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[ sec : intro ] many interesting phenomena in the physical sciences and in applications are characterized by their high dimensionality and the presence of many different spatial and temporal scales . standard examples include atmosphere and ocean sciences  @xcite , molecular dynamics  @xcite and materials science  @xcite . the mathematical description of phenomena of this type quite often leads to infinite dimensional multiscale systems that are described by nonlinear evolution partial differential equations ( pdes ) with multiple scales . often physical systems are also subject to noise . this noise might be either due to thermal fluctuations  @xcite , noise in some control parameter  @xcite , coarse - graining of a high - dimensional deterministic system with random initial conditions  @xcite , or the stochastic parameterization of small scales  @xcite . high dimensional multiscale dynamical systems that are subject to noise can be modeled accurately using stochastic partial differential equations ( spdes ) with a multiscale structure . there are very few instances where spdes with multiple scales can be treated analytically . the goal of this paper is to develop numerical methods for solving accurately and efficiently multiscale spdes . several numerical methods for spdes have been developed and analyzed in recent years , e.g.  @xcite , based on a finite difference scheme in both space and time . it is well known that explicit time discretization via standard methods ( e.g. , as the euler - maruyama method ) leads to a time - step restriction due to the stiffness originating from the discretisation of the diffusion operator ( e.g. the cfl condition @xmath0 , where @xmath1 and @xmath2 are the time and space discretization , respectively ) . the situation is even worse for spdes with multiple scales ( e.g. of the form   and   below ) as in this case the courant - friedrichs - lewy ( cfl ) condition becomes @xmath3 , where @xmath4 is the parameter measuring scale separation . standard explicit methods become useless for spdes with multiple scales . such time - step restriction can in theory be removed by using implicit methods as was shown in @xcite . however the implicitness of the numerical scheme forces one to solve potentially large linear algebraic problems at each time step . furthermore , it was shown in  @xcite that implicit methods are not suited for studying the long time dynamics of fast - slow stochastic systems as they do not capture the correct invariant measure of the system . although this result has been obtained for finite dimensional stochastic systems , it is expected that it also applies to infinite dimensional fast - slow systems of stochastic differential equations ( sdes ) , rendering the applicability of implicit methods to spdes with multiple scales questionable . we also note that a new class of explicit methods , the s - rock methods , with much better stability properties than the euler - maruyama method was recently introduced in  @xcite . although these methods are much more efficient than traditional explicit methods , computing time issues will occur when trying to solve spdes with multiple scales as considered here , since the stiffness is extremely severe for small @xmath5 furthermore , capturing the correct invariant measure of the spde for @xmath6 is still an issue for such solvers .    in this paper we consider spdes of the form @xmath7 posed in a bounded domain of @xmath8 with appropriate boundary conditions . the differential operator @xmath9 is assumed to be a non - positive self - adjoint operator in a hilbert space @xmath10 , @xmath11 denotes space - time gaussian white noise , @xmath12 is the covariance operator of the noise and we take @xmath4 . we assume that the operator @xmath9 has a finite dimensional kernel , @xmath13 . this assumption leads to scale separation between the slow dynamics in @xmath13 and the fast dynamics in the orthogonal complement of the null space @xmath14 where @xmath15 . in this paper we will furthermore assume that noise acts directly only on the orthogonal complement @xmath16 . when noise acts also on @xmath13 , different distinguished limits than the ones considered in this paper should be considered .    