Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain

Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class of stochastic reaction-diffusion partial differential equations with cubic nonlinearity. Dividing space into overlapping finite elements, a special coupling condition between neighbouring elements preserves the self-adjoint dynamics and controls interelement interactions. When the interelement coupling parameter is small, an averaging method and an asymptotic expansion of the slow modes show that the macroscopic discrete model will be a family of coupled stochastic ordinary differential equations which describe the evolution of the grid values. This modelling shows the importance of subgrid scale interaction between noise and spatial diffusion and provides a new rigourous approach to constructing semi-discrete approximations to stochastic reaction-diffusion partial differential equations.


Introduction
In modelling complex systems we often desire to model the effective macroscopic dynamics. Some cases can be extracted from the full, or microscopic, description by methods such as averaging, invariant manifold reduction and homogenization [4, 18, e.g.].
Stochastic partial differential equations (spdes) are widely studied in modeling, analyzing, simulating and predicting complex phenomena in many fields of nonlinear science [2, 3, 15, 23, e.g.]. Recently, macroscopic reduction for dissipative spdes, with two widely separated timescales, has been studied by the dynamical systems theory of stochastic invariant manifolds [8, 9, 17, e.g.] and by averaging methods [19, 20, e.g.]. Moreover, invariant manifold theory also applies to generate a macroscopic discrete model of deterministic and stochastic dissipative pdes [7, 10, 12, e.g.], the so-called holistic finite differences. Roberts [11] recently extended the approach to ensure macroscopic discrete models preserve important self-adjoint properties of the fine scale dynamics for deterministic systems.
Here we address the macroscopic discrete modelling of dissipative spdes and develop a novel rigorous approach. We consider reaction-diffusion in a one dimensional spatial domain driven by a noise which is white in time and with spatial structure. Let the non-dimensional spatial interval I = [0, L] with length L > 0, and let L 2 (I) be the Lebesgue space of square integrable real valued functions on I. Consider the following non-dimensional stochastic reaction-diffusion equation for a stochastic field u(x, t), of period L in space x, ∂ t u = ∂ xx u + α(u − u 3 ) + σ ∂ t W on I, (1) u(0, t) = u(L, t) and u x (0, t) = u x (L, t), where W (x, t) is an L 2 (I) valued Q-Wiener process defined on a complete probability space (Ω, F , P) which is detailed in the next section. The spatial domain I is divided into M elements and consequently a set of M fields are defined, one on each of these elements. In order to preserve the self-adjoint property of the linear operator defined on the elements we impose special interelement coupling conditions with a strength parametrised by γ. The system dynamics are expanded in the interelement coupling γ so that based upon the case of weak coupling, that is, small γ > 0 , an asymptotic approximation and an averaging method derive a family of coupled stochastic differential equations (sdes) which describe the evolution of grid values; that is, the amplitude of the system on each element. These sdes are a discrete stochastic model of the continuous space stochastic system (1)- (2). Our discrete sde model highlights the macroscopic influence of 'subgrid' interactions between noise and spatial diffusion in the spde.
The simplest conventional finite difference approximation of the spde (1) on a regular grid in x, say X j = jh for some constant grid spacing h, is where U j (t) is the grid value of the field u(x, t) at the grid points X j , and similarly W j (t) = W (X j , t). However, our analysis herein recommends that a more accurate closure incorporates stochastic influences from the neighbour grids as in the Ito system of sdes The second terms in the last line of the above sde system reflect interaction between noise and spatial diffusion. Terms inW j andα are due to the microscopic, subgrid scale, stochastic interactions discussed in Sections 5.
In order to generate a macroscopic discrete model we divide the domain into finite overlapping elements and choose special coupling boundary conditions, Section 2. Such interelement coupling rules were first introduced by Roberts [11] to construct spatially discrete models of deterministic dynamics. One important property of this interelement coupling is the preservation of self-adjoint symmetry in the underlying spatial dynamics. Moreover, the strength of the coupling is parametrised by γ, 0 ≤ γ ≤ 1 : when the coupling parameter γ is small, the coupling is weak and the system separates into 'uninterestingly' decaying fast parts and the relevant slow parts, Section 5. Then an averaging method [19] derives a reduced model which describes the evolution of the overall amplitude of (1)-(2) on the whole domain. Further, an analysis on eigenfunctions obtains a reduced model describing the evolution of the local amplitude on each element. This model is the macroscopic discrete approximation to (1)-(2) expressed in (4).
