recursive time relaxed monte carlo methods for rarefied ...lrc/plasma-cargese/... · properties i)...

Recursive Time Relaxed MonteCarlo methods for rarefied gas

dynamicsL. Pareschi∗, S. Trazzi∗, B.Wennberg†

http://utenti.unife.it/lorenzo.pareschi/

(∗) Department of Mathematics

University of Ferrara, Italy

(†) Department of Mathematics

Chalmers Institute of Technology, Goteborg, Sweden

Outline

General notations

Time Relaxed (TR) discretizations

Wild sum expansions

TR schemes

Recursive Time Relaxed Monte Carlo (TRMC) methods

Derivation;

Algorithm for the computation of collisions

Technique for truncation of the collisional trees

Numerical results

Conclusions

Stefano Trazzi, Summer school on Mathematical modelling and computational challenges in plasma physics and applications (Cargese, october 26-30, 2004) – p.1/23

The Boltzmann Equation

The kinetic model is described by a density function f(x, v, t) ≥ 0 in the phase

space which satisfies the equation

∂f

∂t+ v · ∇f =

1

εQ(f, f)

ε > 0 : Knudsen number (proportional to mean free path between collision)Q(f, f) collisional operator: describes the binary collision between particles

Q(f, f)(v) =

Gain term︷︸︸︷

Q+(f, f)(v)−

Loss term︷︸︸︷

L[f ](v)f(v)

Q+(f, f)(v) =

∫

IR3

∫

S2σ(|v − v∗|, ω)f(v

′)f(v′∗) dω dv∗,

L[f ](v) =

∫

IR3

∫

S2σ(|v − v∗|, ω)f(v∗) dω dv∗


The Boltzmann equation II

ω: unitary vector of the sphere S2

(v′, v′∗)→ (v, v∗): collisional velocity

v′ =

1

2(v + v∗ + |v − v∗|ω), v

′∗ =

1

2(v + v∗ − |v − v∗|ω)

σ: scattering cross sectionExample : Model of Variable Hard Sphere (VHS) :

σ(|v − v∗|, θ) = Cα|v − v∗|α, Cα > 0,

α = 0: Case of Maxwellian molecules, α = 1: Hard Sphere.


The Splitting Method

Convection Step :∂f

∂t+ v · ∇f = 0

Collisional Step :∂f

∂t=

1

εQ(f, f)

If f is a probability density (i.e.∫f = 1) and µ > 0 is a suitable constant, it is

possible to rewrite the homogeneous Boltzmann equation in the form

∂f

∂t=1

ε[P (f, f)− µf ]

with P (f, f) = (µ− L[f ]) f +Q+(f, f), nonnegative bilinear operator.


Essential Bibliography

Fundamental references

G.A.BIRD, (1970)

K.NAMBU, (1980)

H.BABOVSKY, (1986)

Recent improvements (some)

...

S.RJASANOW, W.WAGNER, (1996) [SWPM]

R.E.CAFLISCH, L.PARESCHI, (1999) [Hybrid]

L.PARESCHI, G.RUSSO, (2001) [TRMC]

L.PARESCHI, B.WENNBERG, S.TRAZZI, (2004) [Recursive]


Time Relaxed (TR) discretizations

Wild sum expansion

Starting point for the construction of TR schemes.

Consider the differential system seen above

∂f

∂t=1

ε[P (f, f)− µf ]

with initial conditionf(v, 0) = f0(v)

It is possible to show that the function f satisfies the following formal expansion

f(v, t) = e−µt/ε

∞∑

k=0

(

1− e−µt/ε

)kfk(v). (1)

The functions fk are given by the recurrence formula for k = 0, 1, . . .

fk+1(v) =1

k + 1

k∑

h=0

1

µP (fh, fk−h) (2)


Propertiesi) Conservation : If the collision operator preserves some moments, then all the functions fk will

have the same moments, i.e. if, for some function φ(v)∫

IR3P (f, f)φ(v) dv = µ

∫

IR3f φ(v) dv,

then the coefficients fk defined by (2) are nonnegative and satisfy ∀ k > 0∫

IR3fkφ(v) dv =

∫

IR3f0φ(v) dv.

ii) Asymptotic behavior : If the sequence {fk}k≥0 defined by (2) is convergent, then (1) is welldefined for any value of ε. Moreover, if we denote by M(v) = limk→∞ fk then

limt→∞

f(v, t) =M(v),

in which M(v) is the local (Maxwellian) equilibrium.