in order to describe the longtime behavior of the spdes we perform an _ advective _ rescaling set @xmath17 . using the scaling properties of white noise we obtain the following singularly perturbed spde @xmath18 another scaling is of interest , namely the _ diffusive _ rescaling @xmath19 which leads to the spde @xmath20 for concreteness , we will focus on the class of spdes with quadratic nonlinearities that was considered in  @xcite , and assume that @xmath21 where @xmath22 is a quadratic function ( e.g. @xmath23 , a symmetric bilinear form ) , @xmath24 a linear function and the exponent @xmath25 is either @xmath26 or @xmath27 . and @xmath24 involve derivatives of the function @xmath28 . for example , for both the burgers and the kuramoto - shivashinsky equation we have @xmath29 . the linear function @xmath30 is included to induce a linear instability to the dynamics . in the case of the burgers equation we will simply take @xmath31 whereas in the case of the kuramoto - shivashinsky equation we can take @xmath32 . further discussion can be found in section  [ sec : numerics ] and in  @xcite . ] the choice of @xmath25 will depend on the particular scaling . singularly perturbed spdes with quadratic nonlinearities provide a natural testbed for testing the applicability of the heterogeneous multiscale method to infinite dimensional stochastic systems , since a rigorous homogenization theory exists for this class of spdes  @xcite . furthermore , spdes of this form arise naturally in stochastic models for climate  @xcite and in surface growth  @xcite . finally , it has already been shown through rigorous analysis and numerical experiments that these systems exhibit a very rich dynamical behavior , such as noise - induced transitions  @xcite and the possibility of stabilization of linearly unstable modes due to the interaction between the additive noise and the scale separation  @xcite . we believe , however , that the methodology developed in this paper has a wider range of applicability and is not restricted to spdes with quadratic nonlinearities . further comments about the class of spdes for which we believe that the proposed numerical method can be applied can be found in section  [ sec : conclusions ] . our numerical algorithm is based on a combination of a spectral method with micro - macro time integration schemes . we denote by @xmath33 the projection onto @xmath13 and by @xmath34 , @xmath35 the projection onto @xmath16 . we then rewrite   and   as fast - slow system of sdes    [ e : fast_slow_intro_adv ] @xmath36    and    [ e : fast_slow_intro_diff ] @xmath37    where the functions @xmath38 and @xmath39 are the projections of @xmath40 onto @xmath13 and @xmath16 . we remark that an @xmath41 nonlinear term can be added in  . the fast - slow systems   and   resemble fast slow systems for sdes ( however , the fast process @xmath42 is infinite dimensional and the well known averaging and homogenization theorems  @xcite do not apply . averaging and homogenization results for spdes have been obtained recently  @xcite . in particular , provided that the fast process @xmath42 in   has suitable ergodic properties , then the slow variable @xmath43 converges , in the limit as @xmath44 tends to @xmath45 to the solution of the averaged equation @xmath46 where the averaged coefficient is given by the average of @xmath47 with respect to the invariant measure of the ( infinite dimensional ) fast process @xmath42 . when this average vanishes ( i.e. the centering condition from homogenization theory is satisfied ) then the dynamics at the advective time scale becomes trivial and it is necessary to look at the dynamics at the diffusive time scale , equations  . it was shown in  @xcite that the slow variable @xmath43 of this system of equations , the solution of  , converges in the limit as @xmath44 tends to @xmath48 to the solution of the homogenized equation @xmath49 with explicit formulas for the homogenized coefficients  see section  [ sec : multisc ] for details . for finite dimensional fast systems , the coefficients in   and   can be calculated , in principle , in terms of appropriate long - time averages  see  @xcite for details . the numerical method proposed in  @xcite and analyzed in  @xcite , coined under the name of the heterogeneous multiscale method ( hmm ) , relies on the numerical approximation of the coefficients in   and   by solving the original fine scale problem on time intervals of an intermediate time scale and use that data to evolve the slow variables using either   or  . in this paper we show how this methodology , when combined with a spectral method , can also be applied to spdes with multiple scales , that is , to the systems and . the aim of the present paper is to present the algorithm and report numerical experiments . the analysis of the proposed numerical method and the extension to more general classes of spdes with multiscale structure will be presented in a forthcoming paper . the rest of the paper is organized as follows . in section  [ sec : algorithm ] we present our new algorithm . analytical and computational techniques for the analysis of spdes with multiple scales at the heart of the multiscale algorithm are presented in  [ sec : multisc ] . in section  [ sec : numerics ] we present numerical experiments . section  [ sec : conclusions ] is reserved for conclusions and discussion on future work . we propose a numerical algorithm to approximate numerically the solution of based on a micro - macro algorithm , capable of capturing the effective behavior of the spde . we explain the numerical algorithm for the case of diffusive time scale ( the hardest numerically ) and comment on the advective time scale later