Implicit-Explicit schemes for BGK kinetic equations - Politecnico di Torino

Implicit-Explicit schemes for BGK kinetic equations Sandra Pieraccini, Gabriella Puppo

∗

†

Abstract In this work a new class of numerical methods for the BGK model of kinetic equations is presented. In principle, schemes of any order of accuracy in both space and time can be constructed with this technique. The methods proposed are based on an explicit-implicit time discretization. In particular the convective terms are treated explicitly, while the source terms are implicit. In this fashion even problems with infinite stiffness can be integrated with relatively large time steps. The conservation properties of the schemes are investigated. Numerical results are shown for schemes of order 1, 2 and 5 in space, and up to third order accurate in time.

1

Introduction

Navier Stokes equations describe the motion of a fluid when the continuum hypothesis underlying the model is valid: in particular, the ratio between the mean free path (λ) and the characteristic dimensions of the problem (L) is small: Kn = λ/L  1, where Kn is the Knudsen number. When Kn = O(1), the kinetic theory of rarefied gas dynamics comes into play. Here the fundamental equation is Boltzmann’s equation. Traditionally, an important field of application of kinetic theory has been the motion of objects in the rarefied layers of the atmosphere, such as re-entry problems in aerospace engineering. In these cases in fact, the Knudsen number is large, because the mean free path is of orders of magnitude larger than the characteristic length of the space vehicle. Recently however a huge new field of applications of Boltzmann’s equations has begun to develop in the modelling of fluid flows in nanostructures: in this case, the Knudsen number is large because the scale of interest L is so small that the ratio λ/L is of order one and microscopic effects cannot be neglected. The main tool to integrate Boltzmann’s equation numerically is the Direct Simulation Monte Carlo (DSMC) method, see for instance the classical reference [7]; a more recent review can be found in [21], see also references therein. The DSMC scheme however is very slow due to its low convergence rate and to the complexity of the evaluation of the collision term. Moreover its results are polluted by stochastic noise and therefore lack smoothness. Interest has also focused on deterministic methods, see for instance [20]. From a numerical point of view, the BGK [6, 8] model approximating Boltzmann’s equation for moderate Knudsen numbers is particularly attractive. It has a strong theoretical background, see for instance [23], and extensive numerical computations have tested its ability to approximate Boltzmann solutions for moderate Kn and Euler solutions for Kn  1, see [10] and [27]. More recently, extensive research on the elliptic BGK-ES model [13] has drawn attention to the improved approximation of the Navier Stokes regime for low Knudsen numbers, [27, 2, 16]. The BGK model has also been used to evaluate several flows of physical interest, see for instance [3, 4] and [1, 17] for applications to reacting ∗ This

work was supported by INDAM project “Kinetic Innovative Models for the Study of the Behaviour of Fluids in Micro/Nano Electromechanical Systems”, and by PRIN04-Brezzi project † Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italia, e-mail: [email protected], [email protected]

1

gas mixtures. We also mention an application of BGK-like ideas to the development of models for the behaviour of fluids in nanostructures [11]. The importance of the BGK model in applications has prompted a parallel development of numerical methods tailored to the particular structure of BGK equations. We start mentioning the first order numerical scheme proposed in [10]. A second order scheme is described in [4]. In this case, the scheme is constructed with linear second order upwinding: the lack of limiters results in the possible onset of spurious oscillations. The third order in space scheme appearing in [27] exploits ideas derived from high order schemes for conservation laws: here the onset of spurious oscillations is prevented using ENO (Essentially Non Oscillatory) reconstructions from [12]. The scheme relies on a first order operator splitting, and thus its accuracy in time does not match the high accuracy of the space discretization. The high order schemes mentioned so far do not satisfy exact conservation of the macroscopic variables (mass, momentum and energy) at the discrete level. This problem is solved in [15] and [16]. Exact conservation is obtained computing equilibrium at the discrete level. This construction requires the solution of a non linear system of 5 equations for the BGK model and 10 equations for the ES-BGK model at each grid point in space, even for explicit integration in time. All schemes described so far require the solution of large linear systems of equations when the BGK model becomes stiff, i.e. for small relaxation times. Moreover the heavy non linearity of the collision term usually requires further approximations in the evaluation of the Jacobian matrices needed to solve the non linear system arising from the discretization of the BGK equations in the implicit case. In this work we propose a new high order scheme in both space and time to solve the BGK equations. Our method is based on a high order implicit-explicit discretization of the time derivative obtained with Runge-Kutta IMEX schemes, following the work of [14], [5] and [22]. In our scheme, the discretization of the time derivative is carried out before the approximation of the space derivatives and the discretization of velocity space, as in [18]. In particular, the convective term is treated explicitly, while the source term is integrated implicitly. This approach allows to treat even problems with infinite stiffness easily, see [22] and [24] for applications to relaxation systems. In the implicit step, we first evaluate the macroscopic moments and next we update the distribution function: in this fashion, conservation is naturally enforced, and the solution of the implicit step is local, with no need to solve large systems of equations. We think that this approach yields a noticeable improvement in the efficiency of the scheme, and it allows to achieve very high order accuracy. Space discretization is carried out with WENO (Weighted Essentially Non Oscillatory) reconstructions, see [25]. When the grid in velocity space is introduced, exact conservation no longer holds at the discrete level. However, we find in our tests that the error in the conserved variables is very small. Exact conservation can be enforced following the technique introduced in [15]. In this work, we show results obtained for schemes of order 1, 2, and 5 in space and 1, 2, and 3 in time, respectively. Higher order schemes can be derived with the same techniques, provided suitable higher order IMEX Runge-Kutta schemes become available. We start reviewing the BGK model in §2, and we describe the main characteristics of the new numerical scheme in §3. Next we briefly describe the space and time discretizations in §4: this material is well established, but the application of these ideas in the context of kinetic equations is not standard and it allows some simplifications. We end with a few numerical tests in §5.

2

The BGK model

In this section the BGK model is described. For simplicity, we only consider the classical BGK model introduced in [6], using the notation of [10] and [15]. The scheme can be easily extended to more general BGK models.

