Speed of convergence of time Euler schemes for a stochastic 2D Boussinesq model

We prove that an implicit time Euler scheme for the 2D-Boussinesq model on the torus $D$ converges. Various moment of the $W^{1,2}$-norms of the velocity and temperature, as well as their discretizations, are computed. We obtain the optimal speed of convergence in probability, and a logarithmic speed of convergence in $L^2(\Omega)$. These results are deduced from a time regularity of the solution both in $L^2(D)$ and $W^{1,2}(D)$, and from an $L^2(\Omega)$ convergence restricted to a subset where the $W^{1,2}$-noms of the solutions are bounded.


Introduction
The Boussinesq equations have been used as a model in many geophysical applications.They have been widely studied in a both the deterministic and stochastic settings.We take random forcings into account and formulate the Bénard convection problem as a system of stochastic partial differential equations (SPDEs).The need to take stochastic effects into account for modeling complex systems has now become widely recognized.Stochastic partial differential equations (SPDEs) arise naturally as mathematical models for nonlinear macroscopic dynamics under random influences.The Navier-Stokes equations are coupled with a transport equation for the temperature and with diffusion.Here, the system will be subject to a multiplicative random perturbation which will be defined later.Here, u describes the fluid velocity field, while θ describes the temperature of the buoyancy driven fluid, and π is the fluid's pressure.
In dimension 2 without any stochastic perturbation, this system has been extensively studied with a complete picture about its well posedness and longtime behavior.In the deterministic setting, more investigations have been extended to the cases where ν = 0 and/or κ = 0 with some partial results.
Numerical schemes and algorithms have been introduced to best approximate the solution to non-linear PDEs.The time approximation is either an implicit Euler or a time splitting scheme coupled with Galerkin approximation or finite elements to approximate the space variable.The literature on numerical analysis for SPDEs is now very extensive.In many papers the models are either linear, have global Lipschitz properties, or more generally some monotonicity property.In this case the convergence is proven to be in mean square.When nonlinearities are involved that are not of Lipschitz or monotone type, then a rate of convergence in mean square is more difficult to obtain.Indeed, because of the stochastic perturbation, one may not use the Gronwall lemma after taking the expectation of the error bound since it involves a nonlinear term which is often quadratic; such a nonlinearity requires some localization.
In a random setting, the discretization of the Navier-Stokes equations on the torus has been intensively investigated.Various space-time numerical schemes have been studied for the stochastic Navier-Stokes equations with a multiplicative or an additive noise, that is where in the right hand side of (1.1) (with no θ) we have either G(u) dW or dW .We refer to [7,11,5,8,6], where convergence in probability is stated with various rates of convergence in a multiplicative setting for a time Euler scheme, and [1] for a time splitting scheme.As stated previously, the main tool to get the convergence in probability is the localization of the nonlinear term over a space of large probability.We studied the strong (that is L 2 (Ω)) rate of convergence of the time implicit Euler scheme (resp.space-time implicit Euler scheme coupled with finite element space discretization) in our previous papers [2] (resp.[3]) for an H 1 -valued initial condition.The method is based on the fact that the solution (and the scheme) have finite moments (bounded uniformly on the mesh).For a general multiplicative noise, the rate is logarithmic.When the diffusion coefficient is bounded (which is a slight extension of an additive noise), the supremum of the H 1 -norm of the solution has exponential moments; we used this property in [2] and [3] to get an explicit polynomial strong rate of convergence.However, this rate depends on the viscosity and the strength of the noise, and is strictly less than 1/2 for the time parameter (resp.than 1 for the spatial one).For a given viscosity, the time rates on convergence increase to 1/2 when the strength of the noise converges to 0. For an additive noise, if the strength of the noise is not too large, the strong (L 2 (Ω)) rate of convergence in time is the optimal one, that is almost 1/2 (see [4]).Once more this is based on exponential moments of the supremum of the H 1 -norm of the solution (and of its scheme for the space discretization); this enabled us to have strong polynomial time rates.
In the current paper, we study the time approximation of the Boussinesq equations (1.1)-(1.2) in a multiplicative setting.To the best of our knowledge, it is the first result where a time-numerical scheme is implemented for a more general hydrodynamical model with a multiplicative noise.We use a fully implicit time Euler scheme and once more have to assume that the initial conditions u 0 and θ 0 belong to H 1 (D) in order to prove a rate of convergence in L 2 (D) uniformly in time.We prove the existence of finite moments of the H 1 -norms of the velocity and the temperature uniformly in time.Since we are on the torus, this is quite easy for the velocity.However, for the temperature, due to the presence of the velocity in the bilinear term, the argument is more involved and has to be done in two steps.It requires higher moments on the H 1 -norm of the initial condition.The time regularity of the solutions u, θ is the same as that of u in the Navier-Stokes equations.
