Katholieke Universiteit Leuven Department of Computer Science

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1 A mutieve agorithm to compute steady states of attice Botzmann modes Giovanni Samaey Christophe Vandekerckhove Wim Vanroose Report TW 56, January 21 Kathoieke Universiteit Leuven Department of Computer Science Ceestijnenaan 2A B-31 Heveree (Begium)

2 A mutieve agorithm to compute steady states of attice Botzmann modes Giovanni Samaey Christophe Vandekerckhove Wim Vanroose Report TW 56, January 21 Department of Computer Science, K.U.Leuven Abstract We present a mutieve agorithm to compute steady states of attice Botzmann modes directy as fixed points of a time-stepper. At the fine scae, we use a Richardson iteration for the fixed point equation, which amounts to time-stepping towards equiibrium. This fine-scae iteration is acceerated by transferring the to a coarse eve. At this coarse eve, one directy soves for the density (the zeroth moment of the attice Botzmann distributions), for which a coarse-eve equation is known in some appropriate imit. The agorithm cosey resembes the cassica mutigrid agorithm in spirit, structure and convergence behaviour. In this paper, we discuss the formuation of this agorithm. We give an intuitive expanation of its convergence behaviour and iustrate with numerica experiments. Department of Mathematics and Computer Science, Universiteit Antwerpen, Middeheimaan 1, 22 Antwerpen, Begium

3 A mutieve agorithm to compute steady states of attice Botzmann modes Giovanni Samaey 1,, Christophe Vandekerckhove 1, and Wim Vanroose 2 1 Scientific Computing, Department of Computer Science, K.U. Leuven, Ceestijnenaan 2A, 31 Leuven, Begium. 2 Department of Mathematics and Computer Science, Universiteit Antwerpen, Middeheimaan 1, 22 Antwerpen, Begium. Abstract. We present a mutieve agorithm to compute steady states of attice Botzmann modes directy as fixed points of a time-stepper. At the fine scae, we use a Richardson iteration for the fixed point equation, which amounts to time-stepping towards equiibrium. This fine-scae iteration is acceerated by transferring the to a coarse eve. At this coarse eve, one directy soves for the density (the zeroth moment of the attice Botzmann distributions), for which a coarse-eve equation is known in some appropriate imit. The agorithm cosey resembes the cassica mutigrid agorithm in spirit, structure and convergence behaviour. In this paper, we discuss the formuation of this agorithm. We give an intuitive expanation of its convergence behaviour and iustrate with numerica experiments. 1 Introduction For a broad cass of systems, in appications ranging from physics, fuid fow [1,9] and bioogy [2] to traffic fow [4], a macroscopic (coarse) partia differentia equation (PDE) that modes density evoution in a space-time domain, is often insufficient to accuratey describe physica interactions between individua partices (atoms, moecues, bacteria, vehices). For instance, the dynamics of a system of coiding partices with interactions that depend sensitivey on the reative partice veocities can, in genera, not be modeed competey by a reaction-diffusion equation for the partice density [13]. In such cases, one needs to resort to a more microscopic (fine-scae) description, such as a kinetic equation that modes the evoution of the veocity-position phase space distribution density of partices as a combination of advection according the the current veocity and coisions that redistribute veocities [25]. A attice Botzmann mode (LBM) is a simpe and effective space-timeveocity discretization of a fine-scae kinetic equation [2]. Ony a ow number of veocities are considered, and the discrete veocities are reated to the attice spacing and time step in such a way that the corresponding distributions are simpy shifted over an integer number of attice sites over one time step. The GS is a Postdoctora Feow of the Research Foundation Fanders (FWO Vaanderen).

