Hybrid and Moment Guided Monte Carlo Methods for Kinetic Equations

Transcription

1 Hybrid and Moment Guided Monte Carlo Methods for Kinetic Equations Giacomo Dimarco Institut des Mathématiques de Toulouse Université de Toulouse France Joint work with: Pierre Degond (Institut de Mathématiques de Toulouse & CNRS) Lorenzo Pareschi (University of Ferrara, Italy) CEA-Saclay, 25 September 2012 Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

2 Outline 1 Introduction Multiscale problems and methods Kinetic equations 2 Hybrid methods Hybrid representation The hybrid method Numerical results 3 Moment Guided Monte Carlo Basic principles The MG method Numerical results Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

3 Introduction Outline 1 Introduction Multiscale problems and methods Kinetic equations 2 Hybrid methods 3 Moment Guided Monte Carlo Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

4 Introduction Multiscale problems and methods Multiscale problems A broad range of scientific problems involve multiple scales and multi-scale phenomena (material science, chemistry, fluid dynamics, biology...). These involve different physical laws which govern the processes at different scales. On the computational side, several important classes of numerical methods have been developed which address explicitly the multiscale nature of the solutions (wavelets, multigrid, domain decomposition, stiff solvers, adaptive mesh refinements...). For many problems, representation or solution on the fine-scale is impossible because of the overwhelming costs. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

5 Introduction Multiscale problems and methods The different scales Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

6 Introduction Multiscale problems and methods Multiscale methods Couplings of atomistic or molecular, and more generally microscopic stochastic models, to macroscopic deterministic models based on ODEs and PDEs is highly desirable in many applications. Similar arguments apply also to numerical methods 1. A classical field where this coupling play an important rule is that of kinetic equations 2. In such system the time scale is proportional to a relaxation time ε and a strong model (and dimension) reduction is obtained when ε 0. Many examples could depict this situation, rarefied gas dynamics, plasma physics, granular gases, turbulence, W.E, B.Engquist CMS 03, N. AMS 03 2 P. Degond, L. Mieussens JCP 07, A. Alaia, G. Puppo JCP 10,... Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

7 Introduction Kinetic equations Kinetic equations Kinetic equations t f + v x f = 1 ε Q(f, f), x, v Rd, d 1, (microscale) Here f = f(x, v, t) 0 is the particle density and Q(f, f) describes the particle interactions. In rarefied gas dynamics the equilibrium functions M for which Q(M, M) = 0 are local Maxwellian ) ρ u v 2 M(ρ, u, T)(v) = exp (, (2πT) d/2 2T where we define the density, mean velocity and temperature as ρ = f dv, u = 1 vf dv, T = 1 [v u] 2 f dv. R ρ d R dρ d R d Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

8 Introduction Kinetic equations Fluid limit If we multiply the kinetic equation by its collision invariants (1, v, v 2 ) and integrate the result in velocity space we obtain five equations that describe the balance of mass, momentum and energy. The system is not closed since it involves higher order moments of the distribution function f. As ε 0 we have Q(f, f) 0 and thus f approaches the local Maxwellian M f. Higher order moments of f can be computed as function of ρ, u, and T and we obtain the closed system Compressible Euler equations t ϱ + x (ϱu) = 0, t ϱu + x (ϱu u + p) = 0, t E + x (u(e + p)) = 0, (macroscale) where p is the gas pressure. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

9 Hybrid methods Outline 1 Introduction 2 Hybrid methods Hybrid representation The hybrid method Numerical results 3 Moment Guided Monte Carlo Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

10 Hybrid methods Hybrid representation Hybrid representation The solution is represented at each space point as a combination of a nonequilibrium part (microscale) and an equilibrium part (macroscale) f(v) nonequilibrium f(v) nonequilibrium equilibrium equilibrium R w(v)m(v) βm(v) v R R v R Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

