Review of last lecture I t f + v x f = Q γ (f,f), (1) where f (x,v,t) is a nonegative function that represents the density of particles in position x at time t with velocity v. In (1) Q γ is the so-called granular collision operator, which is a quadratic integral operator descrbing the change in the density function due to creation and annihilation of particles in binary collisions: { v Q γ (f,f) = β (θ) w 1 } R R + J f(v )f(w ) v w f(v)f(w) dw Where J is the jacobian of the transformation (v,w ) (v,w). Here we consider (v,w ) (v,w), and v = 1 2 ( v + w ) + 1 2 ( v w ) h
Review of last lecture II w = 1 2 ( v + w ) 1 2 ( v w ) h h = 0 means the two particles become one after the collision and share the same velocity. And h = 1 corresponds the elasticity case. And we give the expression of h In this case J = h. h = 1 1 + θ
Splitting algorithm I The main part of our analysis will be devoted to the space homogeneous equation t f = Q γ (f,f), (2) It is well known, in fact, that by the standard splitting algorithm we may consider seperately the transport t f + v x f = 0, (3) and the relaxation given by (2). The overall accuracy of this simple splitting is first order in time. A second order generalization of this method is given by Strang s splitting [54]. Further discretization techniques can be found in [44].
Splitting algorithm II Since the main difficulties are represented by the discretization of the collision operator, it is clear that after the splitting the most difficult part relies in the approximation of the collision step (2). For the operator Q γ, there is conservation both of the mass ρ(x,t) = f(x,v,t)dv, and of the momentum u(x,t) = while the energy is dissipated. E (x,t) = R R R vf(x,v,t)dv v 2 f(x,v,t)dv
A spectral method I In this section we derive the Fourier spectral method for the homogeneous Boltzmann equation (7.2)-(7.9) following [43,46]. To this aim we define with, the inner product in L 1 (R) and consider the weak form of the equation t f,ϕ = Q γ (f,f),ϕ = β (θ) v w [ ϕ ( v ) ϕ(v) ] R + R 2 f(v,t)f(w,t)dwdθ, for t > 0 and all test functions ϕ. Here (v,w) (v,w ), that is, we have v = 1 2 (v + w) + 1 (v w) h (4) 2 w = 1 2 (v + w) 1 (v w) h (5) 2
A spectral method II A simple change of variables permits to write Q γ (f,f),ϕ = β (θ) q [ ϕ(v + q + ) ϕ(v) ] R + R 2 f(v,t)f(v + q,t)dqdθdv, where q = w v is the relative velocity, and the vectors q + and q that parametrize the post-collisional velocities are given by q + = q ( 1 + 1 ), q = q ( 1 1 ). 2 1 + θ 2 1 + θ We point out that the possibility to integrate the collision operator over the relative velocity is essential in the derivation of the method. We consider now an initial density function f 0 (v,t) with compact support Supp (f 0 (v,t)) [ R,R]. The solution to (2)
A spectral method III has compact support for later times. In fact by (4,5), if v, w R then v 1 2 v (1 + h) + 1 w (1 h) R 2 and similary we get w R. In addition q, q +, q 2R, thus we have the following Lemma 1 If the function f(v,t) is such that Supp(f(v,t)) [ R,R] then i) Supp(Q γ ( f,f)(v,t)) [ R,R], ii) Q γ (f,f),ϕ = β (θ) q [ ϕ(v + q + ) ϕ(v) ] R + v R q 2R f(v,t)f(v + q,t)dqdθdv,
A spectral method IV with v + q +, v + q [ 3R,3R]. Remark: The previous result shows that for compactly supported functions f in order to evaluate Q γ (f,f) by a psectral method without aliasing error we can consider the density function f restricted on the interval [ 2R,2R], and extend it by periodicity to a periodic function on [ 2R,2R]. Spectral projection of the equation To simplify the notation let us take 2R = π. The approximate function f N is represented as the truncated Fourier series f N (v) = N k= N ˆf k = 1 f(v)e ikv dv. 2π [ π,π] ˆf k e ikv, (6)
A spectral method V A Fourier-Galerkin method is obtained by considering the projection of the homogeneous Boltzmann equation on the space of trigonometric polynomials of degree N. Hence, taking f = f N and ϕ = e ikv for k = N,,N we have [ t f N Q γ (f N,f N )]e ikv dv = 0. (7) [ π,π] By substituting expression (6) into (7) we get a set of ordinary differential equations for the Fourier coefficients t ˆfk = N l,m= N l+m=k ˆf l ˆfm ˆβ (l,m) δk 2 ˆfk, k = N,,N,
A spectral method VI where the Boltzmann kernel modes ˆβ (l,m) are given by ˆβ (l,m) = dθβ (θ) q [ cos ( lq + mq ) cos (lq) ] dq. R + q π In fact, by evaluating (7.14) for ϕ = e ikv and f = f N, one obtains [ ] ˆβ (l,m) = dθβ (θ) q e ikq+ 1 e ilq dq, (8) R + q π and (8) follows by using the parities of the trignometric functions. Note that (8) is a real quantity completely independent of the argument v, dpending on just the particular kernel structure. This property is strictly related to the use of a Fourier spectral method. Other spectral methods may be developed, however they do not lead to this simplification.