Remark: If σB(|v − v1|, ω) ≤ B then the BE can be written in the above form, with µ = 4πρB s.t.

µ ≥ LB [f ](v) ≡

∫

IR3

∫

S2σB(|v − v1|, ω)f(v∗) dω dv∗

and P (f, f) = Q+B(f, f) + (µ− LB [f ])f .


TR schemesFrom the previous representation, the following class of numerical schemes is obtained

fn+1(v) = (1− τ)

m∑

k=0

τkfnk (v) + τm+1M(v),

where fn = f(n∆t),∆t is a small time interval, and τ = 1− e−µ∆t/ε (relaxed time).

Properties :

i) conservation : If the initial condition f0 is a non negative function, then fn+1 is nonnegative forany µ∆t/ε, and satisfies

∫

R3fn+1

φ(v) dv =

∫

R3fnφ(v) dv.

ii) asymptotic preservation : For any m ≥ 1, we have limµ∆t/ε→∞ fn+1 =M(v).

iii) accuracy : If supk>m{|fnk −M |} ≤ C for some constant C = C(v) then

|f(v, t)− fn+1

(v)| ≤ Cτm+1

.


Recursive Time Relaxed Monte Carlo (TRMC-R)

Let now consider in a time interval [0,∆t] the whole sum at the basis of TR schemes:

f(v,∆t) = e−µ∆t/ε∞∑

k=0

(

1− e−µ∆t/ε)k

fk(v). (3)

The sum has a very clear probabilistic interpretation:

f(·,∆t) is the velocity distribution of particles at time ∆t. Taking a particle at random from thisdistribution, it might happen that it has not collided one single time. The probability of this eventis e−µ∆t/ε, and the velocity distribution given this is f0(v).

The first term in the sum corresponds to those particles that have been involved in exactly onecollision. The probability that a particle has such a history is e−µ∆t/ε

(1− e−µ∆t/ε

), and the

velocity distribution is exactly f1(v).

In the same way fn is the conditional velocity distribution given that exactly n+ 1 particles havebeen involved in its collision history back to the initial time. And the number of particles involved,k is geometrically distributed.

Remarks: The probabilistic interpretation holds uniformly in µ∆t, at variance with NB, which requires

µ∆t < 1. Furthermore, as µ∆t→∞, f at time n+ 1 is sampled from a Maxwellian. In a space

non homogeneous case, this would be equivalent to a particle method for Euler equations.


Recursive algorithm

Derivation of the Monte Carlo methodStarting now from the TR schemes

fn+1(v) = (1 − τ)m∑

k=0

τkfnk (v) + τm+1M(v).

we have the following algorithm that can be implemented in a recursive way:Given a set of N particles distributed according to fn, then, at time (n+ 1)∆t

one tries to split the particles into

a set of N(1− τ) particles sampled from f0,

for each k = 1, ..,m, a set of N(1− τ)τk particles sampled from fk, and

Nτm+1 particles sampled from a Maxwellian.


Collision trees

f_0 f_0 f_0 f_0 f_0 f_0 f_0 f_0

f_1f_1 f_1

f_2 f_2

f_0 f_0 f_0 f_0 f_0 f_0 f_0 f_0 f_0 f_0 f_0 f_0

f_1

f_2

f_3

f_1 f_1

f_3f_2

f_3

f_1

Generation of particles in the algorithm

Remarks: The very first step in this calculation is to compute a first pair of particles (velocities)distributed according to fm. This involves the collisions of m+ 1 particles, and hence this number ofparticles are drawn at random from the initial distribution. Obviously this sets a limit of the number ofterms of the Wild-sum that can be estimated with a finite number of particles at the initial time. Allm+ 1 particles must be kept in order that the conserved quantities remain exact. From a practicalviewpoint, the number m can be very large. Thus a maximum allowed value mmax is fixed (whichrepresent the maximum depth of a possible tree) at the beginning of the computations.