2

We consider the initial value problem: ∂f (x, v, t) + v · ∇x f (x, v, t) ∂t f (x, v, 0)

1 (fM (x, v, t) − f (x, v, t)) t ≥ 0, τ = f0 (x, v) ≥ 0 given initial data. =

x ∈ R d , v ∈ RN

(1)

Generally d = 3 and N = 3. In the 1D case, d = 1 and N = 3 and ∇x = (∂x1 , 0, 0). In (1) fM is the Maxwellian obtained from the moments of f , namely:   ρ(x, t) kv − u(x, t)k2 fM (x, v, t) = exp − 2RT (x, t) (2πRT (x, t))N/2 The quantities ρ, u and T are respectively the macroscopic density, velocity and temperature of the gas, and they are obtained from the moments of f , which Rare defined as follows. Given any function g : RN 7→ R, let us denote, as in [15], by hgi the quantity RN g(v) dv; if g : RN 7→ Rp , p > 1, we still denote by hgi ∈ Rp the vector whose components are given by hgi i. The moments of f are defined by  *   + 1 ρ  m = f  . v (2) 1 2 E kvk 2 (we dropped the dependence from x and t for simplicity). Here m is momentum, so that the macroscopic velocity is simply u = m/ρ, while E is the total energy, and the temperature is obtained from the internal energy e, through the relations: ρe = E − 21 ρu2 , e = N RT /2. In most applications, N = 3 which corresponds to a monoatomic gas with three translational degrees of freedom. In our tests, for simplicity, we will choose instead N = 1, as in [10]. This corresponds to a gas with a single degree of freedom, so that e = RT /2. The only difference with respect to a physical monoatomic gas appears in this rescaling of the temperature. With this choice, all velocity integrals will be evaluated in R instead of R 3 . Another approach to reduce the computational complexity of the velocity integrals, while maintaining the physical properties of the gas, has been introduced in [9] and used, among others, in [4] and [27]. In [4] the relaxation time τ is defined by τ −1 = Ac ρ, where Ac is a constant, in such a way that Ac ρ represents the collision frequency of the gas molecules. In [15] the BGK model is written with a relaxation time given by τ −1 = CρT 1−ω where ω is the exponent of the viscosity law of the gas (for example, for argon one has ω = 0.81). In the adimensional case one has C τ −1 = . Kn As in [10], we will pick this last choice with C = 1. The macroscopic moments of f are conserved, in the sense that: ∂t hf i + ∇x · hf vi = 0, ∂t hf vi + ∇x · hv ⊗ vf i = 0,     1 1 2 2 ∂t kvk f + ∇x · kvk vf = 0. 2 2

(3a) (3b) (3c)

Moreover, an entropy principle holds, namely: ∂t hf log f i + ∇x hvf log f i ≤ 0, 3

∀f ≥ 0,

(4)

the equality holding only for f = fM . A numerical scheme for (1) should be able not only to yield an accurate solution to equation (1), but also to satisfy the conservation equations and the entropy principle in some discretized form.

3

The numerical scheme

Following [18] we start discretizing the BGK problem (1) in time with a Runge-Kutta IMEX scheme (for the notation, see §4). Next the scheme will be discretized in space and, lastly, we will consider the discretization of velocity space and the evaluation of velocity integrals through quadrature. Finally, we will review the simplified algorithm that will be tested in the numerical problems. Time discretization For the sake of simplicity we consider a unidimensional problem in space, and we introduce a uniform grid in time, with spacing ∆t. Let f n (x, v) = f (x, v, n∆t). The updated density distribution is given by: f n+1 (x, v) = f n (x, v) − ∆t

ν X i=1

w ˜i v ∂x f (i) |(x,v) + ∆t

ν X wi (i) (fM (x, v) − f (i) (x, v)). (i) τ i=1

(5)

(i)

where fM (x, v), i = 1, ..., ν, are the Maxwellian functions obtained from the moments of the intermediate stages f (i) (x, v), which are defined by the following relations (we drop the dependence on (x, v) for simplicity): a11 (1) (f − f (1) ), τ (1) M i−1 i X X ail (l) = f n − ∆t a ˜il v ∂x f (l) + ∆t (f − f (l) ). τ (l) M

f (1) = f n + ∆t f (i)

l=1

(6a) (6b)

l=1

The coefficients ail , a ˜il , wi and w ˜i define the IMEX Runge-Kutta scheme and are specified in §4. Let φ(v) denote the vector of the collision invariants of f , φ(v) = (1, v, 21 kvk2 )T . First we compute the moments of f . For the first stage value we have: D E  E a11 D (1) fM − f (1) φ . f (1) φ = hf n φi + ∆t (1) τ

Since

D  E (1) fM − f (1) φ = 0,

we immediately find the macroscopic variables ρ(1) , u(1) and T (1) corresponding to f (1) . With these quantities, we compute the corresponding Maxwellian:   ||v − u(1) ||2 ρ(1) (1) exp − (7) fM = (2πRT (1) )N/2 2RT (1) and once the Maxwellian is known ∀x, the stage value f (1) can simply be found solving the linear equation (6a), D for each x, v. For E the following stage values we note that, iterating the construction, for all i we have

(i)

fM − f (i) φ = 0. Thus the moments at the i-th stage are given by: D

i−1 E D E X f (i) φ = hf n φi − ∆t a ˜il v ∂x f (l) φ l=1

4

(i)

and can be explicitly computed. Then the Maxwellian fM can be computed as in (7). Now let: B (i) = f n − ∆t

i−1 X

a ˜il v∂x f (l) + ∆t

l=1

i−1 X ail (l) (fM − f (l) ), (l) τ l=1

that is B (i) contains all information coming from the previously computed stages. Finally we obtain (i)

f (i) =

τ (i) B (i) + ∆t aii fM . τ (i) + ∆t aii (i)

Note that if τ (i)  1, f (i) immediately relaxes on the local Maxwellian fM . This completes the description of the time discretization of our scheme. We end this part formally stating the conservation property of the time discretization. Proposition 1 The time discretization (5)-(6) preserves conservation of density, momentum and total energy. Proof. Computing the moments in (5) one has

f

n+1



n

φ = hf φi − ∆t∂x

*

v

ν X i=1

w ˜i f

(i)

+

φ .

