A Stochastic Galerkin Method for the Fokker-Planck-Landau Equation with Random Uncertainties

Transcription

1 A Stochastic Galerkin Method for the Fokker-Planck-Landau Equation with Random Uncertainties Jingwei Hu, Shi Jin and Ruiwen Shu Abstract We propose a generalized polynomial chaos based stochastic Galerkin method (gpc-sg) for the Fokker-Planck-Landau (FPL) equation with random uncertainties. The method can handle uncertainties from initial or boundary data and the neutralizing background. By a gpc epansion and the Galerkin projection, we convert the FPL equation with uncertainty into a system of deterministic equations. A consistency result is proven for the approimation of the collision operator. To compute efficiently the collision kernel under the gpc epansion, we use a singular value decomposition (SVD) combined with a fast spectral method for the collision operator. For high dimensional random inputs, we adopt a sparse basis and use the sparsity of a set of basis related coefficients and the La-Friedrichs splitting to avoid all the SVD involved. Numerical eperiments verify the efficiency of the gpc-sg method. Introduction First derived by Landau [5] as the grazing collision limit of the Boltzmann equation, the Fokker-Planck-Landau (FPL) or Landau equation is a collisional kinetic Jingwei Hu Department of Mathematics, Purdue University, West Lafayette, IN 4797, USA jingweihu@purdue.edu Shi Jin Department of Mathematics, University of Wisconsin-Madison, Madison, WI 5376; Department of Mathematics, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 224, China jin@math.wisc.edu Ruiwen Shu Department of Mathematics, University of Wisconsin-Madison, Madison, WI rshu2@math.wisc.edu

2 2 Jingwei Hu, Shi Jin and Ruiwen Shu model that describes the non-equilibrium dynamics of charged particles in a plasma [4]. Let f = f (t,,v) be the density distribution function of particles, where t is the time, is the space, and v is the velocity. The FPL equation with the mean-field term (also known as the Vlasov-Poisson-Landau equation) reads t f + v f + E(t,) v f = Q( f, f ), t >, Ω R d, v R d v, () where E(t,) is the electric field given by E(t,) = φ(t,), (2) and φ(t,) is a self-consistent electrostatic potential function satisfying the Poisson equation φ(t,) = µ() f (t,,v)dv, (3) R dv where µ() is a neutralizing background satisfying µ()d = R d R d R dv f (t,,v)dvd. (4) Q( f, f ) on the right-hand side of () is the FPL collision operator that models binary interactions among particles: Q( f, f )(v) = v A(v v )[ f (v ) v f (v) f (v) v f (v )]dv. (5) R dv Here A(w) is a semi-positive definite matri defined by ( A(w) = Ψ(w) I w w ) w 2, (6) where I is the identity matri. For inverse-power law potentials, Ψ(w) = w γ+2 with 3 γ. The case γ = 3 corresponds to the Coulomb interaction which is of primary importance in plasma applications. The collision operator Q( f, f ) possesses some important physical properties: it preserves mass, momentum and energy R dv Q( f, f )φ(v)dv =, φ(v) =,v, v 2 ; (7) and satisfies the entropy dissipation inequality (the H-theorem) R dv Q( f, f )log f dv. (8) The equality only holds when f attains the local equilibrium (Mawellian)

