given by F (t, v) = − , t ∈ [0,T ], v ∈ H and obviously 1 + |v| satisfies Assumption 2.3. The linear differential operator A(t) is given by A(t) = (1 + e−t)(∆(.) − ∇.v(.)) , t ∈ [0,T ], (155) where v is the Darcy velocity. We obtain the Darcy velocity field v = (qi) by solving the following system ∇ · v = 0, v = −k∇p, (156) 1 with Dirichlet boundary conditions on ΓD = {0,L1} × [0,L2] and Neumann bound- 1 ary conditions on ΓN = (0,L1) × {0,L2} such that 1 in {0} × [0,L ] p = 2 0 in {L1} × [0,L2] 1 and −k ∇p(x, t) · n = 0 in ΓN . Here, we use a constant permeabily tensor k and obtained almost a linear presure p. Clearly D(A(t)) = D(A(0)), t ∈ [0,T ] and α α D((−A(t)) ) = D((−A(0)) ), t ∈ [0,T ], 0 ≤ α ≤ 1. The function qi,j(t, x) defined −t in (27) is given by qi,i(t, x) = 1 + e , and qi,j(t, x) = 0, i 6= j. Since qi,i(t, x) is bounded below by 1 + e−T , it follows that the ellipticity condition (28) holds and MAGNUS-TYPE INTEGRATOR FOR NON-AUTONOMOUS SPDES 4623 therefore as a consequence of Section 2.2, it follows that A(t) is sectorial. Obviously Assumption 2.2 is fulfilled.

×10-4 12 11 Magnus β=1.5 10 Magnus β=2 9 8 7 6

5 mean error) 2 4 log( L

1 2 3 4 5 ∆ -3 log( t) ×10

Figure 1. Convergence of the Magnus integrator for β = 1, and β = 2 in (154). The order of convergence in time is 0.57 for β = 1, 0.54 for β = 2. The total number of samples used is 100.

In Figure1, we can observe the convergence of the the stochastic Magnus scheme for two noise’s parameters. Indeed the order of convergence in time is 0.57 for β = 1 and 0.54 for β = 2. These orders are close to the theoretical orders 0.5 obtained in Theorem 2.7 for β = 1 and β = 2.

Acknowledgment. The authors would like to thank the anonymous referees for their careful readings and comments that helped to improve the paper.

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