in this section . we consider spdes in a hilbert space @xmath10 with norm @xmath50 and inner product @xmath51 . @xmath9 denotes a differential operator , @xmath11 space - time white noise and @xmath12 the covariance operator of the noise . we assume that @xmath9 is a self - adjoint nonpositive operator on @xmath10 with compact resolvent . we denote its eigenvalues and ( normalized ) eigenfunctions by @xmath52 : @xmath53 the eigenfunctions of @xmath9 form an orthonormal basis in @xmath10 . we assume that @xmath9 and the covariance operator of the noise @xmath12 commute . thus , we can write , formally , @xmath54 where @xmath55 are independent one - dimensional white noise processes , i.e. , mean - zero gaussian processes with @xmath56 . here @xmath57 and @xmath58 are the usual kronecker delta functions . furthermore , we will assume that @xmath9 has a finite dimensional kernel @xmath59 , @xmath60 and write @xmath61 we introduce the projection operators    @xmath62    and write @xmath63 , @xmath64 . finally , we will assume that noise acts only on @xmath16 , i.e. @xmath65 .    * step 1 . decomposition in a fast - slow system . * + using the projection operators defined in and , equation   can be written as a fast - slow stochastic system    [ e : x - y ] @xmath66    where @xmath67 since @xmath68 we order the pairs of eigenfuctions and eigenvalues such that the kernel @xmath13 is spanned by the first @xmath69 eigenfunctions of @xmath9 . we can write @xmath70 moreover , we introduce @xmath71    as mentioned in the introduction we will often consider the case @xmath72 . then the above decomposition reads @xmath73 where we notice that for a linear function @xmath74 we simply have @xmath75 . then , in view of and we can rewrite the system   in the form    [ e : x - y-1 ] @xmath76    equations  , resp .  , are the infinite system of singularly perturbed sdes that we want to solve numerically . +    * step 2 . * + we consider a finite dimensional truncation of the above system and keep @xmath77 fast processesinstead of @xmath78 and similarly for the eigenvalues @xmath79 and the noise intensity @xmath80 . ] :    [ e : x - y-1_fin ] @xmath81    where @xmath82 and @xmath83 and @xmath84 and @xmath85 . for the decomposition , , we will use the notations    @xmath86    where @xmath87 and @xmath88 with components similar as in or . + * step 3 . numerical solution of the reduced system . * +    the reduced system is solved by a micro - macro algorithm following @xcite . it consists of a macrosolver ( we use the notation @xmath89 ) , chosen here to be the euler - maruyama scheme @xmath90 where @xmath91 ( the wiener increment ) is @xmath92 . notice that @xmath1 represents here a macrotime step , i.e. , @xmath1 can be chosen much larger than @xmath44 . the drift function @xmath93 and diffusion function @xmath94 appearing in , recovered from a time - ensemble average , are given by    @xmath95    where @xmath96 are the solutions of a suitable auxiliary system ( given in below ) involving the fast problem . here @xmath97 denotes the number of samples taken for the numerical calculation , @xmath98 the number of micro timesteps and @xmath99 a number of initial micro timesteps that are omitted in the averaging processes to reduce the effect of transients ( see below ) .    * auxiliary system . * as observed in @xcite , for diffusive timescales , computing effective coefficients via time - averaging ( relying on ergodicity ) , may require to solve over time @xmath100 . to overcome this problem , it was suggested again in @xcite to replace the fast process in by ( @xmath101 )    [ e : auxiliary ] @xmath102 the numerical approximations @xmath103 of and , respectively , are the functions appearing in the averaging procedure to recover the macroscopic drift and diffusion functions ( see ) - ) . notice that we fix the slow variables in the system at the current macro state @xmath104 . we use again the euler - maruyama method and compute @xmath103 as    [ e : microsolver ] @xmath105 where @xmath106 and @xmath107 is a @xmath108 random variable . the index @xmath109 refers to the macrotime , @xmath110 . we compute over @xmath111 microtime steps , over @xmath112 microtime steps to compute the time - ensemble average . notice that for the microsolver , the timestep @xmath113 resolves the fine scale @xmath114 . the initial values for the micro solver are taken to be ( for @xmath115 @xmath116 and @xmath117 for @xmath118 . the motivation for computing the above time averages is given in the next section . we notice that the auxiliary system is degenerate , since the noise in   is additive . this implies that the results presented in  ( * ? ? ? b ) are not applicable in this case and a more elaborate analysis is required for proving geometric ergodicity . this analysis , based on the ergodic theory for hypoelliptic diffusions  @xcite , will be presented elsewhere . in the present paper we will assume that the auxiliary process   is ergodic . [ [ advective - time - scale . ] ] advective time scale . + + + + + + + + + + + + + + + + + + + + +    a similar algorithm can be derived for the advective time scale . we consider the fast - slow system   that after projection and truncation reads    [ e : x - y-1_fin_ad ] @xmath119    similarly as . the macrosolver , chosen to be the euler explicit method , is given by @xmath120 where the effective force @xmath121 is given by the time average @xmath122 where @xmath123 is a numerical approximation of the truncated fast system with a slow variable fixed at time @xmath110 . as previously , @xmath97 denotes the number of samples and @xmath112 the number of micro timesteps and @xmath99 is the number of initial micro timestep ommited to reduce the transient effects . for the advective scaling , there is no need for an auxiliary problem for the micro solver @xcite . in this section we summarize recent results on the averaging and homogenization for spdes  @xcite that are the analytical foundation on which the numerical algorithm presented in section  [ sec : algorithm ] is built .      in this section we briefly discuss the analytical form of the effective system corresponding to . under the assumption that the vector field @xmath124 ( see ) is centered with respect to the invariant measure of the fast process , @xmath125 then the slow process converges to a homogenized equation of the form @xmath126 where @xmath127 represent an @xmath128dimensional wiener process and the sde   is interpreted in the it sense . the subscript @xmath77 are used to emphasise the fact that the homogenized coefficients depend on the number of fast processes that we take into account . an analytic expression for the coefficients that appear in is given by    [ e : coeff_homog ] @xmath129    here @xmath130 denotes the invariant measure of the process @xmath131 which is given by   and @xmath132 denotes the invariant measure of the the process @xmath133 . notice that @xmath134 is the solution of the rescaled process corresponding to , i.e. , @xmath135 . alternatively , the calculation of the coefficients @xmath136 and @xmath137 which appear in the homogenized equation can be obtained by the solution of the poisson equation @xmath138 where @xmath139 is the generator of the fast ( truncated ) ornstein - uhlenbeck process . this process is ergodic and its invariant measure is gaussian : @xmath140 where @xmath141 denotes the normalization constant . we notice that the system   is a finite dimensional fast - slow system of sdes for which standard homogenization theory applies  @xcite . for quadratic nonlinearities the poisson equation   can be solved analytically . the calculation of the coefficients in the homogenized ( amplitude ) equation reduces then to the calculation of gaussian integrals that can also be done analytically . this will be done in section  [ sec : diff_time_scale ] . averaging problems for fast - slow systems of spdes were studied recently in  @xcite and their results can be applied to  . one important observation is that in the system  , the fast process is , to leading order @xmath142 , an infinite dimensional ornstein - uhlenbeck process . the ergodic properties of such an infinite dimensional process can be analyzed in a quite straightforward way and the invariant measure , if it exists , is a gaussian measure in an appropriate hilbert space that can be written down explicitly  @xcite . assuming that the process @xmath143 is ergodic with gaussian invariant measure @xmath144 with mean @xmath48 and covariance operator @xmath145 , then the slow process @xmath43 converges to the solution of the averaged equation @xmath146 where the integration is over an appropriate hilbert space . when @xmath147 in   is given in terms of a symmetric bilinear map , i.e. , @xmath148 the calculation of the vector field that appears in the averaged equation reduces to the calculation of gaussian integrals and can be performed explicitly . in this case we have @xmath149 where @xmath150 and where we used the notation @xmath151 and @xmath152 denotes the dimension of the null space of @xmath9 . then , the fast - slow system   becomes    [ e : averag_quadratic ] @xmath153    and the averaged equation for   reads @xmath154 where @xmath155 in the case when the null space is one - dimensional , @xmath156 , the averaged equation becomes @xmath157 with @xmath158 and @xmath159 . this equation can be solved in closed form : @xmath160 we remark that solutions to  , depending on the choice of the initial conditions , do not necessarily exist for all times . we also remark that it is straightforward to consider the case where there is an additional higher order linear term ( in @xmath44 ) in the equation , i.e. @xmath161 . in this case the averaged equation   becomes @xmath162 where @xmath163 . we consider the system obtained after a diffusive time rescaling to  . in order to describe the homogenized equation , we further assume that @xmath164 in is of the form @xmath165 where @xmath166 is a symmetric bilinear map