In this approach one difficulty is to construct from the original spatiotemporal noise W (x, t) a Wiener process W γ j (x, t) on each element. A natural method is expanding W (x, t) by the eigenfunctions of the linear operator L γ on each element, Section 2, which has analogues in the method of finite elements [16]. This construction shows that for small h and small coupling γ, the stochastic force on the first mode of L γ approximates the grid values W (X j , t). Moreover, analysis of the eigensystem of L γ for small γ shows the slow parts of the system dominate: the fast parts converge to quasiequlibrium with rate 1/h 2 . Then by this and the construction of W γ (x, t), the macroscopic discrete reduced model is proved to be consistent to the stochastic reaction-diffusion equations (1)-(2) as the element size h → 0 .
Another difficulty is that the linear operator L γ , defined in (10), varies with the coupling parameter γ. So the continuity of the linear oprator L γ in γ is needed. Section 4 argues that the graph convergence of L γ as γ → 0 ensures the continuity of eigenfunctions and eigenvalues in γ, then an asymp-totic expansion of the first eigenfunction in γ shows that the grid value is the amplitude of the system on the element. Then by averaging and the expansion of the first eigenfunction of L γ , the macroscopic reduced model is in the first eigenspace of L γ , which is varying with respect to γ. So last we project the reduced model to the first eigenspace of L 0 , the basic mode, to derive the macroscopic model. In this approach one interesting phenomenon is that the effect of noise in the subgrid scale fast modes is transmitted into the macroscopic slow modes by the projection. Numerical simulations confirm such transmittal [10].

Overlapping finite elements and coupling boundary conditions
This section divides the spatial domain I into M overlapping elements with grid spacing h. Let the jth element with grid points X j = jh on each element I j , j = 1, 2, . . . , M . Here for the periodic boundary condition we use the notation X j±M = X j . Let u j (x) denote the field on the element I j , j = 1, 2, . . . , M . Denote by f (u) = −u 3 . Then, modified from the spde (1), consider the following system of spdes defined on elements I j , j = 1, 2, . . . , M , with the following interelement coupling conditions on the fields parametrised by γ, and with γ ′ + γ = 1 , and coupling of the first spatial derivative, denoted by subscript x, with, to account for L-periodicity of solutions, The coupling parameter γ controls the flow of information between the two adjacent elements: when the coupling γ = 0 , adjacent elements are decoupled; when γ = 1 , the system is full coupled and (5)-(7) is equivalent to the dynamics of the physical stochastic reaction-diffusion equation (1)-(2), see Section 3. The noise fields W γ j (x, t) defined on each element I j , j = 1, . . . , M , are infinite dimensional Wiener process which are detailed later from W (x, t).
Related but different interelement coupling boundary conditions empowered an earlier exploration of the non-self-adjoint interaction between noise, nonlinear advection and spatial diffusion in discretely modelling the stochastic Burgers' equation [10].
For our purposes, first we introduce a mathematical framework for system (5)- (7). Let H j = L 2 (I j ) be the set of all square integrable function on I j and Here H α j denotes the usual Sobolev space W 2,α (I j ) [14]. Define the inner product ·, · on H as the sum of the element integrals for any u, v ∈ H with u = (u j ) and v = (v j ) . And denote by · 0 the product L 2 -norm on space H. For any α ∈ Z + denote the semi-norm on H α as For the system (5)- (7) we introduce the family of functional spaces and the subspaces Then define the second order differential operator L γ : By a basic calculation [11], −L γ is a self-adjoint second order operator. A direct calculation yields that for any which establishes the positivity of operator −L γ . Then there are coupling dependent eigenfunctions {(e γ j,k (x))} ∞ k=0 which form a standard orthonormal system in space H γ and a sequence of real numbers 0 Moreover, L γ is an infinitesimal generator of a C 0 semigroup; denote this semigroup by {S γ (t)} t≥0 . By the positivity of −L γ we define (−L γ ) α for any exponent α > 0 by By the same calculation as (11), for α ∈ Z + ∪ {0} , Given a complete probability space (Ω, F , {F t } t≥0 , P), define the L 2 (I) valued Q-Wiener process where {β k (t)} k are mutually independent standard Brownian motions and {e k (x)} k is a standard basis of L 2 (I) with e 0 (x) = 1/L and for k ≥ 1 Moreover, assume that the Wiener process is sufficiently well-behaved that Now we define the H γ -valued Wiener process (W γ j (x, t)) on all the overlapping elements in the following series form where {β j,l } ∞ k=0 are mutually independent standard Brownian motions on (Ω, F , P) and there are q j,l ∈ R such that Moreover, by the assumption (14) where the bound B is independent of the coupling parameter γ ∈ (0, 1].