4 advantage is that one can decompose evoution in two separated steps: first, one simpy advects each distribution function according to its corresponding veocity; subsequenty, one performs a coision step in which the veocities are redistributed. A more precise mathematica description of LBMs wi be presented in section 2. In the appropriate diffusion imit [15, e.g.], the above-described fine-scae kinetic modes coapse onto a imiting coarse partia differentia equation. However, their associated computationa cost is much higher, due to the increased dimension. For such cases, severa approaches have been deveoped that expoit the ink between the coarse and fine-scae eves of descriptions to significanty acceerate computations. Of particuar importance in this work is the equation-free framework that was proposed by Kevrekidis et a. [11,21]. The framework introduces ifting and restriction operators to map a coarse state to a fine-scae state and vice versa. These operators are then used to construct a coarse time-stepper for the unavaiabe coarse equation as a three step procedure: (1) ifting, i.e. the creation of appropriate initia conditions for the fine-scae mode, conditioned upon the coarse state at time t ; (2) simuation, using the fine-scae mode, over the time interva [t,t + t]; and (3) restriction, i.e. the extraction of the coarse state at time t + t. The resuting coarse time- t map can then be used in conjunction with projective integration methods [7] to acceerate time integration, or with a matrix-free agorithm to directy compute coarse steady states [17 19]. We refer to [12] for a recent review and references to reated methods. In this paper, we borrow important concepts from the equation-free framework, and use them to construct a new agorithm that directy computes fine-scae steady states of an LBM as fixed points of the time-stepper. The proposed agorithm bears many simiarities with standard mutigrid [1,22]. Mutigrid is a sophisticated iterative method for the computation of a steady state of a PDE. It is most easiy understood by reaizing that the in any initia guess consists of high and ow wavenumber modes. Basic iterative methods (such as a Jacobi or Gauss-Seide iteration, or Richardson s method, which corresponds to time-stepping) are very efficient in removing highfrequency components; such methods are therefore caed smoothers. In mutigrid, one uses the fact that the smoother yieds s (and residuas) that can be accuratey represented on a coarser mesh (since they are smooth). One then transfers the residuas to the coarser grid (restriction) and soves for the. The resuting is then transferred back to the finer grid (proongation). This sequence is caed coarse grid correction, and the combination of smoothing and coarse grid correction yieds a very powerfu agorithm. At the the coarse grid, the same procedure can be appied. One then performs again a few smoothing steps and a coarse grid correction on an even coarser grid, eading to a hierarchy of grids hence the name mutigrid. The mutieve agorithm that we propose here foow the same strategy. At the fine scae, we consider the computation of a fixed point of a attice Botzmann time-stepper; the basic iteration (smoother) on this eve is a

5 Richardson iteration, which amounts to time-stepping towards equiibrium. Instead of moving to a coarser grid, as in mutigrid, however, here we move to a coarser eve of description, i.e. the eve of the coarse PDE. Note that at this eve, we can ony use the PDE that is known in the diffusion imit, which yieds ony an approximation to the evoution of the density of the LBM. The transfer operators that connect the coarse and fine-scae eves of description are exacty the ifting and restriction operators that have been introduced in the equation-free framework; these then repace the proongation and restriction operators that are used in mutigrid. The remainder of the paper is organized as foows. In section 2 we define our mode probems. Section 3 describes the behaviour of Richardson iteration for this probem. We then proceed to formuate our mutieve agorithm in section 4. Section 5 contains an intuitive expanation of the convergence behaviour. We concude in section 6. 2 Mode probem 2.1 Lattice Botzmann modes Throughout this paper, we consider reaction-diffusion attice Botzmann modes (LBMs) [3,2] in one space dimension as the iustrative exampe. We briefy review the principes of LBMs and describe the specific mode probem of this paper. We define a discrete number of partice distribution functions f i (x j,t n ), with 1 j N, q i q and n, with grid spacing δx in space and δt in time, and choose the veocities v i such that they correspond to a movement over an integer number of attice points during one time-step, v i = c i δx δt, c i = q,..., 1,, 1,...,q. Furthermore, we introduce equiibrium distributions f eq i (x, t) =w eq i which correspond to oca diffusive equiibrium. Here, the coefficients w eq w eq i depend on the number of speeds 2q + 1. ρ(x, t), i = Remark 1 (Notation). To avoid potentia confusion, we emphasize that, in this text, we wi use the symbo δt (and, for notationa consistency, δx) to denote the parameters of the fine-scae (attice Botzmann) time-stepper, whereas the symbo t is used to indicate the size of a coarse time step. With these definitions, we can write an evoution aw for the distributions f(x, t) =(f i (x, t)) q i= q as f i (x + c i δx, t + δt) =(1 ω)f i (x, t) ωf eq i (x, t)+β j V ij f j (x, t)+r i (x, t), (1)