3 Fokker-Planck-Landau Equation with Uncertainties 3 f (v) = M(v) = ρ (v u) 2 (2πT ) d v/2 e 2T, (9) where ρ,u,t are the density, bulk velocity and temperature defined as ρ = f dv, R dv u = ρ R dv v f dv, T = d v ρ R dv (v u) 2 f dv. () The equation () needs to be supplemented with appropriate initial condition f (,,v) = f (,v), () where f can be chosen as, for eample, the local equilibrium (9). For boundary condition, a commonly used one is the Mawell boundary condition: for any boundary point Ω, let n() be the unit inward normal vector to the boundary, then the in-flow boundary condition is specified as g(t,,v) := ( α) f (t,,v 2(v n)n) + f (t,,v) = g(t,,v), v n >, (2) α (2π) dv 2 T dv+ 2 w e 2Tw v2 f (t,,v) v n dv, v n< where T w = T w (t,) is the temperature of the wall (boundary). The constant α ( α ) is the accommodation coefficient with α = corresponding to the purely diffusive boundary, and α = the purely specular reflective boundary. In the past decades, the FPL equation () has been studied etensively both theoretically and numerically. The readers are referred to [4] for a review of the main mathematical aspects, and the recent paper [] and references therein for relevant numerical methods. In spite of the vast amount of eisting research, the study of the FPL equation has mostly remained deterministic and ignored uncertainty. In reality, however, there are many sources of uncertainties that can arise in this equation: imprecise measurements for initial boundary conditions and physical parameters; incomplete knowledge of the fundamental interaction mechanism between particles, and so on. Understanding the impact of these uncertainties is critical to the simulations of comple plasma systems, and will allow scientists and engineers to obtain more reliable predictions and perform better risk assessment. The goal of this paper is to develop an efficient stochastic numerical method for uncertainty quantification (UQ) of the FPL equation (). The basic framework of our work is built on a probabilistic approach which models the uncertain parameters as random variables. In the FPL equation (), this amounts to consider the distribution function (3) f = f (t,,v,z), z R d, (4)

4 4 Jingwei Hu, Shi Jin and Ruiwen Shu now depending on an etra argument z a d-dimensional random vector with support collecting all possible uncertainties in the system. For instance, one may consider z = (z µ,z ini,z bdry ), where z µ, z ini, z bdry denote, respectively, the random parameters arising from the neutralizing background (3): µ = µ(,z µ ); the initial condition (): f (,,v,z) = f (,v,z ini ) for Ω; the boundary condition (2): f (t,,v,z) = g(t,,v,z bdry ) for Ω. We will further assume the components of z are already mutually independent random variables obtained through some dimension reduction technique, e.g., arhunen- Loève epansion [7], and do not pursue the issue of random input parameterization in this paper. To properly model the propagation of uncertainties, we adopt the generalized polynomial chaos based stochastic Galerkin (gpc-sg) method, which is widely used in the UQ simulations nowadays [2, 5, 8,, 6, 3]. Simply speaking, this method seeks to approimate the unknown function f via an orthogonal polynomial series: f (t,,v,z) k= f k (t,,v)φ k (z) := f (t,,v,z), f k (t,,v) = f (t,,v,z)φ k (z)π(z)dz, (5) where {Φ k (z)} is a set of d-variate polynomials of degree up to m which satisfy Φ j (z)φ k (z)π(z)dz = δ jk, j,k, with π(z) being the probability distribution of z and δ jk the ronecker delta function. The number of basis functions is = ( m+d) m. Equipped with this gpc representation, one then proceeds as follows: ) Substitute the epansion (5) into the original equation and conduct a Galerkin projection. This usually results in a system of coupled deterministic equations for the gpc coefficients { f k } k= requiring different treatment from the corresponding deterministic equation. 2) Solve the gpc system. 3) Use { f k } k= to reconstruct the solution in via (5), or construct the solution statistics directly, e.g., the mean and standard deviation can be retrieved as k=2 E[ f ] = f, S[ f ] = f 2 k. (6) For the FPL equation, the main difficulty associated with solving the gpc-sg system lies in the evaluation of the nonlinear collision operator. Similar to the work [4], we propose a fast algorithm to efficiently compute the collision operator under the Galerkin projection. The acceleration is achieved by combining a singular value decomposition (SVD) of the collision kernel with the fast spectral method in the deterministic case [9]. In the cases where the random domain is high dimensional, i.e., d is large, the usual gpc-sg method may fail to be affordable since the number of basis functions