satisfying @xmath167 . we recall that the noise does not act directly on the slow variables , @xmath168 , where @xmath69 is the dimension of the null space of @xmath9 . under appropriate assumptions on the quadratic nonlinearity and on the covariance operator of the noise , together with the assumptions on @xmath9 and @xmath12 stated earlier in this section , it is possible to prove  @xcite that the projection of the solution to   onto the null space of @xmath9 , @xmath169 , converges weakly to the solution of the homogenized sde ( the amplitude equation ) @xmath170 where the noise is interpreted in the it sense and the drift @xmath171 given by @xmath172 where the linear map @xmath173 and the trilinear map @xmath174 are defined by    @xmath175    in the above we used the notation @xmath176 and @xmath177 , whereas @xmath178 stands for the symmetric tensor product we denote by @xmath179 its symmetric tensor product . similarly , we use the notation @xmath180 for the symmetric tensor product of two elements and @xmath181 for the symmetric tensor product of two linear operators . furthermore , we extend the bilinear form @xmath182 to the tensor product space by @xmath183 . more details can be found in  ( * ? ? ? 4 ) . ] and where we have defined @xmath184 the quadratic form associated with the diffusion matrix @xmath185 is given by @xmath186 furthermore , the fast process can be approximated by an infinite dimensional ornstein - uhlenbeck process . the precise statement and proof of the above results can be found in  @xcite . the assumption that the @xmath187 term in   is linear is needed in order to go from   to   after rescaling or , equivalently , to  . if our starting point is the already rescaled spde  , then we can apply the results from  @xcite to nonlinearities of the form @xmath188 where @xmath189 is an arbitrary nonlinearity . in this case the drift term in the amplitude equation   becomes @xmath190 where @xmath191 denotes the invariant measure of the fast ou process . when the null space of @xmath9 is one dimensional and , consequently , the homogenized sde is a scalar equation , it is possible to obtain sharp error estimates and to prove convergence in the strong topology . in this case equation becomes    @xmath192    where @xmath193 in the one dimensional case the formulas for the coefficients that appear in the homogenized equation have a simpler form than in the multidimensional case . in particular , we have , with @xmath194 :    [ e : coeffs ] @xmath195    it is worth mentioning that if we are using a non - orthonormal basis , i.e. a basis @xmath196 , then the coefficients that appear on the right hand side of the above equation transform according to @xmath197 we also have @xmath198 . the formulas for the coefficients that appear in the amplitude equation   can be also obtained by writing the spde   in fourier space , truncating and then using singular perturbation theory - type of techniques for the corresponding backward kolmogorov equation  @xcite . more details on this approach can be found in  @xcite . we also remark that , in general , both additive as well as multiplicative noise will appear in the amplitude equation , although only ( degenerate ) additive noise is present on the spde  . in this section we apply our numerical method to several spdes and report its convergence and performance . we consider here several examples of spdes with quadratic nonlinearities and check that the theory developed in  @xcite and summarized in section  [ sec : multisc ] applies . for all of these examples we can derive rigorously the homogenized equation , with explicit formulas for the coefficients and therefore , we can present a rigorous numerical study for our algorithm and test the effectiveness of the proposed numerical algorithm . we will consider variants of the burgers and the kuramoto - shivashinsky ( ks ) equations ( with a linear instability term added ) in one dimension with either dirichlet or periodic boundary conditions . in particular , we will consider the singularly perturbed spdes ( i.e. we have already rescaled to the diffusive time scale ) @xmath199 and @xmath200 respectively , where the noise @xmath11 is as in section  [ sec : multisc ] . the operator @xmath12 , the covariance operator of the noise , has eigenvalues @xmath201 and eigenfunctions @xmath202 , which are also the eigenfunctions of the differential operator that appears in either   or  , i.e. the two operators commute . we will consider these two equations either on @xmath203 $ ] with dirichlet boundary conditions or on @xmath204 $ ] with periodic boundary conditions . for the burgers nonlinearity and for the boundary conditions that we consider it is straightforward to check that the centering condition @xmath167 is satisfied . a more natural equation to consider than would be the ks equation in the small viscosity regime , i.e. @xmath205 where @xmath206 . this equation can be rewritten in the form @xmath207 the theory presented in  @xcite and