6 which we denote in compact form as f n+1 = φ(f n ), (2) where f =(f i (x)) q i= q, x =(x j) N j=1 and the superscript n denotes the time instance t = nδt. Equation (1) decomposes evoution into a coision phase (the right-hand side) and a streaming phase, during which the post-coision vaues are propagated to a neighboring site. The first two terms of the righthand side represent a BGK reaxation to oca diffusive equiibrium on a characteristic time scae τ = 1/ω. The third term is an externa force, which is discretized as proposed by Luo [14], and the terms R i (x, t) mode reactive coisions. Note that, for pure diffusion, a attice Botzmann time-stepper is stabe for ω [, 2]. We now define the first few (non-dimensiona) moments of the distribution function as ρ(x, t) = q f i (x, t), ϕ(x, t) = i= q q c i f i (x, t), ξ(x, t) = 1 2 i= q q c 2 i f i (x, t), i= q (3) which represent density, momentum and energy, respectivey. The attice Botzmann equation can be written equivaenty as a set of couped PDEs for the 2q + 1 moments of the individua distribution functions. A Chapman Enskog expansion shows that, when the density is sufficienty smooth, the attice Botzmann mode (1) (or the set of couped equations for the moments) can be approximated by a singe reaction-advection-diffusion equation ony depending on the density [5,16], t ρ(x, t) = x (D x ρ(x, t))+β x ρ(x, t)+r(ρ(x, t)), D = 2 ω 2ω δx 2 δt i c 2 i w eq i, (4) where the transport coefficients D and β and the reaction rate r(ρ) depend on the reaxation, externa force and coision terms in the attice Botzmann mode. When the reaction terms R i can be written as functions of the density, R i (x, t) =w eq i δtr(ρ(x, t)), we obtain r(ρ) =R(ρ). In genera, however, if R i (x, t) depend on the individua distribution functions, it is cumbersome to eiminate the veocity dependence and derive an expicit formua for r(ρ). From the assumption that a coarse mode for ρ(x, t) exists, it foows that the higher order moments ϕ(x, t) and ξ(x, t) can be written as functionas of ρ(x, t). (We ca this saving.) For a 3-speed attice Botzmann reactiondiffusion mode, i.e. equation (1) with q = 1 and β =, these saving reations can be written down anayticay as an asymptotic expansion in 1/ω. Up to

7 third order, we have [23], ϕ(x, t) = 2δx 3 ω xρ + δxδt (4ω 2) 3ω 2 (ω 2) xr(ρ)+ δxδt ( 2ω 2 +2ω 2) 3ω 2 2 (ω 2) xtρ(x, t) ξ(x, t) = 1 δt ρ(x, t) 3 6ω (r (ρ(x, t)) tρ(x, t)). (5) These expansions can aternativey be written down in terms of ρ(x, t) and its spatia derivatives ony, by making use of (4). 2.2 Fixed point formuation We are interested in the fixed points of equation (2), i.e. soutions of the noninear equation f φ(f )=. (6) This noninear probem can be soved using Newton s method. We wi appy a mutieve agorithm on the inear systems arising in each Newton iteration, which can be written as I Jφ ( f) f = r( f), (7) or, for short, A f x = b. (8) Here, we use x = f and J φ ( f) =( φ/ f)( f) and r( f) = f φ( f) denote the Jacobian of φ and the residua of equation (6), respectivey, evauated at a current guess f. For notationa convenience, we have eiminated the dependence on f in A f and b. The Jacobian J φ is not avaiabe in cosed form, since we can ony evauate the time-stepper, but Jacobian-vector products can be estimated as J φ ( f)v φ( f + v) φ( f), with appropriatey sma. This enabes the use of methods that require ony matrix-vector products. 3 Richardson iteration We first consider a basic iterative method, namey Richardson iteration, x (k+1) = S(x (k),a f,b)=x (k) +(A f x (k) b), (9) which is seected over aternatives such as Jacobi or Gauss-Seide, because it does not invove a matrix spitting of A f, and hence requires ony matrixvector products. This method can be seen to correspond to time-stepping for