11 Hybrid methods Hybrid representation The starting point is the following 3 Definition I - hybrid function Given a probability density f(v), v R d (i.e. f(v) 0, f(v)dv = 1) and a probability density M(v), v R d called equilibrium density, we define w(v) [0, 1] and f(v) 0 in the following way f(v), f(v) M(v) 0 w(v) = M(v) 1, f(v) M(v) and f(v) = f(v) w(v)m(v). Thus f(v) can be represented as f(v) = f(v) + w(v)m(v). 3 L.Pareschi ESAIM 05, L.Pareschi, G.D. CMS 06, MMS 08 Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

12 Hybrid methods Hybrid representation Taking β = min v {w(v)}, and f(v) = f(v) βm(v), we have f(v)dv = 1 β. Let us define for β 1 the probability density f p (v) = f(v) 1 β. The case β = 1 is trivial since it implies f M. Thus we recover the hybrid representation 4 as f(v) = (1 β)f p (v) + βm(v). 4 R.E.Caflisch, L.Pareschi JCP 99 Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

13 Hybrid methods The hybrid method The general methodology Now we consider the following general representation f(x, v, t) = f(x, v, t) }{{} + w(x, v, t)m(ρ(x, t), u(x, t), T(x, t))(v). }{{} nonequilibrium equilibrium The nonequilibrium part f(x, v, t) is represented stochastically, whereas the equilibrium part w(x, v, t)m(ρ(x, t), u(x, t), T(x, t))(v) deterministically. The general methodology is the following. Solve the evolution of the non equilibrium part by Monte Carlo methods. Thus f(x, v, t) is represented by a set of samples (particles) in the computational domain. Solve the evolution of the equilibrium part by deterministic methods. Thus w(x, v, t)m(ρ(x, t), u(x, t), T(x, t))(v) is obtained from a suitable grid in the computational domain. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

14 Hybrid methods The hybrid method The general methodology II The starting point of the method is the classical operator splitting which consists in solving first a homogeneous collision step and then a free trasport step (C) t f r (x, v, t) = 1 ε Q(fr, f r )(x, v, t) (T) t f c (x, v, t) + v x f c (x, v, t) = 0. Next the BGK model is considered for simplicity Q(f, f) = M f, extension to the full Boltzmann operator are possible. From now on we will consider the case β = min v {w(v)}, which leads to the more interesting scheme. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

15 Hybrid methods The hybrid method Sketch of the method We introduce the projection operator P, and, in a time step t, the relaxation operator R t and the transport operator T t. The projection operator computes from the kinetic variable f (or M f ) the macroscopic averages U(x, t) = (ϱ(x,t), ρu(x,t), E(x,t)), thus P(f(x, v, t)) = U(x, t), P(M f (x,v, t)) = U(x, t), since the local Maxwellian M f has the same moments of the distribution function f. The relaxation and transport operators solve the relaxation and transport steps. The first has the form R t(f(x,v, t)) = λf(x, v, t) + (1 λ)m f (x, v, t), where λ = exp( t/ε), whereas the second reads T t(f(x,v, t)) = f(x v t, v, t). Note that by definition R t(m f ) = M f and so P(R t(m f )) = P(M f ). Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

16 Hybrid methods The hybrid method A simple hybrid scheme I Let us start from an hybrid solution in the form f(x, v, t) = (1 β(x, t))f p (x,v, t) + β(x, t)m f (x, v, t), where f p (x,v, t) is represented by samples so that N(t) (1 β(x,t))f p (x, v, t) = m p j=1 δ(x p j(t))δ(v ν j(t)), where p j(t) and ν j(t) represent the particles position and velocity, m p is the mass of a single particle, while M f (x,v, t) is represented analytically. The transport step produces the solution T t(f(x, v, t)) = (1 β(x v t, t))f p (x v t, v, t) + β(x v t, t)m f (x v t, v, t). (1 β(x v t,t))f p (x v t, v, t) corresponds to solve a simple particle shift for the Monte Carlo samples. β(x v t, t)m f (x v t, v, t) corresponds to a Maxwellian shift. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