the numerical scheme developed in this paper apply to this equation . the application of the numerical method developed in this paper to equation   and to related models will be presented elsewhere . some recent analytical and numerical results on the behaviour of solutions to   have been reported in  @xcite . we will use the notation @xmath208 it is possible to check that for the above equations and for the chose boundary conditions the theory developed in  @xcite and summarized in section  [ sec : multisc ] applies . consider first equations   and   on @xmath203 $ ] with dirichlet boundary conditions . in this case the null space of @xmath209 and @xmath210 is one dimensional : @xmath211 with @xmath212 being either @xmath182 or @xmath213 . the normalized eigenfunctions of @xmath209 and @xmath210 are @xmath214 . the corresponding eigenvalues are @xmath215 since the null space is one - dimensional , the homogenized equation is a scalar sde . for the nonlinearity @xmath216 = \frac{1}{2 } \partial_x ( u v ) $ ] it is straightforward to calculate @xmath194 . we have @xmath217 where @xmath218 denotes the kronecker delta . we can then use formulas   to calculate the formulas that appear in the homogenized equation . let @xmath219 of either @xmath220 or @xmath210 with dirichlet boundary conditions . the homogenized equation is given by that we recall here for convenience @xmath221 where @xmath222 the coefficients that appear in   can be computed as and use formula  . ] [ e : coeffs_homog_burgers ] @xmath223    in the case where only the second mode is forced with noise , @xmath224 then the coefficients become @xmath225 in this case only multiplicative noise appears in the homogenized equation and it can lead to intermittent behavior of solutions as well as noise induced transitions  @xcite . we will also consider either the burgers or the ks equation on @xmath204 $ ] with periodic boundary conditions . in this case the null space of both @xmath209 and @xmath210 is two - dimensional and is spanned by @xmath226 with @xmath212 being either @xmath182 or @xmath213 . the homogenized equation is given by , where @xmath227 . it consists of a system of two coupled sdes . we can use formulas   and  , together with the formula for the nonlinearity @xmath216 = \frac{1}{2 } \partial_x ( u \ , v)$ ] to calculate the coefficients that appear in the homogenized equation . we shall now apply our numerical algorithm to the model problems , described in section [ sec : examples ] . as the behavior of our algorithm is similar for the burgers and the kuramoto - shivashinsky equation we will do a thorough numerical study on the burgers equation and comment on the results for the kuramoto - shivashinsky equation . * burgers equation . * we consider equation on @xmath203 $ ] with homogeneous dirichlet boundary conditions . we know from section  [ sec : examples ] that , for @xmath44 sufficiently small , we have that @xmath228 where @xmath229 is the solution of . the function @xmath230 in depends on @xmath231 which for the burgers equation can be computed using formulas   with @xmath232 . they read @xmath233    following the algorithm described in section , we look for a solution to   of the form @xmath234 substitute the expansion in   to obtain the a fast - slow system of sdes as described in . following the algorithm of section [ sec : algorithm ] we compute numerically the slow variable @xmath104 as @xmath235 where @xmath236 are given by and , respectively . we also consider the truncated homogenized problem , i.e. , @xmath237 where @xmath238 and where @xmath239 are obtained from , with the sums truncated at @xmath77 .    for numerical comparison we also compute @xmath240 the euler - maruyama approximation of the sdes and , respectively . the same brownian path will be used in  , and . we emphasize that the numerical solutions for and rely on analytically computed homogenized coefficients , whereas for we implement the multiscale algorithm of section [ sec : algorithm ] , where the coefficients @xmath241 are computed `` on the fly '' and rely on the microsolver and . hence no a - priori analytical knowledge of the amplitude equation is required . we choose the values of the various parameters entering in the averaging process for the computation of @xmath242 as suggested in @xcite , i.e. , @xmath243 , @xmath244 . according to @xcite , this guarantees ( for the case of non - degenerate fast processes ) that the error is bounded by @xmath245 . in our case with a degenerate fast process an error bound is still to be established . here we monitor such convergence numerically . more precisely , we set @xmath246 and monitor the error using    @xmath247    for various values of @xmath248 , where @xmath249 and @xmath250 represent the final time . notice that captures the error between  the homogenized solution of the truncated system  and the numerical solution of the truncated system , while  , where the index @xmath251 stands for limit , captures the error between the