We then study rates of convergence in probability and in L 2 (Ω).The rate of convergence in probability is optimal (almost 1/2); we have to impose higher moments on the initial conditions than what is needed for the velocity described by a stochastic Navier-Stokes equations.Once more we first obtain an L 2 (Ω)-convergence on a set where we bound the L 2 norm of the gradients of both the velocity and the temperature.We deduce an optimal rate of convergence in probability, that is strictly less than 1/2.When H 1 -norm of the initial condition has all moments (for example it is a Gaussian H 1 -valued random variable), the rate of convergence in L 2 (Ω) is any negative exponent of the logarithm of the number of time steps.These results extend those established for the Navier-Stokes equations subject to a multiplicative stochastic perturbation.
The paper is organized as follows.In section 2 we describe the model, the assumptions on the noise and the diffusion coefficients, and describe the fully implicit time Euler scheme.In section 3 we state the global well-posedeness of the solution to (1.1)-(1.2),moment estimates of the gradient of u and θ uniformly in time, the existence of the scheme.We then formulate the main results of the paper about the rates of convergence in probability and in L 2 (Ω) of the scheme to the solution.In section 4 we prove moment estimates in H 1 of u and θ uniformly on the time interval [0, T ] if we start with more regular (H 1 ) initial conditions.This is essential to be able to deduce a rate of convergence from the localized result.Section 5 states time regularity results of the solution (u, θ) both in L 2 (D) and H 1 (D); this a crucial ingredient of the final results.In section 6 we prove that the time Euler scheme is well defined and prove its moment estimates in L 2 and H 1 .Section 7 deals with the localized convergence of the scheme in L 2 (Ω).This preliminary step is necessary due to the bilinear term, which requires some control of the H 1 norm of u and θ.In section 8 we prove the rate of convergence in probability and in L 2 (Ω).Finally, section 9 summarizes the interest of the model, and describes some further necessary/possible extensions of this work.
As usual, except if specified otherwise, C denotes a positive constant that may change throughout the paper, and C(a) denotes a positive constant depending on some parameter a.

Preliminaries and assumptions
In this section, we describe the functional framework, the driving noise, the evolution equations, and the fully implicit time Euler scheme.

The functional framework. Let
2 ) be the usual Lebesgue and Sobolev spaces of vector-valued functions endowed with the norms Let Ã = −∆ acting on L 2 (D).For any non negative real number k let . Moreover, let V −1 be the dual space of V 1 with respect to the pivot space V 0 , and Therefore, the map B satisfies the following antisymmetry relations: Furthemore, since D = [0, L] 2 with periodic boundary conditions, we have (see e.g.[18]) follow from the Sobolev embedding theorem.More precisely the following Gagliardo Nirenberg inequality is true for some constant Cp Finally, let us recall the following estimate of the bilinear terms (u.∇)v and (u.∇)θ.
Projecting the velocity on divergence free fields, we consider the following SPDEs for processes modeling the velocity u(t) and the temperature θ(t).The initial conditions u 0 and θ 0 are F 0 -measurable, taking values in V 0 and H 0 respectively, and ν, κ are strictly positive constants, and v 2 = (0, 1) ∈ R 2 .We make the following classical linear growth and Lipschitz assumptions on the diffusion coefficients G and G.For technical reasons, we will have to require u 0 ∈ V 1 , θ ∈ H 1 and prove estimates similar to (3.1)-(3.2) raising the space regularity of the processes by one step in the scale of Sobolev spaces.Therefore, we have to strengthen the regularity of the diffusion coefficients. (2.10) and 2.3.The fully implicit time Euler scheme.Fix N ∈ {1, 2, ...}, let h := T N denote the time mesh, and for j = 0, 1, ..., N set t j := j T N .The fully implicit time Euler scheme {u k ; k = 0, 1, ..., N } and {θ k ; k = 0, 1, ..., N } is defined by u 0 = u 0 , θ 0 = θ 0 , and for ϕ ∈ V 1 , ψ ∈ H 1 and l = 1, ..., N , (2.16)

3.1.
Global well-posedness and moment estimates of (u, θ).The first results states the existence and uniqueness of a weak pathwise solution (that is strong probabilistic solution in the weak deterministic sense) of (2.7)-(2.8).It is proven in [9] (see also [14]).

3.3.