Truncation of the trees

It is possible to make a better truncation of the trees:

if the tree is well balanced (i.e. a particle from fk comes by a collision oftwo particles sampled by fh, fj , with min(h, j) > n fixed) then the particleis sampled from a Maxwellian.

if the tree is not well balanced the whole tree is kept.

Two additional adaptive technique can be used to choose the right maximumdepth of the collisional trees:

if the solution is known, then it is possible to reconstruct a function bywhich the maximum k allowed is kept in the transient regime, inconsideration to the distance of solution to the local equilibrium;

if the solution is unknown, then it is possible by indicators that measure thedistance of the solution from the equilibrium (for example as in S.Tiwari,S.Rjasanow 1997)


Well balanced trees

f_0 f_0 f_0 f_0 f_0 f_0 f_0 f_0

f_1 f_1 f_1f_1

f_3 f_3

f_7

Well balancedf_7

f_0f_0 f_0 f_0 f_0 f_0 f_0 f_0

f_1

f_2

f_3

f_4

f_1

f_2

Not well-balanced


Space homogeneous case (hard spheres)

Comparison between recursive TRMC with constant (m = 3500) and adaptivetruncation and Bird’s method. The maximum length of the trees is ≈ 7000during the simulation. The adaptive function here used has the form

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

with mmin = 10 and mmax = 3500.Collisional time step: ∆t

ε= 1

Number of particles: N = 1× 105

Initial condition is the sum of two Maxwellian. The reference solution has been

obtained using N = 5× 105 and Bird’s Method.


4th order momentum

0 2 4 6 8 10 12 14 16 18 205.5

6

6.5

7

7.5

8

8.5

ExactBirdTRMC−R3500TRMC−RA


L2 error and number of collisions

0 2 4 6 8 10 12 14 16 18 200.007

0.008

0.009

0.01

0.011

0.012

0.013

BirdTR−RC3500TR−RCAD

2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

4x 105

BirdTRMC−R3500TRMC−RA


Stationary shock waves

Ma = 3 (Mach Number of incoming flux)

Tinf = 1 (Temperature of incoming flux)

uxinf = −M√

γTinf , uy = 0, γ = 2. (Mean velocity of incoming flux)

ρinf = 1 (Total mass)

ε = 0.1

Hard Spheres : Numerical solution obtained with 50 spatial cells and 1000particles in each cell. Reference ’Exact’ solution with Bird’s Method, 50 spatialcells and 5000 particles in each cell.

Remark: In the order to improve the solution for a stationary shock it is possible

to increase accuracy computing averages of solutions for t large enough.


Adaptive truncation

0 5 10 15 20 25 30 35 40 45 505

10

15

20

25

30

35

40

45function for truncation

epsilon = 0.1epsilon = 1


Mass and relative error

−5 −4 −3 −2 −1 0 1 2 3 4 50.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25



Mean velocity and relative error

−5 −4 −3 −2 −1 0 1 2 3 4 5−4.5

−4

−3.5

−3

−2.5

−2

−1.5


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2



Temperature and relative error

−5 −4 −3 −2 −1 0 1 2 3 4 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5



Number of collisions

−5 −4 −3 −2 −1 0 1 2 3 4 50

500

1000

1500

2000

2500

3000



ConclusionsTRMC schemes are very promising in obtaining efficient and accurate MCmethods.

All schemes share the property to be asymptotic preserving, i.e. in thefluid-dynamic limit they will project the distribution function on the localMaxwellian equilibrium.

The main advantage is the possibility of using larger time steps withoutdegrading accuracy. This results in a overall smaller number of collisionsfor a fixed time, and therefore in a greater efficiency over classical MonteCarlo.

They can be further adapted to the particular problem accordingly to themaximum depth of the trees (as a balance between efficiency andaccuracy) that can be different cell by cell.

Future developments include the truncation on well balanced trees and theadaptive truncation by smoothness indicators of of the solution.


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