Stochastic Particle Methods for Rarefied Gases

Transcription

1 CCES Seminar WS 2/3 Stochastic Particle Methods for Rarefied Gases Julian Köllermeier RWTH Aachen University Supervisor: Prof. Dr. Manuel Torrilhon Center for Computational Engineering Science Mathematics Division RWTH Aachen

2 Contents Introduction 2 2 Kinetic description of non-equilibrium gases 2 2. Kinetic equation Collision models Stochastic Motion Equations Discussion of the model 4 3. Relation to Fokker-Planck model Properties and comparison to other models Solution algorithm 6 4. Estimation of statistical moments Particle Evolution Properties of the scheme Boundary conditions Numerical tests 9 5. Energy preservation Relaxation to Maxwellian Conclusion 4

3 Introduction This paper was developed during the CES-seminar at RWTH Aachen University. Its aim is to revise and reformulate the ideas that were originally first mentioned in the main reference [4]. The work of Torrilhon et al. describes a new approach for modeling monoatomic gas flows using stochastic equations. The proposed model is especially valid beyond the limits of the standard Navier-Stokes model and has applications for example in non-equilibrium situations. It also allows for very efficient numerical simulations. 2 Kinetic description of non-equilibrium gases For non-equilibrium flows, the Boltzmann equation usually serves as a starting point for further investigation. It is not based on the evolution of conservative (macroscopic) variables like density or bulk velocity, but on the evolution of the distribution function for the individual particle velocities. According to [4] we therefore assume, that the set of particles can be described by the so called mass density function F(V, x, t) = ρ(x, t)f(v, x, t) with gas density ρ and probability density function (PDF) f of the particle s individual molecular velocity V at position x and time t. It can be shown, that the equilibrium distribution of the particle velocity PDF is a Maxwellian F M = ρ exp (2πkT/m) 3/2 ( (V i U i ) 2 ) 2kT/m Here the mean molecular velocity is the gas velocity U and the deviation from the mean is correlated to the gas temperature T. m is the mass of each particle and k the so called Boltzmann constant. The macroscopic properties (ρ, U, e s ) can be computed for a given mass density function F by integration, e.g. ρ = FdV, ρu = R 3 2. Kinetic equation R 3 V FdV, ρe s + 2 ρu2 = R 3 () 2 V2 FdV (2) The Boltzmann equation determines the evolution of the density function F and is given by F t + V F i + F i F = S(F) (3) x i V i where F can be seen as an external force but will usually be neglected in the following considerations. On the right hand side, the collision operator S(F) models collisions of particles with each other and is crucial for the conservation properties during collision. We usually want to have mass, momentum and energy of the particle ensemble conserved during the collisions. All the following collision models will show this property, which makes them useful for further investigation. 2

4 2.2 Collision models The most famous collision operator was proposed by Boltzmann itself and was derived using the so called Stosszahlansatz S (Boltz) (F) = gb(f (V)F (V ) F(V)F(V )dbdɛdv (4) ρ R 5 It is in principle possible to use this approach for numerical simulations, but its high dimensionality makes already the evaluation of S (Boltz) at the discrete points very costly. For more information and details about this approach, see for example [2]. With some assumptions on the collisions, it is possible to derive a linearized collision operator from the Boltzmann operator that is known as the BGK model [] S (BGK) (F) = τ BGK (F M F) (5) This ansatz basically represents a relaxation towards the equilibrium distribution F M with relaxation time τ BGK. In this paper, we will focus on another approach, which assumes small velocity changes due to collisions and results in a Fokker-Planck operator for the collisions S (F P ) (F) = V i ( τ F P (V i U i )F ) + 2 V k V k ( 2es 3τ F P F with relaxation time τ F P and sensible energy e s = 3 2 mt in three spatial dimensions. The derivation of this operator is shown in detail in [2]. In [4] the collision operator S (F P ) is used to derive a stochastic algorithm that can be implemented very efficiently and still preserves the conservation properties during collision. 2.3 Stochastic Motion Equations According to [4] the Fokker-Planck operator (6) can be inserted into the Boltzmann equation (3) and the resulting equation can be solved using the following stochastic equations of motion dx i = M i dt dm i = dt τ (M i U i ) + k ) (6) (7) ( ) /2 4es dw i (t) + F i (8) 3τ dt Where each particle at position X i has a molecular velocity M i. The interaction of the particles is due to the mean molecular velocity U (equal to the macroscopic velocity) and the energy e s. As these quantities are moments of the density function, they involve all the particles. Therefore, (7) and (8) represent a first order coupled ODE system. The expression dw i(t) dt is basically the derivative of a Gaussian process with zero mean and unit variance, see also [3] for further explanations. The properties of this stochastic approximation model are discussed in the next Section. 3

