RELAXATION SCHEMES FOR NONLINEAR KINETIC EQUATIONS

SIAM J. NUMER. ANAL. Vol. 34, No. 6, pp. 2168–2194, December 1997

c 1997 Society for Industrial and Applied Mathematics

005

RELAXATION SCHEMES FOR NONLINEAR KINETIC EQUATIONS∗ E. GABETTA† , L. PARESCHI‡ , AND G. TOSCANI† Abstract. A class of numerical schemes for nonlinear kinetic equations of Boltzmann type is described. Following Wild’s approach, the solution is represented as a power series with parameter depending exponentially on the Knudsen number. This permits us to derive accurate and stable time discretizations for all ranges of the mean free path. These schemes preserve the main physical properties: positivity, conservation of mass, momentum, and energy. Moreover, for some particular models, the entropy property is also shown to hold. Key words. Boltzmann equation, fluid dynamic limit, Wild sum AMS subject classifications. 35L65, 65C20, 76P05, 82C40 PII. S0036142995287768

1. Introduction. Numerical resolution methods for the Boltzmann equation play an important role in practical and theoretical analysis of the time evolution of a rarefied gas. The widely used and best-known of these methods is the direct simulation Monte Carlo method due to Bird [4]. After Bird’s algorithm, more sophisticated methods related to the Boltzmann equation have been proposed, either based on a random particle approximation [2], [19], [23] or, more recently, fully deterministic [24], [30]. Most of them take into account the fact that the kinetic equations to be solved involve both convection and interactions between particles. Consequently, the schemes rely on an approximation to the solution where the free molecular streaming and the relaxation to equilibrium are dealt with in two separate steps. In effect, it is expected that the solution to the collision step will approach the steady state quite rapidly, i.e., on the scale of time between collisions. On the other hand, streaming produces a change of density on a much longer time scale. Thus, there should be a separation between the time scales on which collisions and streaming occur. From a physical point of view, the collisions, occurring at each point in the physical space, rapidly drive the local velocity distribution very close to the local Maxwellian, conserving the local density, bulk velocity, and energy, before streaming has time to have any appreciable influence. After this short time, the streaming becomes relatively significant, the effect of the collisions being small near the local equilibrium. For the above reasons it becomes clear that any plausible approximation to the collision process must bring the nonequilibrium data near the local Maxwellian. The characteristic length scale of the relaxation is the mean free path between collisions, which we can identify as the Knudsen number. The limit of large relaxation rate (or small Knudsen number) is usually referred to as the fluid dynamic limit and characterizes the passage from a kinetic description of fluid mechanics to a macroscopic description, given, except near shock waves and boundary layers, by the Euler or Navier–Stokes equations. ∗ Received by the editors June 14, 1995; accepted for publication (in revised form) April 3, 1996. This paper has been written within the activities of the National Council for Researches (CNR), Project “Applicazioni della Matematica per la Tecnologia e la Societ` a.” http://www.siam.org/journals/sinum/34-6/28776.html † Universit´ a di Pavia, Dipartimento di Matematica, Via Abbiategrasso 215, 27100 Pavia, Italy ([email protected], [email protected]). ‡ Universit´ a di Ferrara, Dipartimento di Matematica, Via Machiavelli 35, 44100 Ferrara, Italy ([email protected]).

2168

RELAXATION SCHEMES FOR NONLINEAR KINETIC EQUATIONS

2169

The fluid dynamic limit is a challenge for numerical methods, because in this regime the relaxation term becomes stiff. In most methods based on finite volumes, finite elements, or finite differences, the time step, in order to get positivity of the approximated solution, has to be chosen proportional to the mean free path, a condition that makes these schemes unusable near the fluid regime. On the other hand, random particle methods need so many particles near the fluid regime that they are not competitive. To get unconditionally stable schemes, when the number of involved equations (in velocities) is finite and small, it is natural to use implicit or semi-implicit methods for the collision part [9]. When more realistic models, like the full Boltzmann equation, are considered, this treatment is no longer possible, and it appears extremely difficult to construct approximations to the collision step that are well posed for any regime and that preserve the usual conservation laws. Moreover, as observed by Caflisch, Jin, and Russo [9], this problem is not merely a numerical stability problem. In fact, stable numerical discretizations may still produce spurious solutions [25], [26]. Numerical experience shows that a sufficient condition for the correct behavior of the scheme near the fluid regime is that the numerical fluid dynamic limit should be a good discretization of the model Euler equations. This is guaranteed if in this limit the collision scheme projects into the local Maxwellian at every time step. A well-posed first-order discretization (in space and time) for any regime has recently been proposed by Perthame and Coron [12] for the Bhatnagar–Gross–Krook (BGK) model of the Boltzmann equation. Roughly speaking, in [12] the approximation to the relaxation is given by a convex combination of the initial values and of the corresponding local Maxwellian. This simultaneously gives positivity, conservation of mass, momentum, and energy, and an entropy property independent of the mean free path. As we will explain in the following paragraphs, the method presented here can be considered an extension of the previous idea to general nonlinear Boltzmann-type equations. To summarize, the main features of a numerical approximation to the collision dynamic that is in accord with the physical picture would be: • well-posedness of the collision step for arbitrary values of the mean free path; • preservation of the main physical quantities: mass, momentum, energy, and (whenever possible) entropy principle. • correct fluid dynamic limit, i.e., in this limit the numerical approximation is in good agreement with the Euler equations. We propose, in this paper, a time discretization of the relaxation process that satisfies the aforementioned requirements. The starting point is to represent the solution in a power series of mean values of successive iterations of a bilinear operator, as introduced by Wild [32] to express the solution to the homogeneous Boltzmann equation for Maxwell pseudomolecules. The main feature of this representation is that the weights in the arithmetic mean depend on the exponential of the Knudsen number instead of the inverse of it, as usual. The schemes we derive clearly work for all discrete velocity models of the Boltzmann equation that possess the correct number of collision invariants; in addition, they apply to the full Boltzmann equation for Maxwell kernels, a model for which the Wild sum is the exact solution to the problem, and to more general kernels which satisfy numerically essential cut-off hypotheses. To conclude this introduction, we note that although we develop the schemes in the kinetic background, the method can easily be generalized to a much wider class of hyperbolic systems with nonlinear stiff relaxation terms [9], [20], [25], [26].

2170

E. GABETTA, L. PARESCHI, AND G. TOSCANI

2. The setting. In kinetic theory the time evolution of a rarefied gas in a vessel is governed by the Boltzmann equation [11]  ∂f 1     ∂t + v · ∇x f = ǫ J(f, f ), t ≥ 0, (2.1) f (x, v, t = 0) = f0 (x, v), (x, v) ∈ Λ × R3 ,     γ + f (x, v, t) = Kγ − f (x, v, t), (x, v, t) ∈ E + ,

where f = f (x, v, t) is the one particle distribution function of the molecules of the rarefied gas at time t and ǫ is the Knudsen number proportional to the mean free path between collisions. In (2.1), Λ is a bounded spatial domain of R3 , to which the gas under consideration is confined. The boundary ∂Λ is assumed to have a unit inner normal n(x) at almost every x ∈ ∂Λ. The last relation in (2.1) is concerned with boundary conditions, K is a linear integral operator, and γ ± f denote the traces of f on E ± = (x, v, t) ∈ ∂Λ × R3 × (0, T ) : ±v · n(x) > 0 .