Since this is an explicit Runge-Kutta discretization of (3), it is straightforward that density, momentum and energy are conserved by the time discretization.  Space discretization We introduce a uniform grid in space, with grid points xj , where the distribution function f will be evaluated: fjn (v) = f (xj , v, n∆t). The grid spacing will be denoted by ∆x = xj+1 − xj . The main task of the space discretization is to provide an accurate approximation of ∂ x f , satisfying the following requirements: • the discretization of ∂x f should be non-oscillatory to prevent the onset of spurious oscillations, characteristic of high order schemes in the presence of discontinuities or sharp gradients in the solution; • the discretization should be conservative. There are two main strategies to approximate the space derivative appearing in (5) in a conservative form. The most straightforward (finite volume formulation) is to integrate equation (5) on each cell Ij = (xj − ∆x/2, xj + ∆x/2), and to update in time the cell averages of f : Z 1 n f j (v) = f n (x, v) dx. ∆x Ij The advantage of this approach is that the cell average of the space derivative is simply:   1 ∆x ∆x n v∂x f j = v f (xj + ) − f (xj − ) . ∆x 2 2 The disadvantage of this approach is that it requires to compute cell averages of the Maxwellian f M . For first and second order accurate schemes, this is easily accomplished, because: Z 1 fM j (v) = fM (x, v) dx = fM (xj , v) + O(∆x)2 . ∆x Ij 5

For schemes of higher order accuracy however, a more accurate quadrature rule in space is needed, which requires to evaluate fM at quadrature nodes beyond the grid points xj . This need couples the space cells together, and the scheme becomes much more complex. The second approach is to evaluate (5) at the grid points, and approximate the derivative of f with a conservative finite difference formula, as in [26]. This is the approach we will follow in this work. Note that in this case, to preserve high accuracy the grid spacing ∆x must be uniform or at least smoothly varying in x. Suppose that the values of a flux function F (f ) are given at the grid points xj . In the present case, clearly the flux function is linear, with F (f ) = v f . The first step is to look for a function Fˆ that interpolates the data Fj = F (f (., xj , .)) in the sense of cell averages: Z 1 Fˆ dx, Fj = ∆x Ij thus the derivative of F will be written as: ∂ x F |x j =

 1 ˆ F (xj+1/2 ) − Fˆ (xj−1/2 ) , ∆x

where xj±1/2 = xj ±∆x/2. We outline the main steps of the construction: more details will be found in §4 and [25]. The approximation to Fˆ is typically a piecewise polynomial function, with jump discontinuities at the cell borders xj±1/2 . To ensure stability, it is necessary to pick information coming from the correct direction. In other words, it is necessary to introduce upwinding. In this work, we will use flux splitting, which is particularly straightforward for the convective term of the BGK model. Thus we write the flux F as the sum of its positive and negative parts: F = F + + F − where F + and F − have only non negative (respectively: non positive) eigenvalues. Clearly, in our case, the flux splitting will depend on the sign of v, namely:   v f if v > 0 0 if v ≤ 0 F+ = F− = 0 otherwise v f otherwise Next, two reconstructions are computed, one for F + and one for F − , which will be called respectively Fˆ + and Fˆ − . Each of these has a jump discontinuity at xj+1/2 : to use upwinding, we pick the value from + the left for the positive flux, Fˆj+1/2 = Fˆ + (x− j+1/2 ), and we pick the value from the right for the negative + − − = Fˆ (x ). Thus the numerical flux at each cell border will be given by: flux: Fˆ j+1/2

j+1/2

ˆ− + Fˆj+1/2 = Fˆ + (x− j+1/2 ) + F (xj+1/2 ), and the conservative approximation to the space derivative will be given by:  1 ˆ Fj+1/2 − Fˆj−1/2 . ∂x (v f )|j = ∆x

(8)

(9)

In the present case, the structure of F is particularly simple, and one has:

ˆ + Fˆj+1/2 (v) = max(v, 0)fˆ(x− j+1/2 ) + min(v, 0)f(xj+1/2 ),

(10)

R 1 ˆ where fˆ is a piecewise polynomial function such that fj = ∆x Ij f dx. For example, for the first order scheme fˆ is piecewise constant, namely fˆ(x)|Ij ≡ fj . Thus fˆ(x− j+1/2 ) = + ˆ fj while f (x ) = fj+1 and the numerical flux in this case will be given by: j+1/2

Fˆj+1/2 (v) = max(v, 0)fj + min(v, 0)fj+1 . 6

(11)

Substituting the space discretization in (5), we find: fjn+1 (v) = fjn (v) −

ν ν  X ∆t X  ˆ (i) wi (i) (i) (i) (fM,j (v) − fj (v)), w ˜i Fj+1/2 (v) − Fˆj−1/2 (v) + ∆t (i) ∆x i=1 i=1 τ

(12)

j

with a similar modification of (6) for the computation of the stage values f (i) at the grid points xj : (1)

fj (v) = fjn (v) + ∆t (i)

fj (v) = fjn (v) −

a11 (1)

τj

(1)

(1)

(fM,j (v) − fj (v)),

(13a)

i i−1  X ail (l) ∆t X  ˆ (l) (l) (l) (fM,j (v) − fj (v)). a ˜il Fj+1/2 (v) − Fˆj−1/2 (v) + ∆t (l) ∆x τ l=1

l=1

(13b)

j

We end stating the conservation properties of the semi-discrete scheme just described. Proposition 2 The semi-discrete scheme (12)-(13) preserves conservation of density, momentum and total energy. Proof. Computing the moments of (12) and (13) in v space, we obtain a conservative and consistent discretization of the conservation equations (3) for ρ, m and E. In fact, the space discretization is carried out with an algorithm which is conservative by construction. Further, the numerical flux functions given in (11) and in the next §4 ensure consistency. By the Lax-Wendroff theorem, if the numerical solution converges, we conclude that the macroscopic quantities are conserved. 