homogenized solution of the limit problem   and the numerical solution of the truncated system . * 2-mode truncation . * we set @xmath252 in and substitute the expansion in   to obtain the following system of equations    [ e:3mode ] @xmath253    the auxiliary process can be derived as explained in section [ sec : algorithm ] and reads    [ e : auxiliary3 ] @xmath254 we apply the algorithm of section [ sec : algorithm ] to get a numerical approximation of the homogenised problem corresponding to . the final time is @xmath255 and @xmath256 , which corresponds to macro time - step of size @xmath257 . the macro solver for the method is given by . as mentioned above , we compare our results with and . the unknown coefficients @xmath258 in can be computed using and , where the sum is truncated at @xmath259 and read @xmath260 . .,width=226 ]    we observe in figure [ fig3mode ] that we get numerically the expected order of convergence corresponding to @xmath261 . furthermore , as the micro time - step becomes smaller , the numerical scheme gets closer to and slightly deviates from . this is expected as the numerical solution is not converging to that latter solution . we observe nevertheless that with only two fast modes , the numerical scheme already captures quite well the effective behavior of the slow variable of the infinite dimensional system . we also illustrate the time evolution of one trajectory comparing over the time @xmath262 with @xmath263 , the euler - maruyama method for the amplitude equation , the homogenized equation and the macro solver . the same brownian path is used for generating the three trajectories and as well as the same macro time step .    ) , ( solution denoted @xmath264 ) and ( solution denoted @xmath265 ) for three paths ( left @xmath266 for @xmath104 , middle @xmath267 for @xmath104 , right @xmath268 for @xmath104 ) . we use 2-mode truncation for and .,title="fig:",width=170,height=151 ] -8 mm ) , ( solution denoted @xmath264 ) and ( solution denoted @xmath265 ) for three paths ( left @xmath266 for @xmath104 , middle @xmath267 for @xmath104 , right @xmath268 for @xmath104 ) . we use 2-mode truncation for and .,title="fig:",width=170,height=151 ] -8 mm ) , ( solution denoted @xmath264 ) and ( solution denoted @xmath265 ) for three paths ( left @xmath266 for @xmath104 , middle @xmath267 for @xmath104 , right @xmath268 for @xmath104 ) . we use 2-mode truncation for and .,title="fig:",width=170,height=151 ]    we perform this comparison for increasing accuracy of the micro solver used to recover the macro data , namely , @xmath269 we see in figure [ fig : trajectory ] that the trajectory for the amplitude equation and the homogenized equation coincide , while the macro solver gets closer to the true dynamics as we refine the micro time step . for the same trajectory we also give a space - time plot for the approximation of the original spde @xmath270 with @xmath229 solution of the amplitude equation , the homogenized equation or the macro solver . again we see that the numerical method captures the right behavior of the solution .     of the spde ; @xmath271 ( left figure @xmath266 ) , @xmath272 ( middle figure ) and @xmath273 ( right figure).,width=529,height=264 ]    * 3-mode truncation . * we set @xmath274 in and obtain the following system of equations    [ e:4mode ] @xmath275    the auxiliary process can be computed similarly as for the 3-mode truncation . we perform the same set of numerical experiments as for the 3-mode truncation , reported in figure [ fig4mode ] . similar behavior than previously noted can be observed . observe that the discrepancy between the numerical scheme and gets smaller . this is expected as with additional modes , the homogenized equation ( that we aim at approximating with our multiscale scheme ) gets closer to .    . , width=226 ]    [ fig4mode ]    * 4-mode truncation . * we set @xmath276 in and apply the similar procedure as previously . for the sake of brevity , we do not write the system of equations in this case and just report the numerical convergence . we see in figure [ fig5mode ] a similar behavior of our numerical scheme as observed previously . we again notice that the discrepancy between the numerical scheme and is smaller than for lower order truncation . notice that the first numerical result reported is obtained for @xmath277 with @xmath267 . this is due to stability issues with the euler - maruyama scheme for the fast process . as the linear term in the equation for the fifth mode @xmath278 is @xmath279 , the stability restriction @xmath280 implies @xmath281 and thus the time step @xmath282 is unstable .    . , width=226 ]    [ fig5mode ]     +    [ [ the - kuramoto - shivashinsky - equation . ] ]   the kuramoto - shivashinsky equation . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    the equations for the @xmath77-mode truncation of the kuramoto - shivashinsky equation are very similar to the ones for the burgers equation and will not be presented here . the only difference is that the fast process is more dissipative than for the burgers equation , due to the stronger dissipativity of the operator @xmath210 compared to @xmath220 . as the results of the numerical experiments for the ks equation are very similar to the results reported in this section for the burgers equation , they will not be presented here . we have presented a new numerical method for the efficient and accurate solution of stochastic partial differential equations with multiple scales . the new numerical scheme is based on a combination of a spectral method with the hmm methodology and has been tested on spdes with quadratic nonlinearities for which a rigorous homogenization theory exists . this enables us to check the performance of our method . the numerical experiments presented in this paper suggest that the new method performs well and allows to solve accurately multiscale spdes by solving a low dimensional fast - slow system of sdes . the method is suitable for infinite dimensional stochastic systems for which there is clear separation of scales , and for which a low dimensional homogenised ( or averaged ) equation for the slow modes exists . there are still many questions that are left open . first , the rigorous analysis of the proposed method and a careful study of the convergence and stability properties of the proposed method remains to be done . in addition , the optimisation of the proposed method by tuning appropriately the parameters of the algorithm has not been performed yet . this appears to be an open problem even when the hmm methodology is applied to finite dimensional fast / slow systems of sdes  @xcite . the proposed numerical algorithm could be used to study in detail the qualitative and quantitative properties of solutions to spdes with quadratic nonlinearities , since spdes of this form exhibit very rich dynamical behaviour . furthermore , we would like to apply the numerical algorithm to more general classes ( and systems ) of semilinear spdes , for which an averaged or homogenised equation is known to exist . examples include systems of reaction / diffusion equations that were considered in  @xcite as well as the swift - hohenberg spde  @xcite .    in our algorithm , we did not make use of the fact that the form of the amplitude equation ( i.e. a landau equation with additive and multiplicative noise ) is known . knowledge of the functional form of the coefficients that appear in the homogenised or averaged equation can be used in order to simplify the numerical algorithm . the stochastic landau equation appears as the amplitude equation for several infinite dimensional stochastic dynamical systems , not only for spdes with quadratic nonlinearities , e.g.  @xcite . thus , the algorithm proposed in this paper , could be modified to develop an efficient method for studying the dynamics of spdes near bifurcation . all these topics are currently under investigation . part of this work was done while gp was visiting the mathematics section of epfl . the hospitality of the department and of the group of a. abdulle is greatly acknowledged . the authors thank s. krumscheid for an extremely careful reading of an earlier version of the paper and for many useful remarks . gp is supported by the epsrc . _ analysis of multiscale methods for stochastic dynamical systems with multiple time scales , _ siam mul . , 8:944 - 964 , 2010 . a.  j. majda , ch . franzke , and b.  khouider . an applied mathematics perspective on stochastic modelling for climate . , 366(1875):24292455 , 2008 . g.  c. papanicolaou , d.w . stroock , and s.  r.  s. varadhan . martingale approach to some limit theorems . in _ papers from the duke turbulence conference ( duke univ . , durham , n.c . , 1976 ) , paper no . 6 _ , pages ii+120 pp . duke univ . duke univ . , durham , n.c . , 1977 . m.  pradas , d.  tseluiko , s.  kalliadasis , d.  t. papageorgiou , and g.  a. pavliotis . noise induced state transitions , intermittency , and universality in the noisy kuramoto - sivashinksy equation . , 106(6):060602 , feb 2011 .
a new method for solving numerically stochastic partial differential equations ( spdes ) with multiple scales is presented . 
 the method combines a spectral method with the heterogeneous multiscale method ( hmm ) presented in [ w.  e , d.  liu , and e.  vanden - eijnden , comm . 
 pure appl . 
 math . 
 , 58(11):15441585 , 2005 ] . the class of problems that we consider are spdes with quadratic nonlinearities that were studied in [ d.  blmker , m.  hairer , and g.  a. pavliotis , nonlinearity , 20(7):17211744 , 2007 . ] for such spdes an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales . 
 our method , based on micro and macro solvers , allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem . 
 numerical experiments illustrate the behavior of the proposed method . 
 stochastic partial differential equations ; multiscale methods ; averaging ; homogenization ; heterogeneous multiscale method ( hmm )