Rates of convergence in probability and in L 2 (Ω).The following theorem states that the implicit time Euler scheme converges to the pair (u, θ) in probability with the "optimal" rate "almost 1/2".It is the main result of the paper.For j = 0, ..., N set e j := u(t j ) − u j and ẽj := θ(t j ) − θ j ; then e 0 = ẽ0 = 0.
Theorem 3.5.Suppose that the conditions (C-u) and (C-θ) hold.Let u 0 ∈ L 32+ǫ (Ω; V 1 ) and θ 0 ∈ L 32+ǫ (Ω; H 1 ) for some ǫ > 0, u, θ be the solution to (2.7)-(2.8),{u j , θ j } j=0,...,N be the solution to (2.15)- (2.16).Then for every η ∈ (0, 1) we have lim We finally state that the strong (i.e. in L 2 (Ω)) rate of convergence of the implicit time Euler scheme is some negative exponent of ln N .Note that if the initial conditions u 0 and θ 0 are deterministic, or if their V 1 and H 1 -norms have moments of all orders (for example if u 0 and θ 0 are Gaussian random variables), the strong rate of convergence is any negative exponent of ln N .More precisely we have the following result.
4. More egularity of the solution ).In this section, we prove that if u 0 ∈ V 1 and θ 0 ∈ H 0 , the H 1 -norm of the velocity has bounded moments uniformly in time.
Proof of Proposition 3.2 Apply the operator A 1 2 to (2.7) and use (formally) Itô's formula for the square of the .V 0 -norm of A 1 2 u(t).Then, using (2.2), we obtain ), integration by parts, the Cauchy-Schwarz and Young inequalities, we deduce for M > 0 and t ∈ [0, T ] Indeed the stochastic integral in the right hand side of (4.1) is a square integrable -hence centered -martingale.Neglecting the time integral in the left hand side, using (3.1) and the Gronwall lemma, we deduce sup Furthermore, the Davis inequality and Young's inequality imply The upper estimates (4.1), (3.1), (3.2) and (4.2) imply for some constant C depending on E( As M → ∞, we deduce this proves (3.3) for p = 1.Given p ∈ [2, ∞) and using Itô's formula for the map x → x p in (4.1), we obtain Integration by parts, the Cauchy-Schwarz, Hölder and Young inequalities imply Since a p−1 ≤ 1 + a p for any a ≥ 0, the growth condition (2.11) implies V 0 , the upper estimate of the corresponding integral is similar to that of (4.5).Since the stochastic integral is square integrable it is centered.Therefore,(4.3)and the above upper estimates (4.4)-(4.5)imply sup sup Finally, the Davis inequality, then the Hölder and Young inequalities imply The upper estimates (4.3), (3.1) and (4.8) imply sup As M → ∞ in this inequality and in (4.7), the monotone convergence theorem concludes the proof of (3.3).✷ 4.2.Moment estimates of θ in L ∞ (0, T ; H 1 ).We next give upper estimates for moments of sup t∈[0,T ] Ã 1 2 θ(t) H 0 , i.e., prove Proposition 3.3.However, since [u(s).∇]θ(s),Ãθ(s) = 0, unlike what we have in the proof of the previous result, we keep the bilinear term.This creates technical problems and we proceed in two steps.First, using the mild formulation of the weak solution θ of (2.8), we prove that the gradient of the temperature has finite moments.Then going back to the weak form, we prove the desired result.
Let {S(t)} t≥0 be the semi-group generated by −νA, {SS(t)} t≥0 be the semi-group generated by −κ Ã, that is S(t) = exp(−νtA) and SS(t) = exp(−κt Ã) for every t ≥ 0. Note that for every α > 0 Similar upper estimates are valid when we replace A by Ã, S(t) by SS(t) and V 0 by , we can write the solutions of (2.7)-(2.8) in the following mild form where the first equality holds a.s. in V 0 and the second one in H 0 .Indeed, since Furthermore, Therefore, the stochastic integral t 0 S(t − s)G(u(s))dW (s) ∈ V 0 a.s., and the identity (4.11) is true a.s. in V 0 .
A similar argument shows that (4.12) holds a.s. in H 0 .We only show that the convolution involving the bilinear term belongs to H 0 .Using the Minkowski inequality and the upper estimate (2.6) with positive constants δ, α, ρ such that α, ρ ∈ (0, 1  2 ), δ + ρ > 1 2 and δ + α + ρ = 1, we obtain where the last upper estimate is deduced from Hölder's inequality and δ 1−ρ < 1.The following result shows that for fixed t, the L 2 -norm of the gradient of θ(t) has finite moments.