Velocity discretization The discretization of the velocity arises several problems. The main difficulty is that the moments of f are computed approximately through a quadrature formula. As a consequence, it is no longer true that f and fM have the same moments. A second problem is due to the difficulty of selecting a suitable grid in velocity space. On one hand, a fine grid ensures a good accuracy of the velocity integrals which yield the macroscopic variables. On the other hand, f must be updated on each velocity grid point, which means that the scheme increases rapidly in computational cost as the velocity grid is refined. Moreover, the solution is quite sensitive to the choice of velocity grid points, which seems to be highly problem dependent, see for instance [4], where the velocity grid changes at almost each test problem. Choose a grid in velocity space, and let {vk }, k ∈ K, be the set of the grid points, where k is a multi-index if N > 1. Given any function g : RN 7→ R, let: Z hgiK = Q(g, {vk }) ' g(v) dv, RN

i.e. hgiK denotes the approximation of hgi obtained by means of a suitable quadrature rule Q( · , {v k }) built on the nodes {vk }; for a vector function g : RN 7→ Rp we use a convention similar to the one already introduced for hgi, i.e. hgiK ∈ Rp and the application of h · iK is meant componentwise. The macroscopic variables now depend on the quadrature rule and on the grid used in velocity space. Let:   ρK  mK  = hf φi K EK 7

be the moments computed from f with the quadrature Q( · , {vk }). Now construct an approximate Maxwellian with the formula:   ||v − uK (x, t)||2 ρK (x, t) exp − MK (f )(x, v, t) = (14) 2RTK (x, t) (2πRTK (x, t))N/2 The problem is that there is no reason why f and MK (f ) should have the same discrete moments, that is in general: h(f − MK (f )) φiK 6= 0, as pointed out in [15]. However, again in [15], it is proven that it is possible to find a discrete Maxwellian M such that: h(f − M(f )) φiK = 0, M(f )(x, v, t) = exp (α(x, t) · φ(v)) , (15) where α(x, t) is an unknown vector that depends on the macroscopic quantities. Note that α is computed precisely solving the non linear system defined by (15). We can now introduce the discretization in velocity space in the semidiscrete system defined by (12) n and (13). Let fjk = f (xj , vk , tn ). The updated values of f will be given by: n+1 n fjk = fjk −

ν ν  X wi ∆t X  ˆ (i) (i) (i) (i) w ˜i Fj+1/2 (vk ) − Fˆj−1/2 (vk ) + ∆t (M(fj )(vk ) − fjk ). (i) ∆x i=1 τ i=1

(16)

K,j

To compute the stage values, at each level i, for each space grid point xj , first compute the discrete (i) moments (ρK , mK , EK )j . Then solve equation (15) to find the local components of the vector αj , and (i)

thus compute the discrete Maxwellian M(fj ). Then the distribution function can easily be evaluated at each space and velocity grid point: (1)

n + ∆t fjk = fjk

(i)

n fjk = fjk −

a11 (1) τK,j

(1)

(1)

(M(fj )(vk ) − fjk )

(17a)

i−1 i  X ail ∆t X  ˆ (l) (l) (l) (l) a ˜il Fj+1/2 (vk ) − Fˆj−1/2 (vk ) + ∆t (M(fj )(vk ) − fjk ). (l) ∆x τ l=1

l=1

(17b)

K,j

Computing the discrete moments of (16), we find a consistent and conservative discretization of the conservation equations (3). Thus again density, momentum and total energy are conserved. It is worthwhile noting that the discrete Maxwellian M(f ) is always computed explicitly, even if the method is implicit in the collision term. This is simpler than in the conservative scheme in [15], where the implicit version of the scheme requires an implicit evaluation of the discrete Maxwellian. Now we describe a simpler version of the scheme defined by (16) and (17), in which the velocity discretization is only approximately conservative. We start from the semidiscrete equations (12) and (13). The updated values of f are now given by: n+1 fjk

=

n fjk

ν ν  X ∆t X  ˆ (i) wi (i) (i) (i) ˆ − w ˜i Fj+1/2 (vk ) − Fj−1/2 (vk ) + ∆t (fM,jk − fjk ). (i) ∆x i=1 i=1 τ

(18)

K,j

(i)

where fM,jk is the usual Maxwellian computed from the moments evaluated at the i-th stage in the following way: D E (i) fj φ

K

i−1 i  E  E X

 ∆t X D ˆ (l) ail D (l) (l) (l) φ a ˜il Fj+1/2 − Fˆj−1/2 φ M(f ) − f . (19) + ∆t = fjn φ K − j j (l) ∆x K K τ l=1

l=1

8

K,j

Note that since in the equation above the discrete Maxwellian appears, the last term drops out. From (i) the discrete moments, we compute the corresponding Maxwellian fM,jk . Then the stage values are given by: (1)

n fjk = fjk + ∆t

(i)

n fjk = fjk −

a11 (1)

τK,j

(1)

(1)

(fM,jk − fjk ),

(20a)

i−1 i  X ∆t X  ˆ (l) ail (l) (l) (l) a ˜il Fj+1/2 (vk ) − Fˆj−1/2 (vk ) + ∆t (fM,jk − fjk ). (l) ∆x τ l=1

l=1

(20b)

K,j

This scheme requires no solution of the non linear system (15), and is therefore much faster than the conservative scheme described above. In fact, the discrete Maxwellian M(f ), although used, is never computed. We wish to stress the fact that this simplified scheme is not exactly conservative. However, in the numerical tests reported in §5 we will see that the errors on the conserved variables are indeed very small. We end this section with a summary of the simplified scheme defined by (18) and (20). We use a ν stages IMEX scheme among those proposed in [19, 22]. Let Nx be such that Nx + 1 is the number of points in the space grid and let Nv be the number of nodes in velocity space. The overall scheme reads as follows: • Given f 0 := f (xj , vk , 0) ∈ RNv ×Nx +1 • For n = 0, 1, ... – For i = 1, ..., ν (i)

∗ Compute the moments m(i) and the approximate Maxwellian fM corresponding to the point values (xj , vk ), j = 0, ..., Nx , k = 1, ..., Nv , with i = 1 corresponding to the starting data at tn . (i−1) ∗ If i > 1 construct the numerical fluxes Fˆj+1/2 (vk ), k = 1, ..., Nv , with the reconstruction procedure described in section 4, and according to equation (8). ∗ Assemble the Right Hand Side of (20) and solve for f (i) .

– Compute f n+1 via (18).