The bilinear collision operator J(f, f ) describes the binary collisions of the particles,

(2.2)

=

Z

R3 ×S 2

β

J(f, f )(x, v, t) q·σ , |q| [f (x, v1 , t)f (x, w1 , t) − f (x, v, t)f (x, w, t)] dw dσ. |q|

In the above expression, σ is a unit vector, so that dσ is an element of the area of the surface of the unit sphere S 2 in R3 . Moreover, q = v − w is the relative velocity, whereas (v1 , w1 ) represent the postcollisional velocities associated with the precollisional velocities (v, w) and the collision parameter σ, 1 1 (v + w + |q|σ), w1 = (v + w − |q|σ). 2 2 We have conservation of momentum and energy in every collision,

(2.3)

(2.4)

v1 =

v1 + w1 = v + w,

v12 + w12 = v 2 + w2 .

As already mentioned in the introduction, from a formal point of view we expect that in the limit ǫ → 0 the local mass density, momentum, and temperature Z f (x, v, t) dv, ρ(x, t) = R3 Z vf (x, v, t) dv, ρu(x, t) = (2.5) R3 Z 1 [v − u(x, t)]2 f (x, v, t) dv, T (x, t) = 3ρ R3 converge to a solution of compressible Euler equations  ∂ρ    ∂t + div(ρu) = 0,     ∂ρu    + div(ρu × u) + grad(p) = 0, ∂t (2.6)  ∂E   + div(Eu + pu) = 0,   ∂t      p = ρT, E = 3 ρT + 1 ρu2 . 2 2 Rigorous results in this direction can be found in [11].


2171

As we will see below, the discretization we propose preserves this characteristic, becoming, as ǫ → 0, a stable numerical scheme for (2.6). 2.1. Splitting of the time scales. The initial-boundary value problem for the Boltzmann equation described in (2.1) will be approached by means of a splitting in time of the free molecular flow and relaxation due to collisions. To this end, we rewrite equation (2.1) in the form ∂f = Sf + Rf, ∂t

(2.7) where (2.8)

S = −v · ∇x

and R is given by the nonlinear collision operator on the right-hand side of (2.1). Thus, in order to compute the solution, we can solve the coupled equations ∂f = Sf, ∂t

(2.9)

∂f = Rf ∂t

and apply Trotter’s formula (2.10)

et(S+R) = lim

N →∞

t

t

eN SeN R

N

.

The method is said to be convergent if Trotter’s formula (2.10) holds when S and R are the aforementioned operators. In further detail, for any N > 0 and 0 ≤ n < N , we denote for a fixed t > 0 ∆t =

t , N

tn = n∆t.

Then, two sequences f n+1/2 and f n+1 of piecewise continuous functions are defined on [tn , tn+1 ] by induction on n. Now, let f0 be the initial density and suppose that the functions f n approximating the solution at tn have been constructed. Then, the approximation functions f n+1/2 and f n+1 are obtained in two steps. On the time interval [tn , tn+1 ], first we solve the transport phase  n+1/2  ∂f = Sf n+1/2 , (2.11) ∂t  n+1/2 (x, v, t = tn ) = f n (x, v), f and then, denoting by f n+1/2 (x, v) = f n+1/2 (x, v, tn+1 ), the collision phase  n+1  ∂f = Rf n+1 , ∂t (2.12)  n+1 f (x, v, t = tn ) = f n+1/2 (x, v).

Finally, we define (2.13)

f n+1 (x, v) = f n+1 (x, v, tn+1 ).

The method yields approximate solutions and in particular permits the separate treatment of the two subproblems (2.11)–(2.12).

2172


Remark 2.1. Some recent results showed that the splitting can be used in a constructive way to get existence and uniqueness of a local solution to kinetic equations. This has been proved for the planar Broadwell model in a square box in [31]. More recently, a proof of weak convergence of the splitting towards the DiPerna–Lions [14] solution of the Boltzmann equation has been obtained in [13]. In the following sections, we are interested in developing a numerical approximation to the relaxation process (2.12) which is uniformly stable with ǫ. For the approximation of (2.11), which goes beyond our purpose, we refer to the large class of numerical methods for multidimensional systems of conservation laws (see, for example, [8], [22], [27], [29]). In section 5, for simplicity, we will consider some onedimensional schemes. 3. Approximation of the relaxation. In the collision step described in the previous section, we are going to study, for each x ∈ Λ, in the time interval [tn , tn+1 ] the spatially homogeneous problem  n+1 1  ∂f = J(f n+1 , f n+1 ), ǫ ∂t (3.1)  n+1 (x, v, t = tn ) = f n+1/2 (x, v). f

Here, the x-variable acts as a parameter, and it will be omitted in the rest of the paper. During the evolution process described by (3.1), the local mass density, momentum, and energy do not vary with time, and hence we can write (3.2)

ρn+1 (t) = ρn+1/2 ,

un+1 (t) = un+1/2 ,

T n+1 (t) = T n+1/2 .

3.1. Series solutions. The solution to the Cauchy problem for (3.1) can be sought in the form of a power series (3.3)

f n+1 (v, t) =

∞ X

k

(t − tn ) f (k) (v),

f (0) (v) = f n+1/2 (v),

k=0

where, substituting (3.3) into (3.1), we obtain for the functions f (k) the recurrent formula (3.4)

f (k+1) (v) =

k 1 X1 J(f (h) , f (k−h) ), k+1 ǫ

k = 0, 1, . . . .

h=0

As one can easily observe, this representation is unusable for our numerical purposes because it becomes meaningless in the stiff region (ǫ ≪ 1) of (3.1). In particular, truncating expression (3.3) beyond the first-order term, we get the classical Euler formula. We point out that power series expansion of the type (3.5)–(3.6) can be considered as the limit result of iteration processes applied to the integral form of problem (3.1). Let us consider instead of (3.1) a differential system of the type   df = 1 [P (f, f ) − µf ] , dt ǫ (3.5)  f (v, t = 0) = f0 (v), where µ 6= 0 is a constant and P a bilinear operator.


2173

Wild [32] solved equation (3.5) in the case of Maxwell pseudomolecules, but his method can be applied under more a general hypothesis on P . Following Wild’s approach, the solution to (3.5) can be expressed, formally, as an arithmetic mean, with weight changing exponentially with time, of sequences of convolution iterates of P. Let us replace the time variable t and the required function f = f (v, t) using the equations t′ = (1 − e−µt/ǫ ),

(3.6)

F (v, t′ ) = f (v, t)eµt/ǫ .

Then F is easily shown to satisfy   dF = 1 P (F, F ), 0 < t′ < 1, µ dt′ (3.7)  F (v, t′ = 0) = f0 (v).

Now, problem (3.7) has the same structure as (3.1), and hence the solution can be expanded in the form of a power series in t′ : F (v, t′ ) =

(3.8)

∞ X

k

t′ f (k) (v),

f (0) (v) = f0 (v),

k=0

where the functions f (k) are given by the recurrent formula (3.9)

f (k+1) (v) =

k 1 X1 P (f (h) , f (k−h) ), k+1 µ

k = 0, 1, . . . .

h=0

Reverting to the old notation, we obtain the following formal representation of the solution to the Cauchy problem (3.5): f (v, t) = e−µt/ǫ

(3.10)

∞ X

k=0

1 − e−µt/ǫ

k

f (k) (v).