is similar to the definition of that in the finite element method [16,24,25]. Now for fixed h > 0 and for any T > 0 , and {W γ (x, t)} 0<γ≤1 is compact in space C(0, T ; H α γ ) for α < 2 for almost all ω ∈ Ω . Then for almost all ω ∈ Ω, the following limit is well defined in space C(0, T ; H) for some γ n → 1 as n → ∞ . Moreover, by the coupling conditions we have almost surely Remark 2. By the analysis on eigenfunctions (e γ j,k (x)) k in Section 4, the above limit of W γ (x, t) in space C(0, T ; H) is unique in the sense of distribution for any sequence γ n → 1. Further, the distribution of W j (x, t) coincides with that of W j±1 (x, t) on the common overlapping domain.

Limit system for full coupling
Now we show that for full coupling, that is, as γ → 1 , equations (5)- (7) generates a model for the dynamics of the original physical stochastic reactiondiffusion equation (1)-(2). This is followed by a discussion similar to the case of Dirichlet boundary conditions [21], here we just state the result and omit the detailed proof.
with u(0) = u 0 and To prove the above result needs some energy estimates on the solutions u γ (x, t). By the definition of L γ , the spdes (5)-(7) takes the following abstract from where u γ (t) = (u γ j (t)) and F (u γ ) = (f (u γ j )). Writing the temporal dependence explicitly, in a mild sense we have Then by a standard semigroup approach [6], for any u γ (0) ∈ H γ and any we have the following lemma.
Lemma 4. Assume boundedness (16). For any T > 0 and q > 0 there is a positive constant C q (T ) such that Proof. Since −L γ is positive and self-adjoint, Then the result follows from the stochastic factorization formula [6].
By the standard energy estimate to stochastic reaction-diffusion equations with more general nonlinearity [19] we have We show that {D(u γ )} γ , the distribution of u γ in space C(0, T ; H), is tight. For this we need the following lemma by Simon [13]. Lemma 6. Assume E, E 0 and E 1 are Banach spaces such that E 1 ⋐ E 0 , the interpolation space (E 0 , E 1 ) θ,1 ⊂ E with θ ∈ (0, 1) and E ⊂ E 0 with ⊂ and ⋐ denoting continuous and compact embedding respectively. Suppose p 0 , p 1 ∈ [1, ∞] and T > 0 , such that X is a bounded set in L p 1 (0, T ; E 1 ) and ∂X := {∂v : v ∈ X} is a bounded set in L p 0 (0, T ; E 0 ).
Here ∂ denotes the distributional derivative. If 1 − θ > 1/p θ with then X is relatively compact in C(0, T ; E).
By the above lemma, and noticing the relation (13), we have the following theorem.
Similarly if u γ (0) ∈ H 2 γ with u γ (0) 2,γ ≤ C 0 which is independent of γ, for any T > 0 there is a positive constant C T > 0 such that Then by the embedding of H 2 (I) ⊂ C 1 (I) [14], By the above estimates we can treat the boundary value in passing to the limit γ → 1 of full coupling. By Theorem 7, for any κ > 0 there is a compact set K κ ⊂ C(0, T ; H) such that Then there is a function u ∈ C(0, T ; H) and a subsequence γ n → 1 as n → ∞ , such that in probability u γn → u as n → ∞ .
Now we determine the equation solved by the limit u. Define a test function ϕ ∈ C ∞ 0 (0, L) and define ϕ j = ϕ| I j .