17 Hybrid methods The hybrid method A simple hybrid scheme II For the moments U H (x, t + t) we have U H (x, t + t) = P(T t (f(x, v, t))) = P(T t ((1 β(x, t))f p (x, v, t))) + P(T t (β(x, t)m f (x, v, t))) = U p (x, t + t) + U K (x, t + t). In particular U K (x, t + t) corresponds to the approximation of the Euler solution provided by a kinetic scheme. We state the following result 5 Theorem If we denote with U E (x, t + t) the solution of the Euler equations with initial data U E (x, t) = P(β(x, t)m f (x, v, t)) we have U E (x, t + t) = U K (x, t + t) + O( t 2 ). 5 B. Perthame SINUM 92, G.D. L. Pareschi SISC 10 Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

18 Hybrid methods The hybrid method A simple hybrid scheme III We can then replace the hybrid solution for the moments after the transport with Ũ H (x,t + t) = U p (x, t + t) + U E (x,t + t). This values are used to compute the new Maxwellian M H f (x, v, t + t). The next relaxation step takes the form where we set R t(t t(f(x, v, t))) = λt t(f(x,v, t)) + (1 λ)m H f (x,v, t + t) = = (1 β(x, t + t))f p (x, v, t + t) + β(x, t + t)m H f (x,v, t + t), β(x, t + t) = 1 λ, f p (x,v, t + t) = T t(f(x, v, t)). To compute the new particle fraction we need to sample particles from T t(β(x,t)m f (x, v, t)). This can be realized transforming the Maxwellian part β(x, t)m f (x, v, t) into samples and then advecting the whole set of samples. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

19 Hybrid methods The hybrid method Improvements to the simple hybrid scheme. Part I At each time step me must solve the full BGK model with a Monte Carlo scheme together with a suitable deterministic solver for the Euler equation. We can improve the efficiency observing that we do not need to transform into samples the whole Maxwellian part but only a fraction λ max x{λ(x, t + t)}. In fact at the next relaxation step a fraction β(x,t + t) of samples is discarded in each cell. Thus we need only (1 β(x,t + t))t t(β(x, t)m p f (x,v, t)) = λ(x, t + t)t t(β(x,t)mp f (x, v, t)) advected Maxwellian particles. Since λ vanishes as ε/ t 0, then the number of samples effectively used is a decreasing function of the ratio between the Knudsen number and the time step. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

20 Hybrid methods The hybrid method Improvements to the simple hybrid scheme. Part II The method just described does not take into account the possibility to optimize the equilibrium fraction by increasing its value in time and make it independent on the choice of the time step. In fact at each time step the equilibrium structure is entirely lost and the new fraction of equilibrium is only given by the relaxation step. However, it is possible to recover some information from the transported local Maxwellian although we know it through samples rather than analytically. Thus, if we define T t(β(x,t)m f (x,v, t)) w c, T t(β(x, t)m (x, v, t+ t) = Mf H f (x, v, t)) Mf H (x,v, t + t) (x, v, t + t) 1, T t(β(x, t)m f (x, v, t)) Mf H (x,v, t + t) the new function β becomes β c (x,t + t) = min{w c (x, v, t + t)}. v This new value is used in the simple hybrid scheme to improve the results. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

21 Hybrid methods Numerical results Accuracy test Smooth solution in 1D (velocity and space) with periodic boundary conditions. L 1 norm of the errors for temperature with respect to different value of the Knudsen number ε (in units of 10 2 ). Computational cost. ε = 10 2 ε = 10 3 ε = ε = 10 4 MCM FSI FSI N = 200 particles for cell x = ε = 10 2 ε = 10 3 ε = 10 4 ε = 10 5 MCM N= sec 25 sec 27 sec 26 sec FSI N= sec 22 sec 3 sec 0.6 sec FSI1 N= sec 17 sec 2 sec 0.6 sec FSI N=500 9 sec 8 sec 0.4 sec 0.3 sec FSI N=500 7 sec 6 sec 0.4 sec 0.3 sec x = Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

22 N N Hybrid methods N N Numerical results Computational cost II Smooth solution in 1D (velocity and space) with periodic boundary conditions. x 10 4 Number of particles in the domain for Knudsen=0.01 x 10 4 Number of particles in the domain for Knudsen= FSI1 FSI MC 2 FSI1 FSI MC time time x 10 4 Number of particles in the domain for Knudsen= x 10 4 Number of particles in the domain for Knudsen= FSI1 FSI MC FSI1 FSI MC time time Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