8 the equation of the inearized system. To see this, we eaborate x (k+1) = x (k) +(A f x (k) b) = x (k) + (I J φ )x (k) b = J φ x (k) b. The ast equation shows that the Richardson iteration corresponds to timestepping with the inearized time-stepper, modified by the righthand side of the inear system (the residua of the noninear equation). Hence, the spectra properties of the smoothing operator wi be those of the inearized timestepper. Reca that we assume that an equation exists that coses at the eve of the density ρ(x, t) aone. As a consequence, this assumption impies that the higher order moments become functionas of ρ(x, t) on time-scaes which are fast compared to the overa system evoution (saving). This behaviour is refected in the spectra properties of the smoother: Richardson iterations quicky reduce s in the higher order moments, whereas s in modes that correspond to the density are not significanty affected by the smoother. This observation is iustrated by a numerica experiment. We consider the pure diffusion 3-speed attice Botzmann probem, i.e. equation (1) with q = 1, β = and R i using periodic boundary conditions. The domain is [, 1], and the mode parameters are chosen to be D = 1, ω =1.25 and δx =1/128. The time step then foows from equation (4). A fixed point of this time-stepper can be found as the soution of the inear system A f x =. (1) We now ook into the behaviour of Richardson iteration for the inear system (1). Remark 2. Because of the periodic boundary conditions, the time-stepper has a trivia eigenvaue 1, and consequenty, the fixed point probem is singuar. This singuarity can be removed by adding a phase condition and an artificia parameter, see e.g. [19]. In this experiment, we perform a spectra anaysis of Richardson iteration; for this purpose, simpy considering Richardson iteration on the singuar system is sufficient. Because of the homogeneous righthand side, Richardson iterations reduce to time-stepping with the origina inear time-stepper. For the 3-speed mode, there is a inear 1-to-1 reation between the distribution functions and the moments (3). It can easiy be verified (for instance using Mape) that, when working in the equivaent moment description, the eigenvectors have the form (ρ, ϕ, ξ) =(A sin(k x),bcos(k x),csin(k x)), with 1 k N 1. For each frequency, there are 3 eigenmodes. However, the anaytic determination of the constants, as we as the corresponding eigenvaues, does not yied a workabe cosed formua. Therefore, we compute the spectrum numericay.

9 I(λ) V() R(λ) damping factor k V() Fig. 1. Top eft: Spectrum of the attice Botzmann time-stepper/richardson smoother for homogeneous diffusion. Bottom eft: Richardson ampification factors for each of the eigenmodes, as a function of wave number k. The soid ine corresponds to the ampifications factors of the rea positive eigenvaues; the dashed ine corresponds to the ampifications factors of the compex eigenvaues with negative rea part. Right: a typica eigenmode corresponding to a rea positive eigenvaue (top) and a compex eigenvaue (bottom) of the time-stepper. The soid ine contains the rea part; the dashed ine is the imaginary part. The corresponding eigenvaues are depicted by a square, resp. circe in the top eft pane. Eigenmodes are dispayed in moment representation: the eftmost part of each eigenvector corresponds to density, the midde part to momentum and the rightmost part to energy.

10 The resuts are depicted in figure 1 (top eft). It is seen that the spectrum breaks down into two sets of eigenvaues: a set of N 1 rea and positive eigenvaues, and a set of 2N 2 compex eigenvaues with negative rea part. Figure 1 (right) shows an eigenvector corresponding to a particuar mode in each of these two sets, in the moment representation. One can visuay verify that the eigenvectors corresponding to the rightmost set of rea eigenvaues appear to satisfy the saving reations (5); we ca these the saved modes. The saving reations are not satified for the eigenvectors corresponding to the set of compex eigenvaues (non-saved modes). Ceary, the sower a mode is damped, the better the saving reations are satisfied.m We now ook at the behaviour of the Richardson iteration for each of these eigenmodes individuay. Starting from a normaized eigenmode that satisfies max V =1(with1 3N 3 denoting the components), we perform a singe Richardson iteration and ook at the ampification factor. Figure 1 (bottom eft) shows the maxima vaue of the resuting vector as a function of the wave number. As coud be expected from the spectrum of the time-stepper, we can draw two main concusions : Regardess of the wave number, modes that corresponds to non-saved states are damped quicky by the time-stepper; ony modes that corresponds to saved states can persist over onger time scaes. For the saved modes, we observe that ow wavenumber modes are damped much more sowy than high wavenumber waves. Based on these observations, we concude that it woud be beneficia to combine Richardson iterations with a method to directy reduce the s in the density. To this end, we now turn to a mutieve agorithm. 4 Mutieve agorithm To sove the inear system (8) more efficienty, we make use of the ink between the fine-scae mode (1) and the approximate coarse description (4). We first describe ifting and restriction operators that map coarse to fine-scae states, and vice versa. We then proceed to formuate the compete agorithm. An intuitive expanation on it convergence properties is deferred to section Lifting and restriction We consider operators that define a mapping between the distribution functions f i (x, t) and the density ρ(x, t). A restriction operator, mapping the distribution f(x, t) =(f i (x, t)) i to the density ρ(x, t), R : f(x, t) ρ(x, t) =R(f(x, t)), (11) can easiy be obtained using (3). The converse ifting operator, reconstructing distributions from the density, L : ρ(x, t) f(x, t) =L(ρ(x, t)), (12)