Then by the boundary conditions we have in the variational form for the system (5)- (7) u γn (t), ϕ = u γn (0), ϕ + t 0 L γn u γn (s) + αγ 2 n u γn (s), ϕ ds Then by estimate (21), letting n → ∞ , that is γ ′ n → 0 , the last two terms disappear. Notice that f (u γn j ) → f (u j ) weakly in space L 2 (0, T ; L 2 ) and by the assumption on W γn we have, by passing to the limit n → ∞ , with W (t) = (W j (t)) which is well defined by Remark 2. Then a density argument yields that u = (u j ) solves the following stochastic equations with coupling boundary conditions Remark 8. By the boundary condition (19) and Remark 2, the distributions of u j and u j+1 in space C(0, T ; L 2 (X j , X j + h)) coincides.
and an L 2 (I)-valued Wiener process W (x, t) as Then u(x, t) solves the stochastic reaction-diffusion equation (1)- (2) with the noise term W (t) replaced by W (t) without changing the distribution. So (5)-(7) recovers the original system (1)-(2), in distribution, in the limit of full coupling, as γ → 1 .

Amplitudes on the elements
We derive a discrete macroscopic approximation to the system of spdes (5)-(7) based upon small coupling parameter γ > 0 . By the analysis on operator L 0 in section 2, for γ = 0 the dominant mode is (e j0 ), so by hyperbolicity we expect that for small γ > 0, the dominant mode is (e γ j0 ). This is followed by the analysis on the continuity of {(e γ jk )} k and λ k (γ) on coupling parameter γ. Further the asymptotic expansion for (e γ j0 ) in γ shows that the grid value approximates the amplitude on each element.
For this we study the continuity properties of L γ as coupling γ → 0 . We use variational convergence for operators [1]. For any subsequence γ n with γ n → 0 as n → ∞ , we introduce the G-convergence for L γn .
Definition 9 (G-convergence). Operator L γn is said to be graph-convergent (G-convergent) to L 0 as n → ∞ if for every (u, v) with v = L 0 u, there exists a sequence (u n , v n ) with v n = L γn u n such that u n → u strongly in V and v n → v strongly in V * , the dual space of V. Now for any u = (u j (x)) ∈ V 0 , denote by v = L 0 u . First we choose bounded set {v n } ⊂ H γn such that v n → v in the dual space V * . Then solve the following equation L γn u n = v n .
By the relation (13), {u n } is bounded in H 2 which yields that {u n } is compact in V. Then there is a subsequence, which we still denote by {u n }, that converges toũ in V as n → ∞ . Multiplying testing function ϕ ∈ C ∞ 0 (0, L) on both sides of (24) and passing to the limit n → ∞ , we have which yields that u =ũ by the uniqueness of the solution to (25). Then we have L γn is G-convergent to L 0 . Now we draw the following result on the continuity of eigenvalues and eigenfunctions in coupling γ [21].
Let m k be the multiplicity of λ k (γ), the sequence of subspaces L γ k of dimension m k generated by ((e γ,1 jk ), . . . , (e γ,m k jk )) converges in H to the eigenspace of L 0 corresponding to λ k = −k 2 π 2 /h 2 .
Assume we have the following asymptotic expansion for each element, j = 1, . . . , M , where F γ j,3 (x) = O γ 3 . By λ 0 (0) = 0 and the coupling boundary condition (6)-(7) , F γ j,k , k = 1, 2 , are kth order polynomial in x . Then also by the boundary condition (6)-(7) we have and with The above asymptotic expansion shows that for small coupling γ > 0 , the first mode is dominating and the grid value u γ j (X j , t) approximates the amplitude of the field u γ (x, t) on the element I j .

Macroscopic models for small coupling
By the asymptotic expansion in the previous section 4 we derive a discrete macroscopic approximation model to (5)-(7) for small coupling γ > 0 . For this we first apply an averaging method to reduce (5)-(7) onto the slow mode (e γ j0 ). We split (u γ j ) into slow part and fast part. Define map P γ 0 on H γ to H P γ 0 (u γ j ) = u γ j , e γ j,0 e γ j,0 (x)/ e γ j,0 2 0 where I is the identity operator on H γ . And for no coupling, γ = 0 , write P 0 = P 0 0 and P 1 = P 0 1 . Then denote by (u γ j (x, t)) the solution to system (5)-(7) and make the following expansion Now define the slow part and fast part, respectively, Then we have that these satisfy the coupled spdes where B γ 0 (t) = q h j,0 β j,0 (t) and B γ 1 (t) = ∞ k=1 q h j,k β j,k (t)e γ j,k (x) .