5 Fokker-Planck-Landau Equation with Uncertainties 5 = ( m+d) m is too large. To circumvent this difficulty, we use the sparse wavelet basis as in our previous work [3]. We use N level hierarchical piecewise polynomial functions of degree at most m as basis functions in one dimension, and use a standard sparse grid construction to obtain basis functions in d-dimensional random spaces. With this basis one can achieve an accuracy of O(N d 2 N(m+) ) with = O((m + ) d 2 N N d ) basis functions. The accuracy is O( (m+) (log) (m+2)(d ) ) in terms of. This method is much more efficient than the usual gpc-sg method if d is large. When using the sparse grid method, can still be too large to make an SVD of order affordable. Thus the following two difficulties arise. The first one is that one can no longer afford the SVD approach for the collision operator. To avoid it we notice the sparsity of a basis related tensor S b,i jk proved in [3]. As a result, one can compute the collision operator directly with low computational cost. The second difficulty is that a direct computation of the numerical flu for the mean-field term requires a diagonalization of constant flu matrices of order. To avoid this diagonalization, we utilize the local La-Friedrichs splitting [6]. In this way one can compute the second-order upwind flu without diagonalization of the flu matrices. The rest of this paper is organized as follows. Section 2 describes in detail the gpc-sg method for the FPL equation with uncertainty. Section 3 discusses the spatial and time discretization. In section 4 we give a consistency analysis of the gpcsg method for the collision operator. In section 5 we give a sparse wavelet method for problems with high-dimensional random inputs. Etensive numerical results are presented in section 6. Finally the paper is concluded in section 7. 2 The gpc-sg method for the FPL equation with uncertainties In this section we describe the gpc-sg method for the FPL equation with uncertainty. We start by substituting the truncated gpc epansion (5) into equation (). Upon a standard Galerkin projection, this yields a system of equations for the gpc coefficients f k : t f k (t,,v)+v f k (t,,v) + v E(t,,z) f (t,,v,z)φ k (z)π(z)dz (7) = Q k ( f, f )(t,,v), k, where Q k ( f, f ), the k-th mode of the collision operator, is defined as Q k ( f, f ) := Q( f, f )(t,,v,z)φ k (z)π(z)dz. (8) To simplify the forcing term, note that E k (t,) = φ k (t,), φ k (t,) = µ k () f k (t,,v)dv, R dv (9)

6 6 Jingwei Hu, Shi Jin and Ruiwen Shu where µ k () = µ(,z)φ k (z)π(z)dz are the gpc coefficients of the neutralizing background µ. Then the integral term in (7) becomes ( )( ) E i (t,)φ i (z) f j (t,,v)φ j (z) Φ k (z)π(z)dz = A k j (t,) f j (t,,v), i= (2) with A k j (t,) := S i jk E i (t,), S i jk = Φ i (z)φ j (z)φ k (z)π(z)dz. (2) i= To simplify the collision term, we define the bilinear FPL collision operator as Q( f,g)(v) = v A(v v )( f (v ) v g(v) f (v) v g(v ))dv, (22) R dv Then the collision term (8) can be epressed as Q k ( f, f ) = i, S i jk Q( f i, f j ). (23) Due to the double summation in (23), a direct evaluation of the collision operator Q k would be very epensive. To reduce the computational cost, we follow the approach proposed in [4]. Specifically, we precompute the singular value decomposition (SVD) of the matri {S i jk } i j for each k: S i jk = R k r= U k irv k r j, (24) where R k is the numerical rank of the matri. Plugging (24) into (23) and rearranging terms give Q k ( f, f ) = R k r= ( ) Q g k r,h k r, g k r := i= U k ir f i, h k r := V k r j f j. (25) Therefore, we reduce the original double summation into a single one. To compute the bilinear term Q ( g k r,h k r), we apply the fast spectral method introduced in [9] for the deterministic FPL collision operator. See Appendi for a brief description of this method. The numerical compleity of such a computation is O(N d v v logn v ) where N v is the number of mesh points in each velocity direction, and d v is the dimension of the velocity space. Thus, for each k, the cost of computing Q k is O(R k N d v v logn v ) with R k, and = ( m+d) m is the dimension of d-variate polynomials of degree up to m (note that the direct evaluation of Q k based on (23) requires O( 2 N 2d v v ) operations). The initial data is given by