11 is consideraby more invoved. One coud simpy initiaize the distributions as f i (x, t) =w i ρ(x, t), w i =1, (13) where w i = w eq i is an obvious choice, since this corresponds to the oca diffusive equiibrium. Via a Chapman Enskog expansion, it can be shown that this corresponds to a zeroth order approximation of the saving reations [23]. Ceary, initiaizing aso the higher order terms in the Chapman Enskog expansion yieds a more accurate reconstruction. A numerica procedure to obtain a second order approximation to the saving reations, using ony the time-stepper (2) is the constrained runs scheme [8,6]. Here, during a preparatory step, a number of attice Botzmann timesteps is performed, after each of which the density is reset. This scheme can be shown to converge ineary, with a convergence factor 1 ω [23]; hence, it is aso stabe whenever ω [, 2]. Higher order variants, which are more accurate but may become unstabe, have been investigated in [24]. i 4.2 Coarse-eve correction Once the fine-scae state has been transferred to the coarse eve, we need to define the coarse inear system that needs to be soved. For this, we start from the approximate coarse mode (4), and construct a corresponding timestepper, which we write in compact form as ρ n+1 = Φ(ρ n ), where ρ = ρ(x). The fixed-point equation at this eve can then equivaenty be written as which we write in a shorthand notation (I J Φ ( ρ)) ρ = R( ρ), (14) A c X = B. (15) Here, we use X = ρ, and J Φ ( ρ) =( Φ/ ρ)( ρ) and R( ρ) = ρ Φ( ρ) denote the Jacobian of Φ and the residua of the coarse fixed-point equation, respectivey, evauated at a current guess ρ. 4.3 The mutieve agorithm With a buiding bocks in pace, we are now ready to formuate the compete mutieve agorithm. One iteration, starting from x (m), consists of the foowing steps:

12 1. Presmoothing: Perform ν 1 Richardson iterations, x (m) = S ν1 (x (m),b). (16) 2. Coarse-eve correction: Compute the defect: d (m) = b Ax (m). Restrict defect : D ( m)=r(d (m) ). Coarse-eve sove : A c V (m) = D (m). Lift correction : v (m) = L(V (m) ). Update fine-scae soution : ˆx (m) = x (m) + v (m). 3. Postsmoothing: Perform ν 2 Richardson iterations, x (m+1) = S ν2 (ˆx (m),b). (17) Remark 3 (Comparison with mutigrid). The agorithmic structure resembes that of a cassica mutigrid method to find steady states of partia differentia equations [1,22]. For the PDE case, the ampification factors of a coarseeve Richardson iteration are simiar to the soid ine in figure 1 (bottom eft). Mutigrid combines the good smoothing properties for high wavenumber modes with a coarse grid correction. One first performs a number of smoothing steps, after which one transfers the residuas to the coarser grid (restriction) and soves for the. This restriction is possibe, because a smooth residua can be accuratey represented on a coarser mesh. The resuting is then transferred back to the finer grid (proongation), and added to the current guess. Here, exacty the same procedure is foowed; ony the proongation and restriction have been repaced by equation-free ifting and restriction. Cassica mutigrid can be used for the coarse eve correction. Remark 4 (Fu approximation scheme). Note that, as in mutigrid, we transfer the residua to the coarse eve, and sove the equation there. Instead of using Newton s method with a mutieve agorithm to sove the inear systems in each iteration, the noninear equation (6) can aso be soved using a noninear mutieve agorithm directy, in the spirit of a noninear mutigrid method (the fu approximation scheme)[1,22]. 5 Convergence of the two-eve cyce We now proceed to show how smoothing and coarse-eve correction work together to yied fast convergence. To this end, we adapt the exposition that was presented in [1] for the cassica mutigrid method to our setting. 5.1 An intuitive agebraic picture We start with an iustrative numerica exampe. We consider the pure diffusion 3-speed attice Botzmann probem, i.e. equation (1) with q = 1, β =