Letη γ (t ′ ) = (η γ j0 (t ′ )) ∈ P γ 1 H γ be the unique stationary solution of the following linear equation Then by an energy estimate and almost the same discussion to that by Wang and Roberts [19], for any T ≥ 0 , there is a positive constant C T such that We have no explicit expressions of (e γ jk (x)) , k ≥ 1 , so for our purpose we derive another approximation for V γ (t ′ ).
Lemma 12. Assume bound (16). For any T > 0 , there is positive constant C T such that whereη(t ′ ) = (η j0 (t ′ )) is the unique stationary solution of the following linear equation and with distributions independent of coupling parameter γ.
Proof. Expandη γ (t) andη(t) by the eigenfunctions of L γ and L 0 respectively asη , (e γ j,k (x)) (e γ j,k (x))/ (e γ j,k ) 2 0 , η k (x, t ′ ) = η(t ′ ), (e j,k (x)) (e j,k (x))/M . and Using the Itô formula and the stationary property ofη γ k andη k , there is a positive constant C such that for k ≥ 1 and hence Then by the analysis of Section 4 on λ k (γ), k ≥ 1 , assumption (16) and the estimates (35) we have Thus by (34), the proof is complete. Now by (32) and the above result, for V γ on the original time scale, Moreover, For U γ we follow an averaging approach [19] which yields the following averaged equation By the definition of f and thatη is Gaussian with zero mean, we havē Moreover, by a deviation argument [19,20], stochastic effects in these subgrid scale fast modes are fed into the slow modes by the nonlinear interaction. So we have the following averaged equation plus deviation with, for fixedŪ γ , (0)) −F (Ū γ ) ds , andβ(t) = (β j (t)) is an M dimensional standard Brownian motion. For any T > 0 and any κ > 0 , there is a positive constant C κ,T such that Then for the original system (5), by (32), we have the following reduced equation whereβ(t) = (β j (t)) = γ −1β (γ 2 t) is the scaled M dimensional standard Browian motion. Then for any κ > 0 , there is a constant C κ,T > 0 such that Now having the reduced system (40), we use the approximation to the amplitude on each element to derive a further approximate model. Write U γ 0 (x, t) = (u γ j0 (x, t)) and define (U j (t)) = a γ 0 (t) e γ j,0 (X j ) . Then by (27) for (u γ j,0 (x, t)) we have the following asymptotic expansion Putting (41) into (40) yields j,0 , e j,0 η 2 j,0 (0) − Eη 2 j,0 , e j,0 ds .
Here we use the approximation of e j,0 (x) to e γ j,0 (x) for small γ . Notice that system (42) is not a complete discrete model because the noise terms are still described on the mode (e γ j,0 (x)). In order to give a discrete approximating model for small coupling γ, we explore the evolution of the amplitude of the basic mode (e j,0 (x)). For this we project (U j ) onto the basic space E 0 spanned by (e j,0 (x)). However, the fast modes V γ have a nonzero projection in basic space E 0 ; because of the complicated expression for (e γ j,k (x)), we choose to project γη, which approximates V γ up to error of O(γ 2 ). For this we first project γη onto (e γ j,0 (x)), then project to E 0 . Notice that for small coupling γ, 1 γη (t ′ ) behaves as a noise process. By a martingale approach [5,19,22] we have the following lemma.

Conclusion
Stochastic averaging is an effective method to extract macroscopic dynamics from spdes with separated time scale [19,20]. Here by applying the stochastic averaging and dividing the spatial domain into overlapping finite sized elements with special interelement coupling boundary conditions (6)-(7), we derive a macroscopic discrete model (44) for stochastic reaction-diffusion partial differential equations (1) with periodic boundary conditions. The most important property of such interelement coupling boundary conditions is preserving the self-adjoint symmetry which is often so important in application [12]. Furthermore, by the choice of stochastic forcing on each element, this coupling boundary conditions also assures the consistency for vanishing element size, section 6. Moreover, the final discrete model (44), which is different from the usual finite difference approximation model (3), shows the importance of the subgrid scale interaction between noise and spatial diffusion and provides a new rigorous approach to constructing semi-discrete approximations to stochastic reaction-diffusion spdes.