4

Time and Space discretizations

In this section, we give a few details on IMEX Runge-Kutta schemes and on high order conservative finite difference formulas for the evaluation of the space derivatives appearing in (5) and (6). The algorithms we will overview in this section are drawn from [22] for the IMEX part and from [25] for the WENO differentiation formula.

4.1

IMEX Runge-Kutta schemes

To set the notation for the Runge-Kutta IMEX schemes, we consider the autonomous ODE problem: y 0 (t) = f (y) + τ1 g(y) y(t0 ) = y0 .

(21)

We suppose that 0 ≤ τ i, and let w, ˜ w be two coefficient vectors with ν elements. An IMEX Runge-Kutta scheme for autonomous problems is represented by the following double Butcher’s tableau: A˜

A

w ˜T

wT

The corresponding numerical scheme for (21) is: y n+1

= y n + ∆t

ν X

ν

w ˜i f (y (i) ) +

i=1

∆t X wi g(y (i) ), τ i=1

(22)

where the stage values y (i) are given by: y (1) y (i)

∆t a11 g(y (1) ) τ i−1 i X ∆t X a ˜il f (y (l) ) + ail g(y (l) ). = y n + ∆t τ = yn +

l=1

(23) (24)

l=1

The coefficients of the Butcher’s tableaux are computed in order to maximize accuracy. Further, the implicit scheme must be L-stable, to ensure that the numerical solution relaxes on the equilibrium solution, if τ is very small. Moreover, it is desirable that the IMEX scheme becomes a high order explicit numerical scheme for the conserved variables, when τ → 0: for this reason, the matrix A must be invertible, see [19]. In the numerical tests, we will apply IMEX schemes of order 1, 2 and 3. Here we add the corresponding Butcher’s tableaux. IMEX1 1st order scheme, [19] 0

1

1

1

IMEX2 2nd order scheme, [22] 0 0 0

0 0 1

0 0 0

1 2 − 12

0

0

1 2 1 2

0

1 2

1 2

0

1 2

0 0 1 2 1 2

IMEX3 3rd order scheme, [22] 0 0 0 0

0 0 1

0 0 0 1 4

0 0 0 0

α −α 0 β

0 α 1−α η

1 4

0

1 6

1 6

2 3

0

1 6

1 2

0 0 0 0 α 0 −β−η−α α 1 6

being α = 0.24169426078821, β = 0.06042356519705, η = 0.1291528696059. 10

2 3

4.2

Conservative, high order differentiation

The main formula for the computation of the numerical flux is (8). Here we want to report the formulas that are needed to compute the numerical flux with first, second and fifth order accuracy. At first order accuracy, the function fˆ (see §3) is reconstructed as a piecewise constant function; for the second order scheme, fˆ is reconstructed as a piecewise linear function, while fifth order accuracy is achieved reconstructing fˆ as a piecewise parabolic function with WENO coefficients. We have already given the formula for the numerical flux of the first order scheme in (11). In the following, we drop the dependence on t and v. Second order For a second order flux, the reconstruction for fˆ is piecewise linear: ˆ I = fj + σj (x − xj ), f(x)| j where σj is a first order accurate limited slope, as for instance σj = MinMod(fj+1 − fj , fj − fj−1 )/∆x, or σj = (fj+1 − fj−1 )/(2∆x). For the accuracy test in §5 on the smooth case we used this second choice, since the MinMod slope limiter deteriorates accuracy close to extrema. Thus the numerical flux will be: 1 1 Fˆj+1/2 (v) = max(v, 0)(fj + σj ) + min(v, 0)(fj+1 − σj+1 ). 2 2

(25)

WENO reconstruction In the WENO reconstruction fˆ(x)|Ij is obtained through the superposition of the three parabolas Pjl (x), with l = −1, 0, 1 interpolating the data fj+l−1 , fj+l , fj+l+1 in the sense of cell averages. We find three estimates of the left and right values of fˆ at xj+1/2 , respectively: l fˆl (x− j+1/2 ) = Pj (xj+1/2 ),

l fˆl (x+ j+1/2 ) = Pj+1 (xj+1/2 ),

l = −1, 0, 1.

The estimates are easily obtained as 7 11 1 fj−2 − fj−1 + fj 3 6 6 1 5 1 fˆ0 (x− j+1/2 ) = − 6 fj−1 + 6 fj + 3 fj+1 1 5 1 fˆ1 (x− fj + fj+1 − fj+2 j+1/2 ) = 3 6 6

fˆ−1 (x− j+1/2 ) =

and 1 5 1 fˆ−1 (x+ j−1/2 ) = − 6 fj−2 + 6 fj−1 + 3 fj 1 5 1 fˆ0 (x+ fj−1 + fj − fj+1 j−1/2 ) = 3 6 6 11 7 1 1 + ˆ f (xj−1/2 ) = fj − fj+1 + fj+2 . 6 6 3 Next these data are combined with suitable weights to yield the desired values: ˆ − f(x j+1/2 ) =

1 X

ωjl fˆl (x− j+1/2 ),

fˆ(x+ j+1/2 ) =

l=−1

1 X

l=−1

11

ω˜jl fˆl (x+ j+1/2 ).

(26)

The weights are defined by the relations: ωjl = P1

αlj

s s=−1 αj

where

,

αlj =

dl ; ( + βjl )2

ω ˜ jl = P1

α ˜ lj

˜ sj s=−1 α

,

α ˜ lj =

d˜l , ( + βjl )2

3 3 3 3 1 1 , d0 = , d1 = ; d˜−1 = , d˜0 = , d˜1 = 10 5 10 10 5 10 are the accuracy constants: they are computed in order to obtain a fifth order scheme;  prevents dividing by zero, and it is usually chosen as  = 10−6 . The quantities βl ’s are the Smoothness Indicators: they prevent the inclusion in the combination of parabolas with a non smooth stencil: d−1 =

βjl

=

X Z

k=1,2

∆x

2k−1

Ij

dk Pjl dxk

!2

dx.