5 3 Discussion of the model 3. Relation to Fokker-Planck model The stochastic motion equations are not only an approximation to the Boltzmann equation (3) with a Fokker-Planck collision operator (6), the two different approaches are in fact equivalent. The derivation of this will be shown here according to [3]. We start with the most general form of a PDF transport equation which can be interpreted as a kind of Taylor-expansion and is known as the Kramers Moyal equation t F ξ(x, t) = n= ( ) n D (n) (x, t)f ξ (x, t) (9) x with the coefficients D (n) as conditional moments under ξ(t) = x D (n) (x, t) = lim t 0 tn! [ξ(t + t) ξ(t)]n x, t (0) Under the assumption of a Markov-Process (D (n) (x, t) depends only on x and t and not on ξ at earlier times) and continuous sample paths ( ξ = ξ(t+ t) ξ(t) is bounded), it can be shown that D (m) = 0 m 3. The Kramers Moyal equation then simplifies to a Fokker-Planck equation t F ξ(x, t) = D i (x, t)f ξ (x, t) + 2 D ij (x, t)f ξ (x, t) () x i x i x j with coefficients D i and D ij as given in (0): D i (x, t) = lim t 0 D ij (x, t) = lim t 0 ξ(t + t) ξ(t) x, t t (2) [ξ(t + t) ξ(t)][ξ(t + t) ξ(t)] x, t 2 t (3) To shown the equivalence of the Fokker-Planck equation to our stochastic motion equations, we now start from a rather general stochastic differential equation for the motion of fluid particles under a stochastic force f: dξ i dt = a i[ξ(s), s] + f i, 0 s t (4) Assuming a Markovian process and vanishing correlation times of f, we can simplify this to dξ i dt = a i(ξ, t) + b ik (ξ, t) dw k dt or its integrated version (5) ξ i (t + t) ξ i (t) = a i (ξ(t), t) + b ik (ξ(t), t) W k (t) (6) 4

6 The term dw k dt is known as a Wiener process, which is actually the derivative of a Gaussian process. It can be fully determined by its first two moments dw k dt = 0, or for the Gaussian respectively dw k dt (t)dw l dt (t ) = δ kl δ(t t ) (7) W k (t) = 0, W k (t) W l (t ) = ( k) tδ kl (8) Inserting Equation (6) into the definition of the coefficients of the Fokker-Planck equation (2) and (3) yields the relations D i (x, t) = a i (x, t), andd ij = 2 b ik(x, t)b jk (x, t) (9) and the corresponding higher order coefficients vanish to zero because they are of higher order in t, just like it is the case for the Fokker-Planck equation. Hence, we conclude that every stochastic differential equation of the form (5) uniquely determines a Fokker-Planck equation of the form (). In our case, the parameters of the stochastic differential equation are (compare (5) and (8)) a i (x, t) = τ (M i U i ), b ik (x, t) = ( ) /2 4es (20) 3τ resulting (according to (9)) consistently in the following coefficients of the Fokker- Planck equation (compare () and 6) D i (x, t) = τ (M i U i ), D ij = 2e s 3τ In other words, the stochastic motion equation can be used to derive the same Fokker-Planck equation for the particles motion that we have seen before (see (6)). The stochastic motion equations are therefore only a different approach to solve the Boltzmann equation, but contain the same physical and mathematical assumptions as the Fokker-Planck collision operator. 3.2 Properties and comparison to other models For the purpose of a better understanding, we will explain the stochastic motion equations in more detail and recapture the properties of the scheme as well as the differences to other common approaches like direct simulation Monte-Carlo (DSMC). The structure of the stochastic motion equations is quite simple, because we have an ODE for every particle and the different equations couple only through the statistical moments, which are the macroscopic velocity U and the sensible energy e s. Note that these values usually vary in space and time. According to the second equation (8), the molecular velocity M of each particle is driven towards the mean velocity U and is (2) 5