13 and R i using Dirichet boundary conditions. The domain is [, 1], and the mode parameters are chosen to be D = 1, ω =1.1 and δx =1/128. The time step then foows from equation (4). We again ook for a fixed point of the time-stepper as the soution of the inear system (1). To this end, we perform two iterations of the mutieve agorithm using ν 1 = 2 presmoothing and ν 2 = 2 postsmoothing steps, starting from an initia guess f that has the moment representation ρ(x) = ϕ(x) = ξ(x) = sin(3x) + sin(45x). The evoution of the throughout the agorithm is depicted in figure 2. The figure shows the effect of presmoothing, coarse-eve correction and postsmoothing during both iterations. We see that during the presmoothing step the in φ and ξ decreases rapidy, as we as the in the high wavenumber modes of ρ. However, the ow wavenumber modes of ρ remain virtuay unaffected by the presmoothing. In the next step, we sove for the density at the coarse-eve correction, and ift the resuting the fine-scae representation using the zeroth-order term of the Chapman Enskog expansion. We remark that during this ifting step, the in ξ aso decreases substantiay, as a consequence of the form of the saving reations (5)). Finay, in the postsmoothing phase, the in the higher order moments is again decreased. Note that the coarse-eve equation does not correspond exacty to the behaviour of the density of the attice Botzmann mode, and hence the resuting density does not vanish. The attice Botzmann mode converges to the imiting density equation for ω 1, so the density that remains after the coarse-eve sove wi be arger if ω is further away from 1. This is iustrated in figure 3, where the experiment is repeated with ω =1.25. We now proceed to providing an intuitive, graphica expanation of this behaviour, see figure 4. The exposition is very simiar in spirit to the intuitive picture that was given in [1, Chapter 5]. In pane (a), we show the tota e f in the current guess, and decompose this according to the subspaces that correspond to saved and non-saved eigenmodes (the dashed coordinate system). We can aso make a second decomposition that corresponds to the transition between the coarse and fine-scae eves of description. A first subspace contains the fine-scae modes that can be represented given ony the density, i.e. the range of the ifting operator Range(L). Because the ifting operator can ony reconstruct the fine-scae state approximatey, the subspace of the saved modes and Range(L) do not coincide exacty. The compementary space contains a remaining fine-scae modes, i.e. the modes that map to a zero density, the nu space of the restriction operator Nu(R). Aso this space does not correspond exacty to the saved modes. We wi now ook into the behaviour of the mutieve agoritm in these decompositions. The first step in the agorithm is presmoothing, depicted in pane (b). Here, the in non-saved modes decrease quicky, whie the in the saved modes is virtuay unatered. (Note from the experiment before that this is ony ap-

14 Fig. 2. Evoution of throughout the mutieve iteration for a pure diffusion attice Botzmann mode with ω = 1.1. (The other parameters are in the text.) Left: first iteration. Right: second iteration. Dashed, resp. soid, ines represent the before, resp. after the presmoothing phase (top), the coarse-eve correction (midde) and the postsmoothing phase (bottom). Eigenmodes are dispayed in moment representation: the eftmost part corresponds to density, the midde part to momentum and the rightmost part to energy.

15 Fig. 3. Evoution of throughout the mutieve iteration for a pure diffusion attice Botzmann mode with ω =1.25. (The other parameters are in the text.) Left: first iteration. Right: second iteration. Dashed, resp. soid, ines represent the before, resp. after the presmoothing phase (top), the coarse-eve correction (midde) and the postsmoothing phase (bottom). Eigenmodes are dispayed in moment representation: the eftmost part corresponds to density, the midde part to momentum and the rightmost part to energy.

16 Non-Saved Nu(R) (a) Non-Saved Nu(R) (b) enc ef Saved enc ef Saved ens es es ens ec Range(L) ec Range(L) Non-Saved Nu(R) (c) Non-Saved Nu(R) (d) enc ef Saved Saved ens enc ef es ens es ec Range(L) ec Range(L) Fig. 4. Schematic representation of the at different stages of the mutieve agorithm: (a) initia ; (b) the effect of presmoothing; (c) the effect of coarse-eve correction; and (d) the effect of postsmoothing. Shown is how the decomposes in terms of (i) the range of the ifting operator and the nuspace of restriction; and (ii) saved and non-saved eigenmodes of the attice Botzmann time-stepper.

17 proximatey true, and ony for the ow wavenumber components.) Next, we transfer the to the coarse eve and perform a coarse-eve sove, see pane (c). In this phase, the part of the that is in Range(L) is significanty reduced. Due to the fact that the density equation ony approximates the behaviour of the density of the attice Botzmann mode, this part of the is not put exacty back to zero. We aso observe that, in this step, the in the non-saved modes has increased again. This is due to the ifting operator, which reconstructs the fine-scae state based on the density aone, reintroducing artifacts. In a fina postsmoothing step, see pane (d), these non-saved modes are again removed by a number of additiona smoothing steps. 5.2 Numerica convergence tests Let us now proceed to numericay iustrate convergence. We again consider the pure diffusion 3-speed attice Botzmann probem, i.e. equation (1) with q = 1, β = and R i using Dirichet boundary conditions. The domain is [, 1], and the mode parameters are chosen to be D = 1, δx =1/128, and ω =1.1, resp. ω =1.25. The time step then foows from equation (4). We again ook for a fixed point of the time-stepper as the soution of the inear system (1) using the mutieve agorithm with ν 1 = 2 presmoothing and ν 2 = 2 postsmoothing steps, starting from a random initia guess. Figure 5 shows the two-norm of the as a function of the iteration number. We see inear convergence, with a convergence rate that depends on ω. As foows from the agebraic picture above, this can be expained by noting that the coarse-eve correction reduces more in the density when ω is coser to 1. We verified that the convergence rates are mesh-independent. 6 Concusions We presented a mutieve agorithm that acceerates the iterative computation of steady states of attice Botzmann modes. The agorithm expoits the ink between a fine-scae description and a coarse-scae imiting equation. As in mutigrid, the is decomposed into fine-scae and coarse components, which are each handed independenty. In mutigrid, the fine and coarse scaes correspond to high and ow wavenumber modes. In our agorithm, the is decomposed into a coarse-scae component, which is defined by a reduced description in terms of density aone, and the remaining fine-scae components. The fine-scae component of the is quicky damped by straightforward time integration; the coarse component, on the other hand, is reduced via the soution of an approximate coarse-scae equation. The agorithm reies on the definition of appropriate ifting and restriction operators to transfer the from the coarse to the fine eve and vice versa. Such operators have been deveoped in the equation-free framework [11,12]

18 iteration Fig. 5. Two-norm of the of the mutieve iteration as a function of the iteration number for a fixed point computation of a attice Botzmann time-stepper with ω = 1.1 (soid) and ω = 1.25 (dashed), starting from a random initia guess. for a variety of fine-scae systems. In cases where fine-scae steady states exist (such as the kinetic formuations in this paper), our agorithm can provide an efficient means to compute these fixed points. We note that, on the coarse eve, one coud aso use a mutigrid agorithm for the soution of the coarse eve system, resuting in a method with overa inear compexity, aso in higher spatia dimension. In future work, we wi extend this work to more genera discretization of kinetic equations, and perform a more rigourous two-grid anaysis. Acknowedgements This work was supported by the Research Foundation Fanders through Research Projects G.13.3 and G.17.8 and by the Interuniversity Attraction Poes Programme of the Begian Science Poicy Office through grant IUAP/V/22 (GS). The scientific responsibiity rests with its authors. References 1. W Briggs, VE Henson, and S McCormick. Amutigridtutoria.SIAM,2. 2. FACC Chaub, PA Markowich, B Perthame, and Christian Schmeiser. Kinetic modes for chemotaxis and their drift-diffusion imits. Monatsh Math, 142(1-2): , 24.

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