In our case, the Smoothness Indicators are simply given by: βj−1

=

βj0

=

βj1

=

13 (fj−2 − 2fj−1 + fj )2 + 12 13 (fj−1 − 2fj + fj+1 )2 + 12 13 (fj − 2fj+1 + fj+2 )2 + 12

1 (fj−2 − 4fj−1 + 3fj )2 4 1 (fj−1 − fj+1 )2 4 1 (3fj − 4fj+1 + fj+2 )2 . 4

ˆ − ˆ + Once the values f(x j+1/2 ) and f (xj+1/2 ) are computed as in (26), we form the numerical flux as: ˆ + Fˆj+1/2 (v) = max(v, 0)fˆ(x− j+1/2 ) + min(v, 0)f(xj+1/2 ).

5

(27)

Numerical experiments

In this section we report some numerical results obtained with the simplified methods defined in (18) and (20). In particular, we will test the first order scheme, BGK1, obtained using the IMEX1 scheme defined in §4 and the first order flux (11). The second order scheme, BGK2, is defined by the second order IMEX scheme IMEX2 of §4, and the second order numerical flux (25). Finally the high resolution scheme BGK3 is obtained coupling the third order IMEX3 scheme with the fifth order accurate numerical flux (27). Following [10], we consider a problem which is 1D in velocity space. We considered the following test problems. Test 1 We consider an initial distribution of the kind f (x, v, 0) = √

ρ (v − u)2 ), · exp(− 2RT 2πRT

x ∈ [−1, 1],

with constant density ρ = 1 and temperature T = 1 and with u=

    1 2 2 exp − (σx − 1) − exp − (σx + 1) σ

with σ = 10. Thus initially the distribution function f is smooth, with a localized perturbation in velocity, in a gas with a uniform density and temperature.

12

Test 2 We consider as initial data a distribution which is discontinuous in space. ( 2 L −v) ), 0 ≤ x ≤ 0.5, ρL (2πRTL )−1/2 · exp(− (u2RT L f (x, v, 0) = (uR −v)2 −1/2 ρR (2πRTR ) · exp(− 2RTR ), 0.5 < x ≤ 1 with (ρL , uL , TL ) = (2.25, 0, 1.125) and (ρR , uR , TR ) = (3/7, 0, 1/6). This test is derived from [10]. We used free-flow boundary conditions in both test problems. We considered several values of the Knudsen number Kn, choosing τ = Kn; further, we assumed R = 1. The velocity space was approximated by the finite interval [−V, V ] with V = 10; the velocity grid points are uniformly distributed around v = 0. The time step is chosen following the CFL condition for the explicit, convective part, namely: ∆t = 0.9∆x/V . The velocity integrals are approximated by trapezoidal or Simpson rule, or by the following rule Z vi+1 ∆v g(v) dv = (−g(vi+2 ) + 13g(vi+1 ) + 13g(vi ) − g(vi−1 )) + O(∆v 4 ). (28) 24 vi

5.1

Test 1

We integrated the first test problem up to time t = 0.04, when the solution is still smooth, in order to estimate the rate of convergence of the numerical solutions. In Tables 1-3 we report the errors computed, in the case τ = 10−5 , for different values of ∆x and ∆v with the three methods with respect to a reference solution obtained with scheme BGK3 on a finer grid, with Nx = 1280 and Nv = 1281. We computed the errors for the density, mean velocity and temperature. Specifically, we used the quadrature rule (28) to compute the integrals with Nv = Nx + 1. Note that on all grids tested, except the coarsest one, the higher order scheme yields smaller errors. As the grid is fine enough, the convergence rate accelerates. For the accuracy test reported in these tables, in BGK2 we didn’t use the MinMod slope limiter, in order to avoid the clipping phenomenon on local extrema. Density, velocity and temperature obtained with τ = 10−5 , Nx =R 160 and Nv = 161 are shown in figures 1-3. Further, in figure 4 we plot the total entropy H(f ) = R hf log(f )i dx as a function of time with τ = 10−2 (left) and τ = 10−5 (right), still with Nx = 160 and Nv = 161. Integrals in space are computed through the quadrature rule (28). We recall that for τ decreasing to 0, the BGK model approaches the hydrodynamic limit, for which the total entropy remains constant for smooth solutions. It can be seen that in the case τ = 10−2 entropy decays due to kinetic collisional effects, while in the case τ = 10−5 the solution is very close to the regular solution of the Euler equations and the total entropy computed with schemes BGK2 and BGK3 is nearly constant. Finally, in Table 4 we report the errors in conservation for density, momentum and total energy in the case τ = 10−5 . The errors are obtained integrating in space the moments and computing the absolute value of the difference between the values at the final time (t = 0.04) and at the initial time. The results are obtained with Nx = 160 and Nv = 161; integrals in space are approximated by formula (28). As expected, errors in conservation are very small, indeed close to machine precision, even for the simplified scheme (18)-(20) which does not require the evaluation of the discrete Maxwellian.

5.2

Test 2

We present the results obtained for density, velocity and temperature for t = 0.16, as in [10], for different values of the parameter τ . We used Nx = 200. Concerning the velocity space, we used the trapezoidal rule with Nv = 81. Figures 5-7 report the results obtained for the three moments ρ, u, T at the time t = 0.16 with the three schemes BGK1, BGK2, and BGK3, for τ = 10−2 (left) and τ = 10−5 (right). In figure 8 we plot the behaviour of the total entropy H(f ) versus time, again for the two already considered values of the Knudsen number. 13

Scheme BGK1 Nx 20 40 80 160 320 640

error 6.483861E-03 3.905098E-03 2.101849E-03 1.084978E-03 5.530284E-04 2.789262E-04

Nx 20 40 80 160 320 640

error 2.314989E-02 1.241993E-02 6.527985E-03 3.337493E-03 1.682644E-03 8.427201E-04

order 0.7315 0.8937 0.9540 0.9722 0.9875 order 0.8983 0.9279 0.9679 0.9880 0.9976

Scheme BGK2 L1 -errors error order 4.551779E-03 1.071216E-03 2.0872 1.826311E-04 2.5522 3.936843E-05 2.2138 9.228221E-06 2.0929 2.238795E-06 2.0433 L∞ -errors error order 1.170818E-02 2.331520E-03 2.3282 5.026083E-04 2.2138 1.208923E-04 2.0557 2.891851E-05 2.0637 7.087809E-06 2.0286