7 influenced by a stochastic noise term, that is a Wiener process. The higher the energy of the system, the higher the influence of the stochastic term on the velocity changes. The relaxation time τ controls the behavior of the system in the way that a smaller value of τ leads to a fast relaxation to equilibrium and a large value of τ results in a slow and smooth change in the system. According to [4], the equations follow the assumption, that position and velocity of the particle represent a stochastic diffusion process, in which it is easy to obtain the macroscopic moments, because the velocity is explicitly calculated in this velocity model. The model proposed is similar to the direct simulation Monte-Carlo (DSMC). For DSCM the collisions are directly modeled as binary interactions of the particles inside one grid cell. The new approach now reduces the interactions of the particles to a stochastic noise term (Wiener process), that each particle experiences on its path (compare [4]). The implementation of the model is therefore much simpler and it is possible to parallelize a large part of the computations as we will see later. 4 Solution algorithm We apply the solution algorithm originally proposed by Torrilhon et al. in [4] with some changes because the computational domain is only two dimensional. Following a Monte-Carlo approach, we simulate the motion of a certain number of particles and obtain macroscopic quantities by weighted averages of the particle ensemble in each grid cell. For simplicity, we consider a two-dimensional setup. In this case, the sensible energy is computed as e s = k mt, because we have only two degrees of freedom and not three as in the three-dimensional case. The relevant set of equations is therefore dx i dt dm i dt = M i (22) = ( ) /2 τ (M 2es dw i (t) i U i ) + + F i τ dt (23) Note the difference in the stochastic term due to the different definition of the energy. After a proper initialization of the particles the solution algorithm now consists of the following steps for each time step: () Estimation of the macroscopic quantities U and e s at time t in every grid cell and interpolation to the particle positions (2) In the case of adaptive time-stepping, the time step size has to be determined (3) A first half-step is performed to predict the particle positions at t + t/2 (4) Application of boundary conditions (5) Estimation of U and e s at time t+ t/2 and interpolation to the particle positions 6

8 (6) A full step is performed to compute the particle positions at t + t using the predicted macroscopic quantities (7) Application of boundary conditions The division into a half step for the prediction of the particle mid-points and a full step using the predicted values is made for the purpose of a higher spatial accuracy. We will now describe the different steps and explain the computation in detail. 4. Estimation of statistical moments For the estimation of the statistical moments we use a exponentially weighted moving time averaging, because we are only interested in statistically stationary solutions and can dramatically reduce the statistic error and the number of required particles. For N p particles and each grid cell J (,..., N n ), we calculate the macroscopic quantities using a kernel function ĝ J (x) as follows U(x J, t) = U J (t) W J (t) (24) e s (x J, t) = 2 ( E J ) (t) W J (t) UJ (t) U J (t) (25) where we used the time averaged quantities U J (t), E J (t) and W J (t) N p U J (t) = µu J (t + t) + ( µ) ĝ J (X j (t))m j (t) (26) j= N p E J (t) = µe J (t + t) + ( µ) ĝ J (X j (t))m j (t) M j (t) (27) j= N p W J (t) = µw J (t + t) + ( µ) ĝ J (X j (t)) (28) j= Here the parameter µ [0, ] is a memory factor. For µ = /n a, we get results that were averaged over about n a time steps, resulting in statistically stationary solutions with a reasonable accuracy and a relatively low number of particles. After the estimation of the velocity and energy in the cells, the values have to be interpolated to the particle positions. We also use the kernel functions from above to do this as follows N n U(X j (t), t) = ĝ J (X j (t)u(x J, t)) (29) J= 7