It is remarkable that in [17], by assuming only certain quadratic properties on P , the uniform convergence of series (3.10) with respect to t has been proved. Remark 3.1. The solution to (3.5) can be expressed in a similar form provided P is an n-linear operator (n ≥ 2). In this situation, system (3.5) reads   df = 1 [P (f, . . . , f ) − µf ] , n ǫ dt (3.11)  f (v, t = 0) = f0 (v).

By replacing the time variable t and the function f = f (v, t) using relations t′ = (1 − e−µ(n−1)t/ǫ ),

F (v, t′ ) = f (v, t)eµt/ǫ .

Similarly, we obtain f (v, t) = e−µ(n−1)t/ǫ

∞ X

k=0

1 − e−µ(n−1)t/ǫ

k

f (k) (v),

2174


which is, for k = 0, 1, . . . , f (k+1) (v) =

1 k+1

X

h1 +···+hn =k

1 Pn (f (h1 ) , . . . , f (hn ) ). (n − 1)µ

The following general result holds [21]. THEOREM 3.1. Let B be a Banach space and let P : B n → B, n ≥ 2, be an n-linear operator satisfying the inequality (3.12)

kP (g1 , . . . , gn )k ≤ CP kg1 k · · · kgn k

∀ gi ∈ B.

We define g(x, t) = e−t

(3.13)

∞ X

k=0

k bk 1 − e−(n−1)t g (k) (x),

where for k = 0, 1, . . . , (3.14)

g (k+1) (x) =

1 k+1

X

h1 +···+hn =k

bh1 · · · bhn Pn (g (h1 ) , . . . , g (hn ) ), (n − 1)bk

and the numbers bk are the coefficients in the Taylor’s expansion of 1

(1 − x) 1−n =

∞ X

bk xk .

k=0

Let t0 =

1 ln(1 − CP−1 kf0 k1−n ) (1 − n)

and 1 ln(1 − CP−1 kf0 k1−n ), (1 − n) t1 =  ∞,  

CP−1 kf0 k1−n < 1, CP−1 kf0 k1−n ≥ 1.

Then, (3.13) is uniformly convergent on compact subsets of ]t0 , t1 [ and is the unique solution to problem   dg = P (g, . . . , g) − g, dt (3.15)  g(x, t = 0) = g0 (x).

Remark 3.2. Let τ = µt/ǫ; then the system (3.11), and so (3.5), take the form (3.15), where now the n-linear operator is P/µ. In all the kinetic equations we will consider in the next paragraphs, we have µ = kf0 kL1 and n = 2. Thus, choosing B = L1 (R3 ), k

P (g1 , g2 ) 1 kL1 ≤ kg1 k1 kg2 kL1 , µ µ

and condition (3.12) is satisfied with CP = 1/µ. Therefore, the initial value problem has a unique solution in the time interval [0, ∞[ independently on ǫ.


2175

3.2. Numerical schemes. We describe in this section the numerical approximation to problem (3.5). First we state the following lemma. LEMMA 3.1. (a) Let P be a nonnegative bilinear operator such that     Z Z 1 1 f  v  dv P (f, f )  v  dv = µ (3.16) R3 R3 v2 v2

where µ 6= 0 is a constant. Then the coefficients f (k) defined by (3.14) are nonnegative and satisfy, ∀k > 0,     Z Z 1 1 f (0)  v  dv. f (k)  v  dv = (3.17) R3 R3 v2 v2 (b) If for p = 0, 1 we have

(3.18)

f (p+1) = f (p) ,

P (f (p) , f (p) ) = µf (p) ,

then f (q) = f (p) ∀q ≥ p. The proof is a simple exercise, and we leave it to the reader. Moreover, a simple modification of the Toeplitz theorem will be useful [28]. LEMMA 3.2. Let {cnk }n,k≥0 be a double sequence of real, nonnegative numbers which satisfy the following conditions: (3.19)

∞ X

cnk = 1

∀n,

lim cnk = 0

n→∞

k=0

∀k.

If {hk }k≥0 is a convergent sequence of elements of a Banach space B, then the sequence gn =

∞ X

cnk hk

k=0

is well defined ∀n, {gn }n≥0 is convergent, and lim gn = lim hk .

n→∞

k→∞

(k)

defined by (3.9) is convergent. Then (3.10) is Suppose the sequence f k≥0 well defined for any value of the mean free path, and moreover, by Lemma 3.2, if we denote by (3.20)

f ∞ (v) = lim f (k) , k→∞

we have (3.21)

lim f (v, t) = f ∞ (v),

t→∞

with f ∞ (v) being the local equilibrium. Thus, for m > 1, we can introduce the following class of numerical schemes for the approximation of problem (3.5) in the time interval [tn , tn+1 ]: (3.22)

f n+1 (v) = (1 − τ (∆t, ǫ))

m X

τ (∆t, ǫ)k f (k) (v) + τ (∆t, ǫ)m+1 f ∞ (v),

k=0

where τ (∆t, ǫ) = 1 − exp {−(µ∆t)/ǫ} and the terms f (k) are defined by (3.9).

2176


THEOREM 3.2. The time discretization defined by (3.22) is such that (i) If supk>m {|f (k) − f ∞ |} ≤ C for a constant C = C(v), then it is at least an m-order approximation (in µ∆t/ǫ) of (3.10). Moreover, |f (v, t) − f n+1 (v)| ≤ C τ (∆t, ǫ)m+1 . (ii) When (3.16) holds then total mass, momentum, and energy are preserved. (iii) If P is a nonnegative operator then f n+1 (v) is nonnegative independently of the value of ǫ. (iv) For any m, n ≥ 0 we have lim f n+1 (v) = f ∞ (v).

ǫ/∆t→0

Moreover, with the same assumptions as (i), the following holds: |f n+1 (v) − f ∞ (v)| ≤ C [1 − τ (∆t, ǫ)m+1 ]. Proof. Property (i) is a simple consequence of |f (v, t) − f n+1 (v)| = (1 − τ (∆t, ǫ))

∞ X

τ (∆t, ǫ)k |f (k) (v) − f ∞ (v)|

k=m+1 ∞ X

≤ C (1 − τ (∆t, ǫ)) τ (∆t, ǫ)k ( k=m+1 m+1 ) ∆t m+1 = C τ (∆t, ǫ) ≤ C min 1, µ ǫ

∀ m, n.

Applying Lemma 3.1, the conservation properties (ii) are just deduced from (1 − τ (∆t, ǫ))

m X

τ (∆t, ǫ)k + τ (∆t, ǫ)m+1 = 1.

k=0

The nonnegativity (iii) is clear since, by Lemma 3.1, f (k) are nonnegative ∀k and 0 ≤ τ (∆t, ǫ) ≤ 1 ∀∆t, ǫ. Finally, relations (iv) are readily checked by noting that lim τ (∆t, ǫ) = 1 ∀ m,

ǫ/∆t→0

and |f n+1 (v) − f ∞ (v)| = (1 − τ (∆t, ǫ)) ≤ C (1 − τ (∆t, ǫ))

m X

k=0 m X

τ (∆t, ǫ)k |f (k) (v) − f ∞ (v)| τ (∆t, ǫ)k

k=0

= C [1 − τ (∆t, ǫ)m+1 ] ∀ m, n. Remark 3.3. Before considering some applications of the previous approximations to various kinetic equations we point out that: • Property (i) implies that the number Ne of evaluation of the bilinear operator P , which in general is the most P expansive part of the computation, is related to the m order m of the schemes by Ne = p=1 p. As we will see, more accurate error estimates are linked to the structure of the bilinear operator P . In particular, the accuracy of the schemes is related to the rate of convergence of the sequence f (k) towards f (∞) . • Property (iii) states that the numerical schemes are unconditionally stable. • Properties (ii) and (iv) are enough to guarantee that the method has a correct numerical behavior near the fluid regime.


2177

4. Applications. 4.1. Discrete velocity models. The discrete velocity models of the Boltzmann equation supply a clarifying example of the application of (3.22). The discrete kinetic theory [16] is concerned with the analysis of systems of gas particles with a finite set of selected velocities, and provides a useful substitute for the Boltzmann equation, in terms of a system of hyperbolic equations with relaxation. The general model is written in the form   ∂fi + v · ∇f = 1 [G (f , f ) − f L (f )] , i i i i i ǫ ∂t (4.1)  fi (x, t = 0) = ϕi (x), i = 1, 2, . . . , r, where V = {vi , i = 1, 2, . . . , r} is the set of the admissible velocities and f (x, t) = {f1 (x, t), . . . , fr (x, t)} is the r-components vector whose ith component represents the density of the particles with velocity vi in the position x ∈ R3 at time t. In system (4.1) the gain term Gi and the loss term Li , i = 1, 2, . . . , r, are defined through the expressions r X

Gi (f , f )(x, t) =

(4.2)

Akl ij fk (x, t)fl (x, t),

j,k,l=0

(4.3)

Li (f )(x, t) =

r X

Aij kl fj (x, t) .

j,k,l=0

The quantities Akl ij are nonnegative constants, linked to the probability that two particles with velocities vi and vj collide and come out of the collision with velocities vk and vl . Because of symmetry properties, due to the particular choice of the allowed velocities, some of these models possess a number of collision invariants greater than the usual one [16]. In these cases, there is no possibility to define a unique equilibrium state with the same moments of the initial density. For this reason, from now on, it is intended that we will refer only to regular discrete models, namely to models that have only the usual conserved quantities: mass, momentum, and energy. A characterization of the regular models is due to Cercignani [10]. Let us define (4.4)

A = max i,j

r X

Aij kl .

k,l=0

Then we have, for i = 1, 2, . . . , r, Gi (f , f ) − fi Li (f ) =

r X

Akl ij fk fl

−

r X



j,k,l=0

=

r X

Akl ij fk fl +

j=0

j,k,l=0

where the coefficients A −

r X

Pr

k,l=0



A −

r X

Aij kl fi fj

j,k,l=0

k,l=0

 fi fj − Afi Aij kl

r X

fj ,

j=0

Aij kl are nonnegative as a consequence of (4.4).

2178


Thus, with the previous notations, the relaxation step can be written in the equivalent form  n+1  1  ∂fi = Ri (f n+1 , f n+1 ) − Aρn+1 fin+1 , ∂t ǫ (4.5)   f n+1 (t = t ) = f n+1/2 , i = 1, 2, . . . , r, n i i

P where the mass ρn+1 = i fin+1 remains constant during the evolution, and Ri is the positive bilinear operator Ri (f , g)(t)

(4.6)

=

r X

j,k,l=0





A − Akl ij fk (t)gl (t) +

r X

k,l=0





 fi (t)gj (t) . Aij kl

The solution to equation (4.5) will be approximated by the m-order scheme for i = 1, 2, . . . , r: fin+1 = (1 − τ (∆t, ǫ))

m X

k

(k)

(τ (∆t, ǫ)) fi

+ τ (∆t, ǫ)m+1 Mi (f n+1/2 ),

k=0

(4.7) (k+1)

fi

=

1 k+1

k X

h=0

1 ρn+1/2

Ri (f (h) , f (k−h) ),

(0)

fi

n+1/2

= fi

,

where τ (∆t, ǫ) = 1 − exp −(Aρn+1/2 ∆t)/ǫ and Mi (f ) is the ith component of the local Maxwellian with the same moments as f . It is interesting to note that the coefficients f (k) in (4.7) can be computed directly by recursion using (4.6). Thus, contrary to the continuous Boltzmann equation, for discrete velocity models the implementation of the schemes does not need further approximations. Obviously, due to the increasing of the computational cost, highorder schemes (m ≥ 2) are of practical interest when the number of velocities is small or in one-dimensional situations. On the other hand, except for some simple models, an accurate evaluation of the local Maxwellian requires a proper numerical method. 4.1.1. Maxwellian state. As in classical kinetic theory, it is well known that among all velocity distributions corresponding to given macroscopic variables, the densities fi of the associated Maxwellian state are those for which the discrete Boltzmann H-function (4.8)

H=

r X

fi log(fi )

i=1

is minimum. By introducing the space of summational invariants (4.9) Ψ(V ) = ψ : V → R s.t. Akl ij (ψ(vi ) + ψ(vj ) − ψ(vk ) − ψ(vl )) = 0 ,

the Maxwellian state is characterized by the fact that log(f ) belongs to Ψ(V ). As already observed, in the general situation, this space can be larger than expected, i.e., the space spanned by (1, v, v 2 ). For regular models this does not occur, and hence


2179

there are q ≤ 5 coefficients ci , i = 1, . . . , q, depending on the mass, momentum, and energy of the initial data such that ( q ) X ci φi , M (f ) = exp i=1

i

where φ ∈ V, i = 1, . . . , q is a basis of Ψ(V ). Using a different basis one gets the usual expressions for the Maxwellian densities (4.10) Mi (f ) = a exp −(vi − b)2 /c)

whose parameters a, b, c depend on mass, mean velocity, and temperature of the initial data # " r r r X 1 1X 1X 2 2 Mi (f )vi , T = Mi (f )vi − u . Mi (f ), u = (4.11) ρ= 3 ρ i=1 ρ i=1 i=1

Clearly, it is difficult to obtain analytically the expression of Mi (f ) as a function of the macroscopic variables ρ, u, T . The simplest way to get an explicit approximation of the Maxwellian densities is given by [18] ρ exp −(vi − u)2 /2T . (4.12) Mi (f ) = Pq 2 h=1 exp {−(vh − u) /2T }

This approximated solution satisfies exactly the first of (4.11), whereas the remaining relations become exact as the number of velocities approaches infinity. Thus (4.12) is acceptable only for very large values of r. More accurate equilibrium state can be obtained solving numerically the system of nonlinear equations (4.10)–(4.11) with a standard iterative method using (4.12) as initial estimate of the solution. 4.1.2. Broadwell model. The connection of the present approximation with previous ones will now be analyzed for the planar Broadwell model. This model is the simplest discrete velocity model with the correct number of collision invariants (mass and momentum) and represents a prototype of hyperbolic system with relaxation. For this reason the development of specific numerical methods has been considered by several authors [5], [9], [15]. Particles can move with only four velocities, directed along the axes of an orthogonal reference frame, and only head-on collisions are permitted. The system of four equations is  ∂f1 ∂f1 1   +v = (f2 f4 − f1 f3 ),   ∂t ∂x ǫ     ∂f ∂f 1 2 2   +v = (f1 f3 − f2 f4 ),  ∂t ∂y ǫ (4.13) ∂f 1 ∂f  3 3   −v = (f2 f4 − f1 f3 ),   ∂t ∂x ǫ     ∂f 1 ∂f 4 4   −v = (f1 f3 − f2 f4 ). ∂t ∂y ǫ

Since in this case the constant A defined by (4.4) is equal to 1, the relaxation process linked to (4.13) takes the form (4.14)

1 ∂fin+1 = Ri (f n+1 , f n+1 ) − ρn+1 fin+1 , ∂t ǫ

i = 1, 2, 3, 4,

2180


where, using the compact notation fkn+1 = fln+1 , if k ≡ l mod 4, n+1 n+1 Ri (f n+1 , f n+1 ) = (fin+1 + fi+1 )(fin+1 + fi+3 ).

A simple calculation shows that the local Maxwellian with the same moments as f n+1 is given by (4.15)

Mi (f n+1 ) =

1 Ri (f n+1 , f n+1 ). ρn+1

(1)

The first term in (4.7) satisfies fi = Mi (f n+1 ); thus by virtue of Lemma 3.1(b) the Wild sum comes at once into the exact solution and all schemes for m ≥ 1 are equal to the exact solution in the time interval [tn , tn+1 ]: n+1 n+1 n+1/2 (4.16) fin+1 = e−ρ ∆t/ǫ fi + 1 − e−ρ ∆t/ǫ Mi (f n+1/2 ), i = 1, 2, 3, 4.

It is clear that (4.16) satisfies the conservation of mass, momentum, and entropy principle, since the aforementioned approximated solution is nothing else than a convex combination of the initial values and of the local Maxwellian. As usual, implicit schemes, generating nonnegative approximations at any regime, can be obtained from (4.16) by substituting e−(ρ∆t)/ǫ with 1/[1 + (ρ∆t)/ǫ] at the first order or with [1 − (ρ∆t)/2ǫ]/[1 + (ρ∆t)/2ǫ] at the second order. We point out that only the first-order scheme possesses the correct fluid dynamic limit. The first-order approximation has been used in [31] to obtain a constructive proof of local existence and uniqueness of a solution to the plane Broadwell model in a square box, by means of the splitting of the equations. More recently, the same approximation to the relaxation step for Broadwell has been used in [9] to get an unconditionally stable second-order splitting scheme with a correct description of the fluid regime. A splitting of the plane Broadwell model, in which the relaxation step has been solved directly on (4.16), has been employed in [5] to investigate the asymptotic behavior of the solution towards a steady state for small values of the mean free path. Remark 4.1. To end this section let us recall that by virtue of (4.14)–(4.15) the planar Broadwell model has the same structure of the BGK model of the Boltzmann equation, and thus for this model, we get n+1 n+1 f n+1 (v) = e−ρ ∆t/ǫ f n+1/2 (v) + 1 − e−ρ ∆t/ǫ M (f n+1/2 )(v)

with M now being the local Maxwellian with the same moments as f . We remark that in [12] the authors, in order to produce a numerical way to pass from kinetic equations to compressible Euler equations, developed a stable discretization valid for arbitrary values of the mean free path based on the previous approximation to the relaxation step. 4.2. Maxwell models. Maxwell pseudomolecules clearly represent the natural, but not the simplest, situation in which expansion (3.10) can be fruitfully employed to get an approximation to the relaxation dynamics for the Boltzmann equation. In such a gas model, particles interact with a collision rate β = β( q·σ |g| ) depending only on the scattering angle and not on the energies of the colliding molecules. Thus, it is immediate to see that the conservation of density leads to a simplification of the structure of problem (3.1) which takes the form  n+1 1  ∂f ¯ n+1 f n+1 , = Q(f n+1 , f n+1 ) − βρ ∂t ǫ (4.17)  n+1 f (v, t = tn ) = f n+1/2 (v),


2181

where (4.18)

Q(f, g)(v, t) =

Z

β

R3 ×S 2

q·σ |q|

f (v1 , t)g(w1 , t) dwdσ

and β¯ =

Z

S2

β

q·σ |q|

dσ .

Applying the Wild method to problem (4.17), we obtain the following representation of the solution: (4.19)

¯

n+1

f n+1 (v, t) = e−(βρ

t)/ǫ

∞ X

¯

n+1

1 − e−(βρ

t)/ǫ

k=0

k

f (k) (v)

with, for k ≥ 1, (4.20)

f (k+1) (v) =

n 1 X 1 (h) (k−h) )(v), ¯ Q(f , f k+1 βρ h=0

f (0) (v) = f n+1/2 (v).

It is a simple exercise to verify that the coefficients f (k) (v), k ≥ 1, have the same mass, bulk velocity, and temperature of f (0) (v). In particular, this implies that the Cauchy problem (4.17) is well posed in L∞ with weight 1 + v 2 . Note that the coefficients of expansion (4.19) include numerous five-fold integrals like (4.18). Thus, except for one-dimensional situations, the most interesting scheme for practical applications is that for m = 1. We get the first-order approximation to the solution by substituting f (k) , k ≥ 2, with the local Maxwellian. With the notations of section 2, the first-order scheme reads n+1 (v) = 1 − τ (∆t, ǫ)f n+1/2 (v) f 2 1 n+1/2 n+1/2 ,f ) + τ (∆t, ǫ) M (f n+1/2 )(v), (4.21) +(1 − τ (∆t, ǫ))τ (∆t, ǫ) ¯ Q(f βρ ¯ n+1/2 ∆t)/ǫ , and we denote by where τ (∆t, ǫ) = 1 − exp −(βρ (4.22)

M (f )(v) =

ρ exp (2πT )3/2

(v − V )2 2T

the local Maxwellian with the same moments as f (v). Since we are interested in the time integration, we remark that in order to solve system (4.21) numerically, we need a velocity discretization for the evaluation of (4.18) and (4.22). Thus, depending on the numerical method used for the five-fold integral, both deterministic and stochastic, we get different approximations of the solution to (4.17). As predicted by Theorem 3.1, by the properties of Q, it is clear that f n+1 is well defined independently on the Knudsen number and has the same density, mean velocity, and temperature of f n+1/2 . It would certainly be interesting to verify the validity of the entropy condition, but the question is open in its generality.

2182


4.2.1. Planar case. A positive answer can be found for some simpler situations, like the planar case. In a recent paper [6], in fact, a sufficient condition has been derived for a convex functional Γ to satisfy the inequality 1 (4.23) Γ ¯ Q(f, f ) ≤ Γ(f ). βρ For the Boltzmann H-function H(f ) =

Z

f (v) log f (v) dv,

R2

inequality (4.23) is a consequence of the exponential power inequality by Shannon [6]. Thus, by convexity, we obtain H(f n+1 ) ≤ (1 − τ (∆t, ǫ))H(f n+1/2 ) +(1 − τ (∆t, ǫ))τ (∆t, ǫ)H

1 n+1/2 n+1/2 Q(f , f ) + τ (∆t, ǫ)2 H[M (f n+1/2 )] ¯ βρ

≤ (1 − τ (∆t, ǫ)2 )H(f n+1/2 ) + τ (∆t, ǫ)2 H[M (f n+1/2 )] ≤ H(f n+1/2 ), proving a total entropy estimate. The recursivity relation (4.20) permits us to extend the entropy property to m > 1. The Maxwell gas in the plane is interesting because of the propagation of regularity. In fact, in [6] another convex functional satisfying (4.23) was discovered. This functional, known as the Linnik functional, is defined by Z p 2 ∇ f (v) dv. (4.24) L(f ) = 4 R2

In analogy with the Boltzmann H-functional, given a function f (v) with finite mass, momentum, and energy, the Linnik functional is such that L(f ) ≥ L[M (f )]. Thus, by the same argument we used to derive the entropy estimate, we get L(f n+1 ) ≤ L(f n+1/2 ). Since mass is conserved and

it follows that (4.25)

p 1 k f kH 1 = ρ1/2 + L(f )1/2 , 2 q p k f n+1 (·)kH 1 ≤ k f n+1/2 (·)kH 1 .

By means of classical Sobolev embeddings it is possible in this case to get error estimates in sup-norm. 4.3. Boltzmann equation with a general kernel. In this section we deal with a possible extension of the method to the Boltzmann equation with a collision kernel that satisfies the boundedness condition Z q·σ β (4.26) , q dσ ≤ C. |q| S2


2183

This condition seems to be essential for convergence proofs of simulation methods to the solution to the full Boltzmann equation. An exhaustive discussion on this point for Nambu’s scheme can be found in [3]. From a physical point of view, (4.26) means that β will have to be truncated in almost all cases of practical interest. From a numerical point of view, the situation is less dramatic. In effect, what we are looking for is a well-posed approximation to the relaxation step (3.1) for the full Boltzmann equation. After [1], it is well known that the solution to the spatially homogeneous problem exists and is unique for almost all general kernels with cut-off, provided sufficiently many moments exist initially. This means that, when condition (4.26) is not satisfied with the position q·σ q·σ (4.27) βr , q = min β ,q ,r , |q| |q| the solutions frn+1 (x, t) r≥1 to the problems

 n+1 1  ∂f = Jr (f n+1 , f n+1 ), ǫ ∂t  n+1 f (v, t = tn ) = f n+1/2 (v)

(4.28)

obtained by replacing β with βr converge, as r goes to infinity, towards the unique solution to the original problem. Hence, for a general collision kernel with cut-off, by further cutting the kernel to satisfy (4.27), we get a problem whose solution, at least for large r, is a reasonable approximation to the true problem. Thus, for a fixed r, let us consider instead of (3.1) the relaxation problem

(4.29)

where (4.30)

 n+1 1  ∂f = Qr (f n+1 , f n+1 ) − f n+1 Sr (f n+1 ) , ∂t ǫ  n+1 (v, t = tn ) = f n+1/2 (v), f Qr (f, f )(v, t) =

Z

q·σ , |q| f (v1 , t)f (w1 , t) dwdσ |q|

βr

βr

R3 ×S 2

and (4.31)

Sr (f )(v, t) =

Z

R3 ×S 2

q·σ , |q| f (w, t) dwdσ . |q|

Because of definition (4.27), Sr (f )(v, t) ≤ 4πrρ. Therefore, if we define Rr (f, g)(v)

(4.32)

q·σ , |q| f (v)g(w) dwdσ, = Qr (f, g)(v) + r − βr |q| R3 ×S 2 Z

2184


Rr (f, g) is a nonnegative bilinear operator, and the relaxation part can be rewritten in the form  n+1 1  ∂f = Rr (f n+1 , f n+1 ) − 4πrρn+1 f n+1 , ∂t ǫ (4.33)  n+1 (v, t = tn ) = f n+1/2 (v), f

which is of the type (3.5). Thus the solution fr to (4.33) can be represented in the form of a Wild sum like (4.19)–(4.20) with β¯ = 4πr. We can easily extend to this situation the considerations of the previous section. In particular, as for the Maxwellian case, the first-order scheme is the main interesting one and reads frn+1 (v) = 1 − τ (∆t, ǫ)frn+1/2 (v) (4.34) +(1 − τ (∆t, ǫ))τ (∆t, ǫ)

2 1 Rr (frn+1/2 , frn+1/2 ) + τ (∆t, ǫ) M (frn+1/2 )(v) , 4πrρ

where τ (∆t, ǫ) = 1 − exp −(4πrρn+1/2 ∆t)/ǫ . Hence frn+1 defined by (4.34) is well defined independently of the Knudsen number, it has the correct moments, and it converges towards the correct fluid-dynamic limit. No proofs of the entropy principle are presently available. 5. Implementation and numerical tests. As already observed, the implementation of the schemes is dependent on the different choices of the relaxation operator P . In particular, the applications to the full Boltzmann equation are strictly related to the integration rule used for the evaluation of the collision operator. In this case, except for one-dimensional problems (in velocity), the most interesting schemes are the first-order ones. Numerical results in this direction for the homogeneous Boltzmann equation in two dimensions can be found in [24]. On the contrary, for the discrete velocity models of the Boltzmann equation, the computation of the schemes can be performed directly by recursivity without further approximations. For the reason given above, the tests we have performed refer to some discrete models of the Boltzmann equation at different regimes. 5.1. Relaxation of some nonlinear models. The purpose of this section is to show, through some practical examples, how different structures of the relaxation terms lead to different computable algorithms, and that the numerical schemes are unconditionally stable and work with uniform accuracy with respect of ǫ. Example 1. Carleman model. The Carleman model of the Boltzmann equation describes a set of particles moving with velocity ±1 along the x-axis. The relaxation process reads

(5.1)

 ∂f1 1   = (f22 − f12 ), ∂t ǫ   ∂f2 = 1 (f 2 − f 2 ), 2 ǫ 1 ∂t

where f1 (t) and f2 (t) are the density functions of the particles traveling with velocity +1, −1.


2185

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2 (k)

FIG. 1. Coefficients f1

(k)

(◦) and f2

3

4

(∗) for Carleman model.

Due to mass conservation, ρ = f1 + f2 , system (5.1) can be rewritten in the equivalent form  ρ 1 ρ ∂f1   + f1 = (f22 + f2 f1 ) = f2 , ∂t ǫ ǫ ǫ (5.2) 1 ρ ρ ∂f  2 2  + f2 = (f1 + f1 f2 ) = f1 . ǫ ǫ ǫ ∂t

It is easy to check that the local Maxwellian is given by f1∞ = f2∞ = ρ/2 and that (k) all terms fi , k > 1, in the Wild representation of the solution to the initial value problem for (5.2) are equal to the local Maxwellian. Figure 1 shows the behavior of (k) (k) (0) (0) the sequences f1 , f2 for ρ = 1.0 and f1 − f2 = −0.5. Thus, by making use of the notations introduced in the previous sections, all the schemes for m ≥ 1, similarly to the plane Broadwell model situation, provide the exact solution to (5.2) in the time interval [tn , tn+1 ] i n+1/2 1 h n+1/2 n+1/2 ∆t) n+1/2 (5.3) fin+1 = ρ + (−1)i+1 e−2(ρ (f1 − f2 ) , i = 1, 2. 2

Example 2. Broadwell models. Here we consider the relaxation process for a set of modified one-dimensional Broadwell models defined by  1 ∂f1  = (f32 − f1 f2 ),    ∂t ǫ   ∂f2 1 2 (5.4) = (f3 − f1 f2 ),  ∂t ǫ      ∂f3 = 1 (f f − f 2 ), 1 2 3 αǫ ∂t

2186


with f1 , f2 , and f3 now being the density of particles with velocity 1, −1, 0, respectively. The parameter α, such that α ≥ 1 is an integer, is proportional to the number of particle densities moving with zero velocity. In particular, for α = 1, (5.4) is nothing but the reduced four velocity Broadwell model we have analyzed in section 4.1, whereas for α = 2, it corresponds to the reduced six velocity Broadwell model. The space of the collision invariant has dimension q = 2 corresponding to conservation of mass and mean velocity ρ = f1 + f2 + 2αf3 ,

ρu = f1 − f2 .

From the conservation of mass we can write (5.4) in the form

(5.5)

 ∂f1 ρ 1  + f1 = (f12 + f32 + 2αf1 f3 ),    ∂t ǫ ǫ   1 2 ρ ∂f2 + f2 = (f2 + f32 + 2αf2 f3 ),  ǫ ∂t ǫ      ∂f3 + ρ f = 1 (f 2 (2α2 − 1) + f f + αf f + αf f ). 2 3 1 3 1 2 3 αǫ 3 ǫ ∂t

Clearly, if α = 1, it is possible to linearize system (5.5), and hence the schemes coincide once more with the exact solution (4.16). On the contrary, for α > 1, system (5.5) preserves the nonlinearity and thus represents an interesting test case. The Maxwellian state is characterized by two constants a and b such that (5.6)

M1 = a exp{b},

M2 = a exp{−b},

M3 = a.

In particular, it is possible to get the analytic expression of a and b as a function of ρ and u: (1 − e) ρ(1 − e) , , b = log (5.7) a= 2α α(e − u) where e = e(u) is given by  1  2   e = (1 + u ), α = 1, 2 (5.8) p 1   α α 2 u2 + 1 − u2 − 1 , e = 2 (α − 1)

α > 1.

By noting with Ri (f , f )/ǫ, i = 1, 2, 3, the right-hand side of equations (5.5), the system has the same structure as (4.5) with A = 1, and hence the general m-order scheme is given by (4.7) with r = 3. The numerical computations refer to the initial nonequilibrium data characterized by relations (5.6)–(5.7) but not (5.8) with (5.9)

ρ = 1.0,

u = 0.1,

e = 0.5.

In all computations we used a fixed ∆t = 0.1. (k) (k) Figures 2 and 3 show the convergence of the sequence e(k) = f1 + f2 towards the Maxwellian state for different values of α. It is remarkable that, in the nonlinear situation, after a fast convergence rate of low-order terms, the asymptotic convergence rate becomes extremely slow.

2187


0.55 0.5 0.45 0.4 0.35 0.3 0

1

2

3

4

5

6

7

8

9

FIG. 2. Convergence of the sequence e(k) for α = 1 (◦), α = 2 (△), α = 3 (∗).

FIG. 3. The convergence rate of the sequence e(k) for a wide range of α.

Figure 4 represents the error function Em = supi |[fi,m+1 −fi,m ]/fi,m+1 | obtained by comparing after the first time step the solution fi,m given by the m-order scheme to that given by the (m + 1)-order scheme for a wide range of Knudsen numbers. The numerical results seem to confirm that the schemes are uniformly accurate with respect to ǫ. Obviously, since the schemes become exact as ǫ → 0 and ǫ → ∞, there is a slight deterioration of the accuracy in the intermediate regime ǫ = O(∆t).

2188


0.020 0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 0.000

0.001

0.010

0.100

1.000

10.000

100.000

FIG. 4. Relative error Em with ∆t = 0.1 for m = 1 (×), m = 2 (◦), m = 3 (⋄), m = 4 (∗).

FIG. 5. Relaxation process for α = 2 versus ǫ.

Finally, in Figure 5, we present the trend towards equilibrium of the Boltzmann Hfunction in time for α = 2 at different regimes. Since different schemes give essentially the same results, we have shown only the first-order one. It is clear that the relaxation time is proportional to the Knudsen number. Numerical computations for other initial data confirmed the qualitative behavior described by Figures 1–5. 5.2. Shock wave profiles. The previous kinetic models can be used to show the influence of small Knudsen number on the structure of shock waves.


2189

The spatial dependent situation reads  ∂f1 1 ∂f1  + = (f32 − f1 f2 ),    ∂t ∂x ǫ   ∂f2 ∂f2 1 2 (5.10) − = (f3 − f1 f2 ),  ∂t ∂x ǫ      ∂f3 = 1 (f f − f 2 ). 1 2 3 αǫ ∂t

After the splitting of (5.10) we need a numerical scheme for solving the transport phase for f1 and f2 . Because we are dealing with a hyperbolic system it is natural to use upwind schemes. In our numerical experiments we considered the first-order upwind scheme with uniform mesh spacing ∆x in the spatial grid points xi given by

(5.11)

n+1/2

fj

(xi ) = fjn (xi ) + η[fjn (xi+ij ) − fjn (xi )],

j = 1, 2

and its second-order TVD (total variation diminishing) extension [22] n+1/2

(5.12)

fj

(xi ) =fjn (xi ) + η[fjn (xi+ij ) − fjn (xi )]− ij

η(1 − η) n [Fj (xi+ij )∆x − Fjn (xi )∆x], 2

j = 1, 2,

where η = ∆t/∆x, ij = (−1)j , [fj (xi−j+2 ) − fj (xi−j+1 )] φ(θj (xi )), Fj (xi ) = ∆x

fj (xi ) − fj (xi−1 ) θj (xi ) = fj (xi+1 ) − fj (xi )

ij

,

and φ is the particular slope-limiter function. In particular, we test two different slope limiters, the “superbee” limiter of Roe, φRS (θ) = max{0, min{1, 2θ}, min(θ, 2)}, and Van Leer’s limiter function, φV L (θ) = (|θ| + θ)/(1 + |θ|). The initial data is characterized by two local Maxwellians with mass and velocity (5.13)

ρ, u,

x ≤ 0,

ρ, u,

x > 0,

where the macroscopic quantities ρ, u are computed in terms of ρ, u from the classical Rankine–Hugoniot relations. The test case we consider is the infinite Mach number shock wave problem for α = 2 characterized by ρ = 4.0,

u = 0;

ρ = 1.0,

u = 1.0.

In this situation, corresponding to a shock wave traveling with speed s = 1/3, the problem can be solved exactly [7]: (5.14) where ξ = [3x − t]/2.

ρ(x, t) =

4 + eξ/ǫ , 1 + eξ/ǫ

u(x, t) =

eξ/ǫ , 1 + eξ/ǫ

2190


4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 -0.50 -1.00 -0.90 -0.80 -0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10

0.00

FIG. 6. Numerical solution for the shock wave problem of ρ (◦) and u (⊗) for the first-order scheme with η = 0.5.

4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 -0.50 -1.00 -0.90 -0.80 -0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10

0.00

FIG. 7. Numerical solution for the shock wave problem of ρ (◦) and u (⊗) for the first-order scheme with η = 1.0.

In Figures 6, 7, 8, and 9 we compare with the exact solution the computed density and mean velocity profiles of the first- and second-order splitting schemes (transport and collision) for different values of ǫ. All the graphs refer to a fixed ∆x = 0.01 at t = 1.5, whereas ∆t depends on the choice of the Courant–Friedrichs–Levy parameter

2191


4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 -0.50 -1.00 -0.90 -0.80 -0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10

0.00

FIG. 8. Numerical solution for the shock wave problem of ρ (◦) and u (⊗) for the second-order scheme with φ = φV L and η = 0.5.

4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 -0.50 -1.00 -0.90 -0.80 -0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10

0.00

FIG. 9. Numerical solution for the shock wave problem of ρ (◦) and u (⊗) for the second-order scheme with φ = φRS and η = 0.5.

η. The density and mean velocity profiles are plotted in three different regimes, ǫ = 0.1,

0.05,

10−6 .

Figure 6 shows the result of the first-order scheme, (5.11) and (4.7) with m = 1, for

2192


η = 0.5. Near the intermediate regime the description of the shock is quite accurate, whereas the result in the fluid limit is very diffusive. In Figure 7 it is seen that the choice η = 1.0 for the first-order scheme yields a better description of the sharp shock of fluid mechanics but loses the accuracy of the intermediate regime. Next we consider the second-order schemes that we obtain from (5.12) and (4.7) for m = 2. Figures 8 and 9 show the computed solution for η = 0.5 of both limiters φV L , φRS . As can be seen, both the second-order schemes show a significant improvement in resolution over the first-order scheme. In particular, the sharpness of the shock obtained with Roe’s limiter is slightly improved with respect to Van Leer’s limiter. Finally, we conclude by remarking that for larger values of ǫ the convection process becomes dominant, and thus the computation is just characterized by (5.11) and (5.12). For a comparison of the present results with other numerical schemes, we refer to [9]. 6. Conclusion and perspectives. In the present paper we proposed a way to approximate the relaxation step for a wide class of hyperbolic systems with nonlinear relaxation terms (the Boltzmann equation and related kinetic equations). To our knowledge the schemes we have built here are the only schemes that can be easily designed such that the most important physical properties, positivity, conservation of mass, momentum, energy, and in some cases the entropy property as well, are guaranteed for arbitrary values of the mean free path. As a consequence these schemes represent a numerical way to pass from the Boltzmann equation to Euler equations or, at least, to guarantee an efficient coupling with the Euler equations. The main limitation in the applications to the full Boltzmann equation is given by the increasing of the computational cost of high-order schemes. It would be interesting to investigate the possibility of obtaining accurate schemes like (3.22) but with a lower number of evaluations of the collision term. We observe that our analysis can be applied to a general relaxation system of the type (3.5), where the corresponding limit state is less well understood than in kinetic theory. In this situation, in fact, it is possible to use the last term of a truncation to series (3.10) as an approximation of the limit state. In perspective, we remark that this representation of the solution is not unique. As derived in [21], the solution to (3.5) can be written also in the form (6.1)

f

n+1

−λµt/ǫ

(v, t) = e

∞ X

k=0

where

cλ,k =

k cλ,k 1 − e−µt/ǫ f (λ,k) (v), k+λ−1 λ−1

and

(6.2)

f (λ,k) (v) =

k X

h=0

k+λ−l−2 λ−2 f (h) (v) k+λ−1 λ−1

for every λ and k = 0, 1, . . . . For λ = 1, (6.1) coincides with (3.10), whereas for λ 6= 1, these representations give rise to different approximations like (3.22) with a free parameter of relaxation λ.


2193

REFERENCES [1] L. ARKERYD, On the Boltzmann equation I and II, Arch. Rational Mech. Anal., 45 (1972), pp. 1–34. [2] H. BABOVSKY, On a simulation scheme for the Boltzmann equation, Math. Meth. Appl. Sci., 8 (1986), pp. 223–233. [3] H. BABOVSKY AND R. ILLNER, A convergence proof for Nambu’s simulation method for the full Boltzmann equation, SIAM J. Numer. Anal., 26 (1989), pp. 45–65. [4] G.A. BIRD, Molecular Gas Dynamics and Direct Simulation of Gas Flows, Clarendon Press, Oxford, UK, 1994. [5] A.V. BOBYLEV, E. GABETTA, AND L. PARESCHI, On a boundary value problem for the plane Broadwell model. Exact solutions and numerical simulation, Math. Models Methods Appl. Sci., 5 (1995), pp. 253–266. [6] A.V. BOBYLEV AND G. TOSCANI, On the generalization of the Boltzmann H-theorem for a spatially homogeneous Maxwell gas, J. Math. Phys., 33 (1992), pp. 2578–2586. [7] J.E. BROADWELL, Shock structure in a simple discrete velocity gas, Phys. Fluids, 7 (1964), pp. 1243–1247. [8] Y. BRENIER, Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal., 21 (1984), pp. 1013–1037. [9] R. CAFLISCH, S. JIN, AND G. RUSSO, Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. Numer. Anal., 34 (1997), pp. 246–281. [10] C. CERCIGNANI, Sur des crit` eres d’existence glˆ obale en th´ eorie cin´ etique discr` ete, C. R. Acad. Sci. Paris I, 301 (1985), pp. 89–92. [11] C. CERCIGNANI, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988. [12] F. CORON AND B. PERTHAME, Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal., 28 (1991), pp. 26–42. [13] L. DESVILLETTES AND S. MISCHLER, About the splitting algorithm for Boltzmann and BGK equations, Math. Model Methods Appl. Sci., 6 (1996), pp. 1079–1101. [14] R. DIPERNA AND P.L. LIONS, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), pp. 321–366. [15] E. GABETTA AND L. PARESCHI, Approximating the Broadwell model in a strip, Math. Models Methods Appl. Sci., 2 (1992), pp. 1–19. [16] R. GATIGNOL, Th´ eorie cin´ etique d’un gas ` a r´ epartition discr` ete de vitesses, Lecture Notes in Phys., 36, Springer-Verlag, Heidelberg, 1975. [17] J.V. HEROD, Series solutions for nonlinear Boltzmann equations, Nonlinear Anal., 7 (1983), pp. 1373–1387. [18] T. INAMURO AND B. STURTEVANT, Numerical study of discrete velocity gases, Phys. Fluids A, 2 (1990), pp. 2196–2203. [19] R. ILLNER AND H. NEUNZERT, On simulation methods for the Boltzmann equation, Transport Theory Statist. Phys., 16 (1987), pp. 141–154. [20] S. JIN, Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comp. Phys., 122 (1995), pp. 51–67. [21] Z. KIELEK, An application of the convolution iterates to evolution equations in Banach space, Univ. Iagel. Acta Math., XXVII, (1988). [22] R.J. LEVEQUE, Numerical Methods for Conservation Laws, Birkhauser-Verlag, Basel, 1992. [23] K. NAMBU, Direct simulation scheme derived from the Boltzmann equation, J. Phys. Soc. Japan, 49 (1980), pp. 2042–2049. [24] L. PARESCHI AND B. PERTHAME, A Fourier spectral method for homogeneous Boltzmann equations, Transport Theory Statist. Phys., 25 (1996), pp. 369–383. [25] R.B. PEMBER, Numerical methods for hyperbolic conservation laws with stiff relaxation, I, Spurious solutions, SIAM J. Appl. Math., 53 (1993), pp. 1293–1330. [26] R.B. PEMBER, Numerical methods for hyperbolic conservation laws with stiff relaxation, II, High order Goudonov methods, SIAM J. Sci. Stat. Comput., 14 (1993), pp. 824–859. [27] B. PERTHAME AND Y. QIU, A variant of Van Leer’s method for multidimensional system of conservation laws, J. Comp. Phys., 1/2 (1994), pp. 370–381. [28] A. PEYERIMHOFF, Lecture Notes on Summability, Lecture Notes in Math. 107, Springer-Verlag, Heidelberg, 1969. [29] P.L. ROE AND D. SIDILKOVER, Optimum positive linear schemes for advection in two and three dimensions, SIAM J. Numer. Anal., 29 (1992), pp. 1542–1568.

2194


[30] F. ROGIER AND J. SCHNEIDER, A direct method for solving the Boltzmann equation, Transport Theory Statist. Phys., 23 (1994), pp. 313–338. [31] G. TOSCANI AND W. WALUS, The initial-boundary value problem for the four velocity plane Broadwell model, Math. Models Methods Appl. Sci., 1 (1991), pp. 293–310. [32] E. WILD, On Boltzmann’s equation in the kinetic theory of gases, Proc. Camb. Philos. Soc., 47 (1951), pp. 602–609.