23 Hybrid methods Numerical results BGK equation: flow past an ellipse Comparison of results for T, DSMC (left), FSI (right) ε is such that 50% of the solution is represented by particles in HM. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

24 Hybrid methods Numerical results BGK equation: flow past an ellipse Comparison of results for T, DSMC (left), FSI (right) The fluid limit ε 0. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

25 Moment Guided Monte Carlo Outline 1 Introduction 2 Hybrid methods 3 Moment Guided Monte Carlo Basic principles The MG method Numerical results Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

26 Moment Guided Monte Carlo Basic principles Basic principles The basic idea consists in obtaining reduced variance Monte Carlo methods forcing particles to match prescribed sets of moments given by the solution of deterministic macroscopic fluid equations 6. These macroscopic models, in order to represent the correct physics for all range of Knudsen numbers include a kinetic correction term, which takes into account departures from thermodynamical equilibrium. We introduce an hybrid setting which permits to entirely remove the resolution of the kinetic equation in the limit of infinite number of collisions and to consider only the solution of the compressible Euler equation. We deal with the full Boltzmann operator. Extensions to other collisional kernels are possible (ex. Fokker-Planck). 6 G.D., P.Degond, L.Pareschi, 09 Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

27 Moment Guided Monte Carlo Basic principles The setting We define U = m(v)f(v)dv = (ρ, ρu, E). R 3 The starting point of the method is the following micro-macro decomposition f = M[U] + g. The function g represents the non-equilibrium part of the distribution function. From the definition above, it follows that g is in general non positive. Moreover since f and M[U] share the same first three moments we have mg = 0. Now U and g satisfy the coupled system of equations t U + x F(U) + x vmg = 0, t f + v x f = 1 Q(f, f). ε Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

28 Moment Guided Monte Carlo Basic principles The setting II Some remarks Solving the Boltzmann equation is sufficient to compute the solution of the moment system. However, the solution of the Boltzmann equation is very expensive and thus we seek for very fast solvers, as for instance DSMC. The counterpart is that the solution is known with very low accuracy. If the departure from equilibrium g is small, the low accuracy and the high level of numerical noise with which g and f are known by using Monte Carlo methods, will still permit to compute the moments with an error which is smaller than the error with which the kinetic equation is solved. We will numerically show that this is indeed the case. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

29 Moment Guided Monte Carlo Basic principles The setting III The method can be summarized as following: at each time step t n 1 Solve the kinetic equation with a Monte Carlo scheme and obtain a first set of moments U = mf. 2 Solve the fluid equations with a finite volume/difference scheme using particles to close the system and obtain a second set of moments U n+1. This means evaluate the term x vmg with particles. 3 Match the moments of the kinetic solution with the fluid solution through a transformation of the samples values f n+1 = T(f ) so that mf n+1 = U n+1 and continue. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

30 Moment Guided Monte Carlo The MG method The Monte Carlo method The starting point is an operator splitting of in a time interval [0, t] between collision and free transport. In Monte Carlo simulations the distribution function f is discretized by a finite set of N particles f = mp N N α i(t) δ(x X i(t))δ(v V i(t)), i=1 where α i(t) is the weight to associate to each particle and constant m p is defined at the beginning of the computation as m p = 1 f(t = 0)dvdx. During the transport N Ω R 3 step, the particles move to their next positions according to X i(t + t) = X i(t) + V i(t) t while α i(t) remains constant. The role of the function α will be clarified in the moment matching procedure. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

31 Moment Guided Monte Carlo The MG method Solution of the Boltzmann collisional operator We seek for a method which will be stable for all choices of t independently of ε. This will permit to avoid the stiffness of the equation and at the same time to reduce the statistical fluctuations. We consider the homogeneous kinetic equation tf = 1 (P(f, f) µf), ε where P(f, f) = Q(f,f) + µf and µ > 0 is a constant such that P(f,f) 0. Typically µ is an estimate of the largest rate of the negative term in the Boltzmann operator. The homogeneous equation can be rewritten now, adding and subtracting M[U], as tf = µ ( ) P(f, f) M[U] + µ (M[U] f). ε µ ε We apply now to the reformulated equation an approach based on the so-called exponential integrators. We rewrite the homogenous equation as (f M[U])e µt/ε t = 1 ε (P(f, f) µm[u])eµt/ε. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

32 Moment Guided Monte Carlo The MG method Solution of the Boltzmann collisional operator II We consider now the family of methods characterized by F (i) = e ciµ t/ε f + µ t i 1 ( ) P(F (j), F (j) ) A ij(µ t/ε) M[U n ] ε µ j=1 ( ) + 1 e c iµ t/ε M[U n ], i = 1,..., ν f n+1 = e µ t/ε f + µ t ν ( ) P(F (i), F (i) ) W i(µ t/ε) M[U n ] ε µ i=1 ( + 1 e µ t/ε) M[U n ], where f is the distribution function value after the transport step, F (i) are called stages, c i 0, while the coefficients A ij and the weights W i are such that A ij(0) = a ij, W i(0) = w i, i, j = 1,..., ν with coefficients a ij, c i and weights w i given by a standard explicit Runge-Kutta method called the underlying method. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

33 Moment Guided Monte Carlo The MG method Solution of the Boltzmann collisional operator III In the sequel, among all possible choices, we use the a first order in time scheme which reads f n+1 = e µ t ε f + µ t ( ) µ t ε e ε P(f, f ) ) M[U n ] + (1 e µ t ε M[U n ], µ which is based on the first order explicit Euler Runge-Kutta method. Observe that this method permits to avoid the stiffness of the collisional operator. This is, in fact, unconditionally stable for all choices of time step t. The Monte Carlo interpretation of the scheme is the following. With probability A = e µ t ε collision step. the velocity of the particle does not change during the With probability B = µ t ε ε the particle occurs in one collision, the details of the collision are described by the operator P. e µ t With probability C = 1 A B the velocity of the particle is replaced by a particle sampled from the Maxwellian distribution M[U]. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

34 Moment Guided Monte Carlo The MG method Solution of the moment equations We consider the function g n known at time n, we have The scheme is based on tu + x F(U) + x vmg = 0. }{{} Euler equations Solving the set of compressible Euler equations with a second order MUSCL scheme. Introduce a filter to eliminate some fluctuations for the non equilibrium distribution g n. The smoothing permits to reduce the fluctuations but on the other hand it causes a degradation of the accuracy in the solution. In practice, a weak moving average filter is used. The non equilibrium flux is then discretized with the same second order MUSCL scheme used for computing the flux of the Euler equations. However, in this case, the numerical diffusion is taken equal to zero. This is because this term is at the leading order a diffusion term, and thus it does not need numerical diffusion for stability to be assured. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

35 Moment Guided Monte Carlo The MG method The moment matching-density We first consider the matching of the mass. Let us consider the set of particles X 1,..., X NIj inside the cell I j after the transport step. The corresponding mass is computed with a piecewise constant reconstruction as ρ n j = N Ij i I j m pα n 1 i, j = 1,.., N x In order to restore the mass we assign to each particle inside the cell j the new value α n i = ϱn j N Ij i = 1,.., N Ij, where ϱ n j is density computed from the solution of the moments equations. The above renormalization force the density to be closer to the value furnished by the exact solution of the problem. In fact, the moment equations furnish solutions which contain less fluctuations with respect to the DSMC method. The reason for this lower level of noise of the macroscopic equations is that only the perturbation from the equilibrium g is computed statistically and not the full solution as in the original DSMC method. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

36 Moment Guided Monte Carlo The MG method The moment matching-momentum and energy In the Monte Carlo setting these moments can be obtained by a piecewise constant reconstructions ũ n j = 1 ρ n j NI j N Ij i I j V i ẽ n j = 1 ρ n j NI j N Ij i I j 1 2 Vi 2. To match mean velocity and energy with those coming from the solution of the moment equations we apply the transformation which gives the new set of velocities Vi defined by Vi = (V i ũ n j )/c + u n ẽ n j j c = (ũn j )2 e n j, i = 1,..., NI (un j )2 j which gives the desired quantities. N 1 Ij V ρ n j NI j i I j i = u n j, 1 ρ n j NI j N Ij i I j 1 2 V i 2 = e n j. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

37 Moment Guided Monte Carlo The MG method Computing g from f. Part I We now discuss the last point of the method: the computation of g n from f n. We have g n j = f n j M[U n j ] = = e µ t ε f n 1 + µ t µ t P(f n 1, f n 1 ) ε e ε µ The above expression tells us that ( e µ t ε + µ t ε e The moments of g n j can be obtained as a contribution of two terms. ) µ t ε M[Uj n ]. The first term is obtained by reconstructing from the particles the moments. The second term is obtained by integrating over the velocity space the analytic expression of the Maxwellian distribution. This implies directly that the moments related to the second part does not contain any statistical error. In the limit ε 0 the contribution of the perturbation g goes to zero. As a consequence, the method resolves a macroscopic system which is as closer as the Knudsen number diminishes to the compressible Euler equation. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

38 Moment Guided Monte Carlo The MG method Computing g from f. Part II Finally, the discretized moment equations reads Uj Uj n t Uj t U n+1 j + ψ j+1/2(u n ) ψ j 1/2 (U n ) x Ψ j+1/2 (< vm = = 0, = Ψ j+1/2(< vm g n >) Ψ j 1/2 (< vm g n >) = x (e µ t ε ε + Ψ j 1/2 (< vm (e µ t ε + µ t ε x + µ t ε x e µ t µ t e ε ) ( g M) n >) ) ( g M) n >). where ψ and Ψ represents respectively a second order MUSCL flux discretization with and without numerical diffusion, while µ t (g ) n j = e ε f n 1 + µ t ε (e µ t ε µ t e ε P(f n 1,f n 1 ) µ + µ t ε µ t e ε As the system approaches the equilibrium, the contribution of the kinetic term vanishes even though it is evaluated through particles. This does not happen if we just compute the kinetic term from the particles. Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42 ) +

39 Moment Guided Monte Carlo Numerical results Sod Shock Tube Test: Knudsen number ε = Velocity, Knudsen number= Temperature, Knudsen number=0.01 MC MC rif MC MC rif 2 EULER rif 7 EULER rif u(x,t) T(x,t) x x 2.5 Velocity, Knudsen number= Temperature, Knudsen number=0.01 MG MC rif MG MC rif 2 EULER rif 7 EULER rif u(x,t) T(x,t) x x Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

40 Moment Guided Monte Carlo Numerical results Sod Shock Tube Test: Knudsen number ε = Velocity, Knudsen number= Temperature, Knudsen number=0.001 MC MC rif MC MC rif 2 EULER rif 7 EULER rif u(x,t) T(x,t) x x 2.5 Velocity, Knudsen number= Temperature, Knudsen number=0.001 MG MC rif MG MC rif 2 EULER rif 7 EULER rif u(x,t) T(x,t) x x Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

41 Moment Guided Monte Carlo Numerical results Sod Shock Tube Test: Knudsen number ε = Velocity, Knudsen number= Temperature, Knudsen number= MC MC rif MC MC rif 2 EULER rif 7 EULER rif u(x,t) T(x,t) x x 2.5 Velocity, Knudsen number= Temperature, Knudsen number= MG MC rif MG MC rif 2 EULER rif 7 EULER rif u(x,t) T(x,t) x x Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42

42 Moment Guided Monte Carlo Numerical results Accuracy test-density Smooth solution in 1D (velocity and space) with periodic boundary conditions Density statistical error for MC method 10 1 Density statistical error for MG method 10 0 ε=1e 2 ε=1e 3 ε=1e ε=1e 2 ε=1e 3 ε=1e Σ (ρ) Σ (ρ) particles number N particles number N Giacomo Dimarco (Inst. Math. Toulouse) Hybrid and Moment Guided Monte Carlo Methods CEA-Saclay, 25 September / 42