9 N n e s (X j (t), t) = ĝ J (X j (t)e s (x J, t)) (30) J= For our simulations we used different kernels. It is possible to average over all particles in one cell with equal value of the kernel function ending up with a simple average of the desired quantities inside the cell. We also implemented a hat function for the kernel with ĝ J ((x I ) = δ JI. The results are pretty much the same except for statistical errors. Therefore, we will only present the results obtained with the simple cell average. 4.2 Particle Evolution Similar to [4], we use a energy preserving time stepping algorithm to solve the equations (22) and (23). For more information about special algorithms for stochastic ODEs, see [5]. The values for position and velocity at the new time level t + t are computed using the following scheme ( ) Mi n+ Mi n = e t/τ (Mi n U i ) + C 2 B ξ,i + A C2 B ξ 2,i + F i t (3) ( ) Xi n+ Xi n = U i t + (Mi n U i ) τ e t/τ + Bξ,i + F i 2 t2 (32) Here ξ,i and ξ 2,i are independent, normal distributed random variables modeling the Wiener process. The values of these variables have to be the same for the first half step and the full step. A = e s ( ) e 2 t/τ (33) ( 2 t ( ) ( )) B = e s τ 2 e t/τ 3 e t/τ (34) τ C = e s τ ( ) e t/τ 2 (35) The difference to the original work of Torrilhon et al. is again due to the different definition of the energy term. 4.3 Properties of the scheme The scheme is designed such that it preserves the energy of the system if the external force vanishes. Furthermore, the exact first and second conditional moments are predicted, see [4] for details. The evolution of the particles is very easy to parallelize because the only coupling is due to the macroscopic values. 8

10 4.4 Boundary conditions In our tests, we only consider the application of periodic boundaries. Particles that vanish through one boundary enter the domain on the opposite side again. This is very easy and efficient to implement because there is no need to take reflections into account. As soon as a particle position is outside the domain the position is corrected. For more details about other types of boundaries see [4]. We will not address the difficulties of wall or open boundaries here, because this is a topic of its own. 5 Numerical tests In this section, we want to show some of the properties of the model as well as of the numerical algorithm, mentioned in the previous sections. We start with the preservation of the energy of the system. 5. Energy preservation We consider a unit box with periodic boundaries in each direction. As we are not interested in temporal accurate solutions, we use the time moving average to reduce the stochastic noise and the computational effort. We use a total of 0 5 particles in one singe cell and monitor the energy of the cell during the calculations. As for the initial condition, we choose a simple Maxwellian with zero mean and internal energy kg m2 e s =. We know that this is already an equilibrium distribution and we expect s 2 the solution not to change on the macroscopic level, including the energy. Figure shows two graphics. On top we see the positions together with their associated molecular velocity vectors of some 00 sample particles. Below that, we see the energy of the system. Besides some very small stochastic fluctuations, the energy is constant for the time of the simulation. This is a property of the numerical algorithm. For a sufficiently high number of particles, we always observed this property. For very few particles, the stochastic fluctuations become stronger and spoil the solution. Furthermore, the initial conditions are not able to sample the Maxwellian good enough, so that we might end up with a different initial condition and thus a different energy of the system. Corresponding results are obtained for the macroscopic mean velocity of the cell which is also constant with some larger oscillations. We performed the same test with equal number of particles but more cells, so that each cell contains a smaller portion of the total number of particles. Apart from stochastic fluctuations, the energy is still constant, if we take a total number of 0 5 particles. For less particles the initial sampling becomes worse and fluctuations increase again. 5.2 Relaxation to Maxwellian We now want to demonstrate, how a different and non Maxwellian initial condition leads to a more or less slow relaxation towards another Maxwellian equilibrium distribution. 9