Scheme BGK3 error 3.609352E-03 3.968098E-04 3.505630E-05 1.189247E-06 2.521739E-08 1.993130E-09 error 1.150409E-02 1.694517E-03 1.391502E-04 5.542816E-06 1.211303E-07 8.720581E-09

order 3.1852 3.5007 4.8816 5.5595 3.6613 order 2.7632 3.6062 4.6499 5.5160 3.7960

Table 1: Errors on ρ

We note that for τ = 10−2 there is a good agreement with the results shown in [10]. Since τ is relatively large, the profiles are still smooth. Note however that the higher order schemes yield somewhat sharper transitions. The entropy decays with time for all the three schemes tested. The fastest entropy decay is obtained with the first order scheme, since this is of course the most diffusive. In the case with τ = 10−5 , the Knudsen number is so small that the solution has already converged to the solution of Euler equations for a monoatomic gas with a single degree of freedom. In the figures we can clearly detect the presence of a contact discontinuity and a shock wave. The results obtained with BGK3 are much sharper than the analogous results shown in [10] with a double number of grid points in space, and even the solution obtained with BGK1, which has an order of accuracy comparable with the scheme used in [10], has a better resolution and does not exhibit spurious oscillations. Again, the entropy decreases in time for all schemes tested. Acknowledgments. The authors would like to thank Roberto Monaco and Giovanni Russo for several fruitful discussions and helpful suggestions.

References [1] P. Andries, K. Aoki, and B. Perthame, A consistent BGK-type model for gas mixtures, Journal of Statistical Physics, 106 (2002), pp. 993–1018. [2] P. Andries, J. F. Bourgat, P. Le-Tallec, and B. Perthame, Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases, Computer and Methods in Applied Mathematics and Engineering, 31 (2002), p. 3369. [3] K. Aoki, K. Kanba, and S. Takata, Numerical analysis of a supersonic rarefied flow past a flat plate, Physics of Fluids, 9 (1997), pp. 1144–1161. [4] K. Aoki, Y. Sone, and T. Yamada, Numerical analysis of gas flows condensing on its plane condensed phase on the basis of kinetic theory, Physics of Fluids A, 2 (1990), pp. 1867–1878. 14

Scheme BGK1 Nx 20 40 80 160 320 640

error 4.706440E-03 2.908094E-03 1.724832E-03 9.576551E-04 5.019068E-04 2.571487E-04

Nx 20 40 80 160 320 640

error 1.288922E-02 7.820597E-03 4.572828E-03 2.613475E-03 1.379249E-03 7.111019E-04

order 0.6946 0.7536 0.8489 0.9321 0.9648 order 0.7208 0.7742 0.8071 0.9221 0.9558

Scheme BGK2 L1 -errors error order 3.860496E-03 1.517495E-03 1.3471 4.298498E-04 1.8198 1.117550E-04 1.9435 2.820160E-05 1.9865 7.089511E-06 1.9920 L∞ -errors error order 9.087437E-03 3.166105E-03 1.5212 1.131411E-03 1.4846 2.909468E-04 1.9593 7.584702E-05 1.9396 1.979247E-05 1.9381

Scheme BGK3 error 5.308953E-03 9.648877E-04 6.859511E-05 2.646778E-06 7.314238E-08 6.933381E-09 error 1.188032E-02 2.904380E-03 2.544219E-04 1.051056E-05 2.972873E-07 2.580405E-08

order 2.4600 3.8142 4.6958 5.1774 3.3991 order 2.0323 3.5129 4.5973 5.1438 3.5262

Table 2: Errors on u

[5] U. Asher, S. Ruuth, and R. J. Spiteri, Implicit-explicit Runge-Kutta methods for time dependent partial differential equations, Appl. Numer. Math., 25 (1997), pp. 151–167. [6] P. L. Bhatnagar, E. P. Gross, and M. Krook, A model for collision processes in gases. Small amplitude processes in charged and neutral one-component systems, Physical Reviews, 94 (1954), pp. 511–525. [7] G. A. Bird, Molecular Gas Dynamics, Oxford University Press, London, 1976. [8] C. Cercignani, Rarefied Gas Dynamics, from Basic Concepts to Actual Calculations, Cambridge University Press, Cambridge, 2000. [9] C. K. Chu, Kinetic-theoretic description of the formation of a shock wave, Physics of Fluids, 4 (1965), pp. 12–22. [10] F. Coron and B. Perthame, Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal., 28 (1991), pp. 26–42. [11] Z. Guo, T. S. Zhao, and Y. Shi, Simple kinetic model for fluid flows in the nanometer scale, Physical Review E, 71 (2005), pp. 035301–1, 035301–4. [12] A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes III, Journal of Computational Physics, 71 (1987), pp. 231–303. [13] L. H. Holway, Kinetic Theory of Shock Structure using an Ellipsoidal Distribution Function, Academic Press, 1966, pp. 193–215. [14] C. A. Kennedy and M. H. Carpenter, Runge-Kutta schemes for convection-diffusion-reaction equations, Applied Mathematics and Comp., (2002). [15] L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Mathematical Models and Methods in Applied Sciences, 10 (2000), pp. 1121–1149. 15

Scheme BGK1 Nx 20 40 80 160 320 640

error 1.348105E-02 9.346559E-03 5.396658E-03 2.894390E-03 1.500760E-03 7.628044E-04

Nx 20 40 80 160 320 640

error 5.079614E-02 3.027057E-02 1.670170E-02 8.732693E-03 4.450104E-03 2.240976E-03

order 0.5284 0.7924 0.8988 0.9476 0.9763 order 0.7468 0.8579 0.9355 0.9726 0.9897

Scheme BGK2 L1 -errors error order 1.012190E-02 2.834595E-03 1.8363 5.109318E-04 2.4719 9.085205E-05 2.4915 1.876646E-05 2.2754 4.268115E-06 2.1365 L∞ -errors error order 2.643118E-02 6.780829E-03 1.9627 1.388356E-03 2.2881 2.773496E-04 2.3236 6.232958E-05 2.1537 1.475972E-05 2.0783

Scheme BGK3 error 8.722357E-03 1.083529E-03 8.679295E-05 2.760939E-06 8.040301E-08 6.982879E-09 error 2.504089E-02 4.651424E-03 3.441841E-04 1.153828E-05 2.920835E-07 2.259559E-08

order 3.0090 3.6420 4.9743 5.1018 3.5254 order 2.4285 3.7564 4.8987 5.3039 3.6923

Table 3: Errors on T

ρ m E

Scheme BGK1 5.684342E-14 2.130964E-16 2.220446E-14

Scheme BGK2 6.306067E-14 2.501528E-16 3.197442E-14

Scheme BGK3 4.440892E-15 4.555868E-16 2.664535E-15

Table 4: Errors in conservation

[16]

, Schemes for Boltzmann-BGK equation in plane and axisymmetric geometries, Journal of Computational Physics, 162 (2000), pp. 429–466.

[17] R. Monaco, M. Pandolfi Bianchi, and A. J. Soares, BGK-type models in strong reaction and kinetic chemical equilibrium regimes, Tech. Rep. 24, Dipartimento di Matematica, Politecnico di Torino, 2005. Submitted. [18] L. Pareschi, G. Puppo, and G. Russo, Central Runge-Kutta schemes for conservation laws, SIAM Journal of Scientific Computing, 26 (2005), pp. 979–999. [19] L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta schemes for stiff systems of differential equations, vol. 3 of Recent Trends in Numerical Analysis, L. Brugnano and D. Trigiante eds., Nova Science, New York, 2000, pp. 269–289. [20]

, Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator, SIAM J. Numerical Analysis, 37 (2000), pp. 1217–1245.

[21]

, An introduction to Monte Carlo methods for the Boltzmann equation, in ESAIM: Proceedings, vol. 10, 2001, pp. 35–75.

[22]

, Implicit-explicit Runge-Kutta methods and applications to hyperbolic systems with relaxation, J. Scientific Computing, 25 (2005), pp. 129–155.

16

1.03

1.03 BGK1 BGK2 BGK3 reference solution

1.028

1.02

1.026

1.01 1.024

1 1.022

0.99

1.02

1.018

0.98

BGK1 BGK2 BGK3 reference solution

0.97

1.016

1.014

0.96

0.95

1.012

1.01

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

Figure 1: Test 1, density. Left: scheme BGK1 (dashed line), scheme BGK2 (dash-dot line), scheme BGK3 (continuous line), reference solution (dotted line). Right: detail of the right peak. [23] B. Perthame, Global existence to the BGK model of Boltzmann equation, Journal of Differential Equations, 82 (1989), pp. 191–205. [24] G. Puppo and G. Russo, Staggered finite difference schemes for balance laws. To appear on: Proceedings of HYP 2004, 2005. [25] C.-W. Shu, Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, vol. 1697 of Lecture notes in Mathematics, Springer, Berlin, A. Quarteroni ed., 1998, pp. 325–432. [26] C.-W. Shu and S. Osher, Efficient Implementation of Essentially Non-oscillatory Shock-Capturing Schemes, Journal of Computational Physics, 77 (1988), pp. 438–471. [27] J. Y. Yang and J. C. Huang, Rarefied flow computations using nonlinear model Boltzmann equations, Journal of Computational Physics, 120 (1995), pp. 323–339.

17

0.06

0.06

0.058

0.04 0.056

0.054

0.02 0.052

0

0.05

0.048

BGK1 BGK2 BGK3 reference solution

−0.02

BGK1 BGK2 BGK3 reference solution

0.046

0.044

−0.04 0.042

−0.06

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.04 0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

Figure 2: Test 1, velocity. Left: scheme BGK1 (dashed line), scheme BGK2 (dash-dot line), scheme BGK3 (continuous line), reference solution (dotted line). Right: detail of the right peak.

1.06

1.06 BGK1 BGK2 BGK3 reference solution

1.04 1.055

1.02

1 1.05

0.98

1.045

0.96

BGK1 BGK2 BGK3 reference solution

0.94

1.04

0.92

0.9

1.035

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

Figure 3: Test 1, temperature. Left: scheme BGK1 (dashed line), scheme BGK2 (dash-dot line), scheme BGK3 (continuous line), reference solution (dotted line). Right: detail of the right peak.

18

−2.8378

−2.8378

−2.8378

−2.8378

−2.8379

−2.8379

−2.8379

−2.8379

−2.838

−2.838

−2.838

−2.838 BGK1 BGK2 BGK3

−2.8381

0

0.005

BGK1 BGK2 BGK3 0.01

0.015

0.02

0.025

0.03

0.035

0.04

−2.8381

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Figure 4: Test 1, entropy; τ = 10−2 (left) and τ = 10−5 (right). Scheme BGK1 (dotted line), scheme BGK2 (dash-dot line), scheme BGK3 (continuous line). In both figures the y axis ranges from −2.8381 to −2.8378.

2.4

2.4 BGK1 BGK2 BGK3

2.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

BGK1 BGK2 BGK3

2.2

1

0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5: Test 2, density: τ = 10−2 (left) and τ = 10−5 (right). Scheme BGK1 (dashed line), scheme BGK2 (dash-dot line), scheme BGK3 (continuous line)

19

BGK1 BGK2 BGK3

0.9 BGK1 BGK2 BGK3

0.8

0.7 0.5

0.6

0.5

0.4

0.3 0

0.2

0.1 0

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

Figure 6: Test 2, velocity: τ = 10−2 (left) and τ = 10−5 (right). Scheme BGK1 (dashed line), scheme BGK2 (dash-dot line), scheme BGK3 (continuous line)

1.2

1.2 BGK1 BGK2 BGK3

BGK1 BGK2 BGK3

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 7: Test 2, temperature: τ = 10−2 (left) and τ = 10−5 (right). Scheme BGK1 (dashed line), scheme BGK2 (dash-dot line), scheme BGK3 (continuous line)

20

−1.04 BGK1 BGK2 BGK3

−1.05

−1 BGK1 BGK2 BGK3

−1.06

−1.07

−1.08

−1.09

−1.1

−1.11

−1.12

−1.13

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

−1.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Figure 8: Test 2, total entropy plotted versus time: τ = 10−2 (left) and τ = 10−5 (right). Scheme BGK1 (dashed line), scheme BGK2 (dash-dot line), scheme BGK3 (continuous line)

21