7 Fokker-Planck-Landau Equation with Uncertainties 7 f k (,,v) = f k (,v) = f (,v,z)φ k (z)π(z)dz. (26) The Mawell boundary condition is given by f k (t,,v) = g k (t,,v), Ω, v n >, (27) with n the inward normal of Ω, and g k (t,,v) := g(t,,v,z)φ k (z)π(z)dz. (28) We consider the case where the wall temperature T w and the accommodation coefficient α may depend on z. We assume that α(z) = k= α kφ k (z). Then g(t,,v,z) :=( α(z)) f (t,,v 2(v n)n,z) α(z) + e v2 (2π) dv 2 T w (,z) dv+ 2Tw(,z) 2 Substitute into (28) one gets g k = + v n< f (t,,v,z) v n dv. ( ) ( α(z))φ k (z)φ j (z)π(z)dz f j (t,,v 2(v n)n) D k j (,v) f j (t,,v) v n dv, v n< (29) (3) where D k j (,v) = α(z) (2π) (dv )/2 T w (,z) (d v+)/2 e 2Tw(,z) Φ k (z)φ j (z)π(z)dz, (3) is a matri that is time independent hence can be pre-computed. v 2 3 The spatial and time discretization In order to solve the Galerkin system (7), we split it into three steps: t f k + v f k =, t f k + A k j (t,) v f j =, t f k = Q k ( f, f ). (32) Note that each A k j is a vector of length d v. To achieve second-order accuracy in time, we use the Strang splitting, and the second-order Runge-utta method for each step.

8 8 Jingwei Hu, Shi Jin and Ruiwen Shu For the transport step, we employ a second order MUSCL scheme with the minmod slope limiter [6]. For the forcing step, we discuss the case d v = for simplicity. The general case follows by computing the flues dimension by dimension. In the case d v =, for each fied, since (A k j ) is a symmetric matri depending on but not on v, the equation becomes a system of linear hyperbolic equations in v with constant characteristic speeds which can be solved by upwind schemes. Thus we can diagonalize the matri A, find the Riemann invariants, and use the MUSCL scheme on each Riemann invariant. To be precise, suppose A is written as A = P DP, where P = (P k j ) is an invertible matri, and D is a diagonal matri. Then the forcing step equations can be written as t f k + D kk v f k =, where f k = P k j f j. These equations in f k are hyperbolic with constant characteristic speeds, therefore can be solved by the MUSCL scheme. And then f k is computed by f k = (P ) k j f j. For the collision step, we use the fast algorithm mentioned above to compute Q k. To choose the time step t, we notice first that it has to satisfy the CFL condition from the transport step, which is t R v, where R v is the largest possible characteristic speed. In addition, it has to satisfy the CFL condition from the forcing step, which is t v C, where the constant C = ma,z E(t,,z) is the maimum of the electric field. Furthermore, due to the parabolic nature of the FPL collision operator, one has the following constraint for the collision step t v2 C 2, where the constant C 2 ma,z R dv A(v v ) f (t,,v,z)dv is the maimum of the strength of diffusion of the collision operator. Thus one should choose t to satisfy the three restrictions. 4 Consistency analysis of the gpc-sg method for the collision operator Here we give a consistency analysis of the gpc-sg method for the FPL collision operator. For simplicity, the random variable z is assumed to be one-dimensional in this section. Suppose the eact solution to the spatial homogeneous FPL equation is t f = Q( f, f ), (33)

9 Fokker-Planck-Landau Equation with Uncertainties 9 f (t,v,z) = k= Given the gpc approimation of f : f k (t,v)φ k (z), f k (t,v) = f (t,v,z)φ k (z)π(z)dz. (34) f f (t,v,z) = k= f k (t,v)φ k (z), (35) To analyze the consistency of the gpc-sg method, one substitutes the eact solution f into the scheme t f k Q k ( f, f ), (36) and estimate the difference of the LHS and the RHS. Since f solves the equation (33), one has t f k = Q k ( f, f ). (37) Thus it suffices to analyze Q k ( f, f ) Q k ( f, f ), the numerical truncation error of the collision operator. We will use the following lemma proved by Pareschi et al. []: Lemma. Let g,h L 2 v, then We estimate the error of collision operator as follows: Q(g,h) L 2 v C h L v g H 2 v. (38) Q k ( f, f ) Q k ( f, f ) 2 = [Q( f, f ) Q( f, f 2 )]Φ k (z)π(z)dz Q( f, f ) Q( f, f ) 2 π(z)dz Φ k (z) 2 π(z)dz. Notice Q( f, f ) Q( f, f ) 2 = Q( f, f f ) Q( f f, f ) 2 2[ Q( f, f f ) 2 + Q( f f, f ) 2 ], Then one gets Q k ( f, f ) Q k ( f, f ) 2 [ 2 Q( f, f f ) 2 + Q( f f, f ) 2] π(z)dz. Integrating in v and using the lemma, we get (39)

10 Jingwei Hu, Shi Jin and Ruiwen Shu Q k ( f, f ) Q k ( f, f ) 2 Lv 2 C ( f 2 L I f f 2 v H 2 + f f 2 v L f 2 v H 2)π(z)dz v z C ( f f 2 H I 2 + f f 2 v L )π(z)dz, v z where C will be a generic positive constant in the sequel. The second inequality above is obtained by assuming that the L v and H 2 v norms of f are bounded, and those norms of f are uniformly bounded in. Also, notice that f f C N N, N, (4) in which the norms are Lv or Hv 2. The term C N N comes from the spectral accuracy of the projection operator, assuming that f Hv N+2. Plug into (4), we end up with the estimate Q k ( f, f ) Q k ( f, f ) 2 Lv 2 C N 2N. (4) which shows the spectral consistency of the gpc-sg method for the collision operator. 5 A remark on high dimensional random spaces If the random space is high dimensional, the usual gpc epansion, which requires = ( m+d) d basis functions where m is the maimal degree of polynomials and d is the dimension of the random space, can be prohibitively epensive. To handle this difficulty, we adopt the sparse technique we proposed in [3]. Using locally supported piecewise polynomials and a hierarchical construction, this technique gives a basis with = O((m + ) d 2 N N d ) basis functions, where N is the number of hierarchical levels, and m is the maimal degree of polynomials. The accuracy is O(N d 2 N(m+) ), which is O( (m+) (log) (m+2)(d ) ) in terms of. With this sparse basis, the number of basis can still be moderately large so that the SVD method for the collision operator as well as the diagonalization of the forcing term matri A in (32) are no longer affordable. To avoid the SVD for the collision operator computation, we follow [3] and compute Q k = i, S i jkq( f i, f j ) directly. The following sparsity result was proven: the number of pairs (i, j) for which there is at least one k with S i jk is no more than O((m + ) 2d 2 2N N d+ ), compared to the total number of pairs O((m + ) 2d 2 2N N 2d 2 ). Only for such pairs it is required to compute Q( f i, f j ), thus the computational cost for Q k is still greatly reduced if N and d are large. To avoid the diagonalization of the forcing term matri A, we discuss the case d v = for simplicity. The cases with larger d v can be treated dimension by dimension. In the case of d v = we use the local La-Friedrichs splitting for the second equation of (32) as follows:

11 Fokker-Planck-Landau Equation with Uncertainties t f + 2 (A() β()i ) vf + 2 (A() + β()i ) vf =, (42) where f = ( f,..., f ), I is the identity matri of order, and β() is a local (in each cell) upper bound of the absolute values of the eigenvalues of the symmetric matri A(). The eigenvalues of the first flu matri (A() β()i ) are all negative, while those of the second one are all positive. Thus one can use a second order upwind scheme with the minmod slope limiter on each flu terms without diagonalizing the matrices. 6 Numerical results In all the numerical eamples here, ecept for the Landau damping, we take the physical domain to be the one-dimensional (d = ) interval [,] and the velocity domain to be two-dimensional (d v = 2). In all the eamples, ecept for the 6-dimensional random domain eample, we take the periodic boundary condition. We discretize the physical domain into N grid points uniformly: ( j = j ), =, j =,...,N. 2 N The velocity domain is truncated into [ R v,r v ] 2 and discretized into N v points in each dimension: v j, j 2 = ( R v + ( j 2 v), R v + ( j 2 2 ) v), v = 2R v, j, j 2 =,...,N v. N v R v is big enough so [ R v,r v ] 2 contains the support of the solution. We assume the random variable z obeys uniform distribution on [,] d. In the first three eamples, we take d =. In the fourth eample we take d = 2. These eamples are computed by the gpc-sg method with the gpc basis being the normalized Legendre polynomials. For the last eample, d = 6, and we use the sparse method given in the previous section. 6. Random initial data: a shock tube problem We take the random initial data to be the equilibrium with macroscopic quantities { ρl = +.2 ( ) z+ 2, ul =, T l =,.5, ρ r =.25, u r =, T r =.25, >.5. We take N =, N v = 32, R v = 6, = 7, t =.,

12 2 Jingwei Hu, Shi Jin and Ruiwen Shu and compute the solution at t =. by the sg method. The result is compared with the solution by the stochastic collocation (sc) method with the same parameters and N z = Gauss-Legendre quadrature points, see Figure. To implement the sc method, we take N z Gauss-Legendre quadrature points {z j } N z in the random domain, and then solve the (deterministic) FPL equation at each z j. Finally the mean and standard deviation of any quantity f are computed by E[ f ] = N z f (z j )w j, S[ f ] = N z f (z j ) 2 w j (E[ f ]) 2, (43) where w j is the quadrature weight of the point z j. For the sc method, we verified that the solution with N z = 2 quadrature points is indistinguishable with the solution with N z = quadrature points. Therefore, the N z = solution is good enough as a reference solution. This is also true for other numerical eamples ecept the last one. One can see from Figure that the results of two methods agree well. This shows that the sg method has good accuracy.. E(rho).9 E(u) E(T) S(rho).25 S(u).2.6 S(T) Fig. Random initial data: epectation and standard deviation of macroscopic quantities. Solid line: sc, N =, N v = 32, R v = 6, N z =, t =.. Dots: sg, N =, N v = 32, R v = 6, = 7, t =..

13 Fokker-Planck-Landau Equation with Uncertainties The Landau damping We use the Landau damping to test our sg method for the forcing term. For simplicity we omit the collision term. The physical space is the interval [,4π] with periodic boundary condition, and the velocity domain is one-dimensional. The random initial condition is We take f (,v) = 2π ( + (.5 +.z)cos(.5))e v 2 2. N = 64, N v = 28, R v = 6, = 7, t =.3, for the sg method, and compare with the sc method with the same parameters and N z =. We compare the epectation and standard deviation of the magnitude of the electric field for t from to 9. It can be seen from Figure 2 that the results from two methods agree well, which shows the accuracy of the sg method. Since the uncertainty is small, the epectation is similar to the result of [2]. The standard deviation in both eamples also shows oscillation in time, and this needs further theoretical eplanations. Fig. 2 Landau damping: epectation and standard deviation. Solid line: sc, N = 64, N v = 28, R v = 6, N z =, t =.3. Dots: sg, N = 64, N v = 28, R v = 6, = 7, t =.3. Upper curve: epectation. Lower curve: standard deviation. L 2 (E) t 6.3 A random neutralizing background We take the deterministic initial data as the equilibrium with macroscopic quantities ρ = (2 + sin(2π))/3, u = (.2,), T = (3 + cos(2π))/4, (44) and the random background as µ(,z) = 2 ( +.2sin(4π)). (45) 3

14 4 Jingwei Hu, Shi Jin and Ruiwen Shu We take N =, N v = 32, R v = 8, = 7, t =., (46) and compare the solution by the sc method with the same parameters and N z = Gauss-Legendre quadrature points at t =.. One can see from Figure 3 that the results of two methods agree well, even for the standard deviations whose magnitude is small. This shows that the sg method can efficiently handle the uncertainties from the neutralizing background. E(rho).5 E(u).95 E(T) S(rho).2 4 S(u).55 4 S(T) Fig. 3 Random background: epectation and standard deviation of macroscopic quantities. Solid line: sc, N =, N v = 32, R v = 8, N z =, t =.. Dots: sg, N =, N v = 32, R v = 8, = 7, t = An eample with a two-dimensional random variable To demonstrate that our sg method is efficient for more than one random dimension, we give a test of our method on an eample with 2-dimensional random domain,z 2 = [,] 2. The gpc basis is taken to be {Φ k (z )Φ k2 (z 2 )} where Φ k (z) is the normalized Legendre polynomial of degree k, and k +k 2 m. The initial data is given by ( ) f (,v) = ρ () 4πT e v u () 2 2T () + e v+u () 2 2T (), ()

15 Fokker-Planck-Landau Equation with Uncertainties 5 where ρ (,z) = 3 u = (.2,), T (,z) = 4 ( 2 + sin(2π) + 2 sin(4π)z + ) 3 sin(6π)z 2, ( 3 + cos(2π) + 2 cos(4π)z + 3 cos(6π)z 2 ). The numerical parameters are N =, N v = 32, R v = 6, m = 5, t =., and the result is compared at t =. with the sc method with the same parameters and N z = collocation points in each dimension. The result is shown in Figure 4. It can be seen that the results of the two methods agree well, which shows the accuracy of the sg method for 2d random domains. E(rho).3 E(u).95 E(T) S(rho).4.6 S(u).55.8 S(T) Fig. 4 2-dimensional test with random initial data: epectation and standard deviation of macroscopic quantities. Solid line: sc, N =, N v = 32, R v = 6, N z =, t =.. Dots: sg, N =, N v = 32, R v = 6, m = 5, t =..

16 6 Jingwei Hu, Shi Jin and Ruiwen Shu 6.5 An eample with a 6-dimensional random domain We finally give an eample with a 6-dimensional random domain. To deal with the high dimensionality we use the sparse sg method mentioned in Section 5. We take the initial data as the equilibrium with ρ(,z) = + ep( (.5) 2 )sin((.5))(.5 +.z 2 ), u(,z) =, T = +.5ep( (.4.z ) 2 ), (47) and boundary data as the Mawell boundary with The random background is given by T w = +.2z 3, α =.5 +.3z 4. (48) µ(,z) = +.z 5 sin(2π) +.2z 6 cos(2π). (49) We choose numerical parameters as N =, N v = 32, R v = 8, t =.. We use the sparse basis with m =, N = 3 to solve the equation, and the result is shown in Figure 5. One can clearly see that near the center of the domain, the mean and standard deviation of the density and the temperature are diffused due to the kinetic transport term, and those of the velocity ehibit more complicated behavior due to the forcing term. The most interesting phenomena is that near the left boundary, the effect of the boundary condition is not influential on the mean, but is dominating the standard deviation. In fact, for all the three standard deviations, one can see that the uncertainty comes from boundary and propagates into the domain. Note that for this eample with 6 random dimensions, with the sparse approach, only 38 basis functions are needed. 7 Conclusion In this paper we propose a gpc based stochastic Galerkin method for the Fokker-Planck-Landau equation with random uncertainties. By a gpc epansion and Galerkin projection, we convert the FPL equation with uncertainty into a system of deterministic equations. We prove the consistency of the gpc-sg method for the collision operator, as well as accelerate the computation of the collision kernel by a singular value decomposition combined with a fast spectral method. We adopt the sparse method from [3] to handle high dimensional random inputs. To avoid the epensive SVD operations, we take advantage of the sparsity of the tensor S b,i jk for the computation of the collision operator and use a flu splitting for the mean-field term. Numerical results show the efficiency of the stochastic Galerkin method.

17 Fokker-Planck-Landau Equation with Uncertainties 7.25 E(rho). E(u).45 E(T) S(rho) t=.2 t=.4 t=.6 t= S(u).8.7 S(T) Fig. 5 6-dimensional random domain eample, using sparse sg: epectation and standard deviation of macroscopic quantities. N =, N v = 32, R v = 8, t =., m =, N = 3. Acknowledgements This research was supported by NSF grants DMS and DMS- 729: RNMS I-Net, by NSFC grant No , and by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation. The first author s research was partially supported by NSF grant DMS-6225 and a startup grant from Purdue University. References. Dimarco, G., Li, Q., Pareschi, L., Yan, B.: Numerical methods for plasma physics in collisional regimes. J. Plasma Phys. 8, 3 (25) 2. Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer- Verlag, New York (99) 3. Gunzburger, M.D., Webster, C.G., Zhang, G.: Stochastic finite element methods for partial differential equations with random input data. Acta Numer. 23, (24) 4. Hu, J., Jin, S.: A stochastic Galerkin method for the Boltzmann equation with uncertainty. J. Comput. Phys. 35, 5 68 (26) 5. Landau, L.: The transport equation in the case of the Coulomb interaction. In: Collected Papers of L.D. Landau, pp Pergamon press (98) 6. LeVeque, R.J.: Finite volume methods for hyperbolic problems. Cambridge university press (22) 7. Loève, M.: Probability Theory, fourth edn. Springer-Verlag, New York (977) 8. Maitre, O.P.L., nio, O.M.: Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Springer (2) 9. Pareschi, L., Russo, G., Toscani, G.: Fast spectral methods for the Fokker-Planck-Landau collision operator. J. Comput. Phys. 65, (2)

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