The following lemma is an extension of Lemma 3.3, p. 316 in [19].For the sake of completeness its prove is given at the end of this section.Lemma 4.2.Let ǫ ∈ (0, 1), a, b, c be positive constants and ϕ be a bounded non negative function such that Then sup t∈[0,T ] ϕ(t) ≤ C for some constant C depending on a, b, c, T and ǫ.
Proof of Proposition 3. 3 We next prove that the gradient of the temperature has bounded moments uniformly in time.
We only prove (3.4) for p ∈ [2, +∞); the other argument is similar and easier.
Applying the operator Ã 1 2 to equation (2.8), and writing Itô's formula for the square of corresponding H 0 -norm.we obtain Then apply Itô's formula for the map x → x p .This yields, using integration by parts, The Gagliardo-Nirenberg inequality (2.4) and the inclusion Then using the Hölder and Young inequalities, we deduce The growth condition (2.14), Hölder's and Young inequalities imply and a similar computation yields Using the Cauchy-Schwarz inequality, Fubini's theorem, (3.3) and (4.13), we deduce The Davis inequality, the growth condition (2.14), the Cauchy-Schwarz, Young and Hölder inequalities imply Therefore, the upper estimates (3.2), (4.13) and (4.21) imply for some constant C independent of N .
As N → +∞, we deduce (3.4); this completes the proof of Proposition 3.4.✷ We conclude this section with an extension of the Gronwall Lemma.Proof of Lemma 4.2 For t ∈ [0, T ], iterating (4.16) and using the Fubini theorem, we obtain for positive constants A 1 (depending on a, b, c, T, ǫ), B 1 (depending on b, c, T, ǫ), and C 1 (depending on c and ǫ).One easily proves by induction on k that for every integer k ≥ 1 for some positive constants A k , B k and C k depending on a, b, c, T and ǫ.Indeed, a change of variables implies for some constant Ck depending on k and ǫ.Let k * be the largest integer such that kǫ < 1, that is for some positive constants A and B depending on the parameters a, b, c, T and ǫ.The classical Gronwall lemma concludes the proof of the Lemma.✷

Moment estimates of time increments of the solution
In this section we prove moment estimates for various norms of time increments of the solution to (2.7)-(2.8).This will be crucial to deduce the speed of convergence of numerical schemes.We first prove the time regularity of the velocity and temperature in L 2 .Proposition 5.1.Let u 0 , θ 0 be F 0 -measurable, and suppose that G and G satisfy (C-u) and (C-θ) respectively. (i Proof.Recall that S(t) = e −νtA is the analytic semi group generated by the Stokes operator A multiplied by the viscosity ν and SS(t) = e −κt Ã is the semi group generated by Ã = −∆.
We next prove some time regularity for the gradient of the velocity and the temperature.
Step 1 For technical reasons we consider a Galerkin approximation.Let {e l } l denote an orthonormal basis of V 0 made of elements of V 2 which are orthogonal in V 1 (resp.{ẽ l } l denote an orthonormal basis of H 0 made of elements of H 2 which are orthogonal in H 1 ).For m = 1, 2, ... let V m = span (e 1 , ..., e m ) ⊂ V 2 and let P m : V 0 → V m denote the projection from V 0 to V m .Similarly, let Hm = span (ẽ 1 , ..., ẽm ) ⊂ H 2 and let Pm : H 0 → Hm denote the projection from H 0 to Hm .
In order to find a solution to (2.15)-(2.16)we project these equations on V m and Hm respectively, that is we define by induction {u k (m)} k=0,...,N ∈ V m and {θ k (m)} k=0,...,N ∈ Hm such that u 0 (m) = P m (u 0 ), θ 0 (m) = Pm (θ 0 ), and for k = 1, ..., N , ϕ ∈ V m and ψ ∈ Hm For almost every ω set R(0, ω) := u 0 (ω) V 0 and R(0, ω) := θ 0 (ω) H 0 .Fix k = 1, ..., N and suppose that for j = 0, ..., k −1 the F t j -measurable random variables u j (m)and θ j (m) have been defined, and that for almost every ω.We prove that u k (m) and θ k (m) exists and satisfy sup Then the Cauchy-Schwarz and Young inequalities imply If page 279, which can be deduced from Brouwer's theorem, we deduce the existence of an element s.Note that these elements u k (m, ω) and θ k (m, ω) need not be unique.Furthermore, the random variables u k (m) and θ k (ω) are F t k -measurable.
Therefore, for fixed k and almost every ω, the sequence The random variable φ k is F t k -measurable.Similarly, for fixed k and almost every ω, the sequence {θ k (m, ω)} m is bounded in H 1 ; it has a subsequence (still denoted {θ k (m, ω)} m ) which converges weakly in H 1 to φk (ω) which is Step 2 We next prove that the pair (φ k , φk ) is a solution to (2.15)- (2.16).By definition u 0 (m) converges strongly to u 0 in V 0 , and θ 0 (m) converges strongly to θ 0 in H 0 .We next prove by induction on k that the pair (φ k , φk ) solves (2.15)- (2.16).Fix a positive integer m 0 and consider the equation (6.1) for k = 1, ..., N , ϕ ∈ V m 0 , and m ≥ m 0 .As m → ∞ we have a.s.
A similar argument proves that φk is a solution to (2.16).This concludes the proof.✷ 6.2.Moments of the Euler scheme.We next prove upper bounds of moments of u k and θ k uniformly in k = 1, ..., N .Proposition 6.1.Let G and G satisfy the condition (C-u)(i) and (C-θ)(i) respectively.Let K ≥ 1 be an integer, and let u 0 ∈ L 2 K (Ω; V 0 ) and θ 0 ∈ L 2 K (Ω; H 0 ) respectively.Let {u k } k=0,...,N and {θ k } k=0,...,N be solution of (2.15) and (2.16) respectively.Then Proof.Write (2.15) with ϕ = u l , (2.16) with ψ = θ l , and use the identity (f, Using the Cauchy-Schwarz and Young inequalities, the antisymmetry (2.1) and the growth condition (2.9), this yields for l = 1, ..., N Fix L = 1, ..., N and add both equalities for l = 1, ..., L; this yields Let N be large enough to have h = T N ≤ 1 8 .Taking expected values, we deduce Neglecting both sums in the left hand side and using the discrete Gronwall lemma, we deduce sup where is independent of N .This implies sup this proves (6.4) for K = 1.For s ∈ [t j , t j+1 ), j = 0, ..., N − 1, set s = t j .The Davis inequality, and then the Cauchy-Schwarz and Young inequality imply for any ǫ > 0 Taking expected values, we deduce for every L = 1, ..., N and h for some constant C depending on K i , Ki , Tr(Q), Tr( Q) and T , and which does not depend on N .Let N be large enough to have 3 h < 1 2 .Neglecting the sums in the left hand side and using the discrete Gronwall lemma, we deduce for E u 0 4 which implies sup that is (6.4) for K = 2.The argument used to prove (6.9) implies Taking the maximum for L = 1, ..., N and using (6.17), we deduce (6.3) for K = 2.The details of the induction step, similar to the proof in the case K = 2, are left to the reader.

Conclusions
This paper provides the first result about the rate of convergence of a time discretization of the Navier-Stokes equations coupled with a transport equation for the temperature, driven by a random perturbation; this is the so-called Boussinesq/Bénard model.The perturbation may depend on both the velocity and temperature of the a fluid.The rates of convergence in probability and in L 2 (Ω) are similar to those obtained for the stochastic Navier-Stokes equations.The Boussinesq equations model a variety of phenomena in environmental, geophysical, and climate systems (see e.g.[12] and [13]).Even if the outline of proof is similar to that used for the Navier-Stokes equations, the interplay between the velocity and the temperature is more delicate to deal with in many places.This interplay, which appears in Bénard systems, is crucial to describe more general hydrodynamical models.The presence of the velocity in the bilinear term describing the dynamics of the temperature makes more difficult to prove bounds of moments for the H 1 -norm of the temperature uniformly in time and requires higher moments of the initial condition.Such bounds are crucial to deduce rates of convergence (in probability and in L 2 (Ω)) from the localized one.
This localized version of the convergence is the usual first step in a non linear (non Lipschitz and not monotonous) setting.Numerical simulations, which are the ultimate aim of this study since there is no other way to "produce" trajectories of the solution, would require a space discretization, such as finite elements.This is not dealt with in this paper and will be done in a forthcoming work.This new study is likely to provide results similar to those obtained for the 2D Navier-Stokes equations.Also note that another natural continuation of this work would be to consider a more general stochastic 2D magnetic Bénard model (as discussed in [9]), which describes the time evolution of the velocity, temperature and magnetic field of an incompressible fluid.
It would also be interesting to study the variance of the L 2 (D)-norm of the error term, in both additive and multiplicative settings, for the Navier-Stokes equations and more general Bénard systems.This would give some information about the accuracy of the approximation.Proving a.s.convergence of the scheme for Bénard models is also a challenging question.
the Leray projection, and let A = −Π∆ denote the Stokes operator, with domain Dom
Global well-posedness of the time Euler scheme.The following proposition states the existence and uniqueness of the sequences {u k } k=0,...,N and {θ k } k=0,...,N .