A spectral method VII In practice all the information characterizing the kinetic eqation is now contained in the kernel modes. Clearly, these quantities can be computed in advance and then stored in a two-dimensional matrix of size 2N. Thanks to symmetry considerations the effective number of kernel modes that need to be computed and stored for the implementation of the method is reduced in practice since ˆβ (l,m) = ˆβ ( l, m). Note that an analytic expression for ˆβ(l,m) can be readily comuted as [ ˆβ (l,m) = β (θ)π 2 2Sinc (p) Sinc ( p) 2 2 R + 2Sinc (l) +Sinc ( ) l 2 ]dθ 2
A spectral method VIII where p = ((l m) + (l + m)h) /2 and Sinc(x) = sin(πx) /(πx). Finally we can rewrite scheme (7.18) as t ˆfk = N m= N ˆf k m ˆfm ˆβ (k m,m), k = N,,N, In the previous expression we assume that thefourier coefficents are extended to zero fo k > N. The eavaluation of (7.22) requires exactly O ( N 2) opeartions which is smaller than the cost of a standard method based on N parameters for f in the velocity space since we gain the integration over the variable Thus the straightforwrad evaluation of (7.22) is slightly less expensive than a usual discrete-velocity algorithm.
Monte Carlo Method I Prof. Pareschi s ppt PartIII.pdf.
Homework I where t f = Q(f,f) Q(f,f) = R R + β (θ) { v w 1 } J f(v )f(w ) v w f(v)f(w) dwd β (θ) = exp( θ), h = 1 θ 1 + θ, J = h v = 1 2 (v + w) + 1 (v w) h 2 w = 1 2 (v + w) + 1 (v w) h 2
Homework II ( f (v,0) = exp (2v 2) 2) ( + exp (2v + 2) 2). t = 0.5, t = 8 f (v) v [ π,π]. Monte Carlo
Monte Carlo Method for this hw I tf = Q(f,f) can be written as t f = Q(f,f) = Q + (f,f) µf where Q + (f,f) = β (θ) v w 1 R R + J f(v )f(w )dw = β (θ)dθ v w f(v )f(w )dw R + R π = β (θ)dθ v w f(v )f(w )dw R + π π µ = β (θ)dθ v w f(w)dw (9) R + π
Monte Carlo Method for this hw II R v w f(w)dw O ( N 2) Using Euler s method to discretize the Boltzmann equation yields f n+1 = (1 µ t)f n + µ t Q+ (f n,f n ) µ where µ is given by (9). But µ is a integral and directly computing will result into O ( N 2) operation for all the v. v [ π,π]. v w 2π = κ. And we can give an estimation of µ π µ = β (θ)dθ v w f(w)dw R + π π β (θ)dθκ f(w)dw R + π
Monte Carlo Method for this hw III = κρ β (θ)dθ R + And we use the notation = κρ β (θ)dθ R + Now we can modify the Algorithm [Nanbu-Babovsky for VHS molecules] on page 21 of PartIII.pdf (Prof. Pareschi) to solve our problem. 1. Compute the initial velocity of the particles, { v 0 i,i = 1,,N } by sampling them from the initial density f 0 (v) 2. set time counter t c = 0 3. for n=1 to n tot - Set N c = Iround ( N 2 t)
Monte Carlo Method for this hw IV - select N c random pairs (i,j) uniformly within all possible N (N 1) /2 pairs for each pair, find a θ by sampling from the density β (θ)/ R β (θ)dθ. + - Compute the relative velocity B ij = v i w j - get a random number r by sampling a uniformly distributed density function on the interval [0,κ]. - if r < B ij perform the collision between i and j, and compute v i and according to the collision law v j v = 1 2 (v + w) + 1 (v w)h 2 w = 1 2 (v + w) 1 (v w)h 2
Monte Carlo Method for this hw V set vi n+1 = v i, and vn+1 j = v j - set vi n+1 = vi n for all the particles that did not collide. end for