11 Figure : Particle visualization and preserved energy It is important to note, that we are interested in time accurate solutions in this test case. Therefore, we cannot make use of the moving time average described above. We may equivalently set the memory factor in Eqn. (26) - (28) to zero. That way, we do not use previous values and always average the current data to come up with a time accurate solution. The domain is the same as in the test case above and we use, particles. For the initial condition, we consider three Maxwellians as follows: one half of the particles sampled from a Maxwellian with mean velocity ( 2, 0) one third of the particles sampled from a Maxwellian with mean velocity (2, 2) one sixth of the particles sampled from a Maxwellian with mean velocity (2, 4) The initial conditions are chosen such that the overall average velocity is (0, 0) so that we can expect the steady state solution to approach a Maxwellian with zero mean, too. We visualize the velocites of the particles in a 2D histogram plot, so that color indicates how many particles with the specific velocity are currently present. 0

12 The relaxation towards equilibrium is strongly influenced by the Knudsen number Kn which is in our case equivalent to the dimensionless variable τ. A high value of Kn means, that we are in the kinetic region with very low density close to free flight conditions and almost no collisions. A low value on the other hand corresponds to the continuous case with relatively large density and many collisions. In Figure 2 we plotted the resulting densities for the velocities according to a relatively large Knudsen number of Kn = 2 at increasing time levels. We see that the initial parts dissolve more and more and form a large Maxwellian in the end. The Maxwellian is centered around the origin which corresponds to the zero mean of the velocities. Note that for the early pictures, some medium values of the histogram have been mapped to the highest color value to ensure a consistent scaling of all six pictures. We did a similar test with the same settings but with a different Knudsen number of Kn = 0.2 which then corresponds more to the continuous region where many collisions enforce a fast relaxation of the flow solution towards equilibrium. The pictures are therefore the same but the equilibrium Maxwellian can be observed at much earlier timesteps. We therefore omit the resulting densities for the velocity in this case. To get an overview of the different macroscopic moments for this test setting, we calculated energy e s, the heat flux q (in both directions) and the stress tensor π. Our test domain and the initial conditions remain unchanged, we take 2000 particles and simulate 200 timesteps until the equilibrium distribution is obtained. The results can be seen in Figure 3. We here plot the normalized quantities and see their behavior as the particles are moving. The energy e s and the stress tensor π are constant throughout the simulation, expect for small statistical fluctuations. The constant stress tensor has only the static pressure part, as the macroscopic velocity is zero and thus the off-diagonals vanish. The heat flux is not zero in the beginning, because the relaxation towards equilibrium changes the velocities of the particles. It approaches zero as soon as the equilibrium distribution is achieved.

13 Figure 2: Relaxation of three distinct Maxwellians to equilibrium distribution during simulation, increasing time from left to right and top to bottom 2

14 e s q q π π 2 0 π timestep Figure 3: Calculated moments for relaxation to Maxwellian, heat flux decreases until equilibrium is obtained 3

15 6 Conclusion After an introduction, we have shown the equivalent physical assumptions of the new model and the more common Fokker-Planck approximation. We have discussed the important properties of the model as well as the energy preserving property of the numerical algorithm. The main aim was to implement the stochastic particle method originally proposed by Torrilhon et al. in [4]. This was successfully done using a two dimensional setting and a box domain with periodic boundaries. Different kernels and parameters where used for the calculations. We want to note, that the computational framework we developed during our work is in general open to other test cases and/or boundary conditions and can be seen as a starting point for further investigation of the stochastic model. 4

16 References [] P. L. Bhatnagar, E. P. Gross, and M. Krook. A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems. Physical Review, 94:5 525, 954. [2] C. Cercignani. The Boltzmann Equation and its Application. Springer-Verlag, New York, 988. [3] S. Heinz. Statistical Mechanics of Turbulent Flows. Springer-Verlag, Berlin, [4] S. Heinz P. Jenny, M. Torrilhon. A solution algorithm for the fluid dynamics equations based on a stochastic model for molecular motion. Journal of Computational Physics, 229: , 200. [5] E. Platen P. Kloeden. Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin,