ISSN 09655425, Computational Mathematics and Mathematical Physics, 2012, Vol. 52, No. 4, pp. 615–636. © Pleiades Publishing, Ltd., 2012. Original Russian Text © Yu.A. Anikin, 2012, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2012, Vol. 52, No. 4, pp. 697–719.
On the Accuracy of the Projection Computation of the Collision Integral Yu. A. Anikin Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141700 Russia email:
[email protected] Received September 26, 2011
Abstract—The accuracy of the projection method as applied to the computation of the collision inte gral is analyzed. It is shown that the method has an error of the second order of smallness with respect to the mesh size. An optimal method for choosing additional nodes that minimizes the computational error is found. The theoretical conclusions and the optimality of the method are confirmed in a series of numerical experiments. DOI: 10.1134/S0965542512040021 Keywords: rarefied gas, Boltzmann kinetic equation, projection method, collision integral computation.
INTRODUCTION A promising numerical method for rarefied gas dynamics is the numerical solution of the Boltzmann kinetic equation with the collision integral directly evaluated by applying the projection method [1, 2]. A major advantage of the projection method is that the resulting collision integral is completely conserva tive with respect to the energy, momentum, and mass. Experience gained in gasdynamic simulation shows that the violation of conservativeness frequently leads to negative effects, including obtaining invalid results. This issue is especially important in rarefied gas dynamics, in which any problem first has to be solved at the microscopic level of description (i.e., a distri bution function has to be determined) followed by the computation of macroscopic quantities. However, in the case of three dimensions, the distribution function depends on six variables. As a result, despite the rapid development of computational technologies over the last decades, computations in practice are usu ally based on rather coarse velocity grids, which may lead to gross violations of the conservativeness of the collision integral as calculated by a traditional method. Problems with intermediate Knudsen numbers are conventionally computed via direct Monte Carlo simulation, in which case the conservativeness issue is naturally resolved. However, the amount of com putations in these methods grows with decreasing Knudsen number [3]. Consequently, the finite differ ence approach can be considered an alternative to the Monte Carlo technique with its own domain of application. The projection method makes it possible to implement this alternative in practice. In seminal papers [1, 2], the rigorous determination of the error in the method and its order of accuracy was beyond their scope. Only the accuracy of interpolations of distribution functions was indicated in [1, 2], while most attention was focused on the substantiation of the method and its practical application. This gap is filled in the present paper, which offers a detailed analysis of the error in the projection approximation, assuming that the interpolation error of the distribution function is known or zero. Based on the conclusions drawn in the second part of this paper, we find an optimal—in terms of the minimiza tion of the total error—method for choosing additional interpolating nodes (by the latter, we mean nodes interpolating a contribution to the integral sum rather than the distribution function). At the end of this paper, numerical results are presented that confirm the theoretical conclusions. 1. STATEMENT OF THE PROBLEM In the projection method [1, 2], the collision integral is written as 2
Nν
b max V sph N 0 I γ = [ δ αν, γ + δ βν, γ – ( 1 – r ν ) ( δ λν, γ + δ μν, γ ) – r ν ( δ λν + sν, γ + δ μν – sν, γ ) ]Ω ν , 4 2 Nν ν = 1
∑
interp
Ω ν = [ ( f 'f 1' ) ν
(ν) (ν)
– f αν f βν ] ξ αν – ξ βν . 615
(1.1)
616
ANIKIN
Here, N0 is the number of velocity nodes lying inside the cutoff sphere of radius ξcut, 3
3
V sph = N 0 Δξ ≈ ( 4π/3 )ξ cut is the volume of the sphere, bmax is the upper limit of target distances, Nν is the number of nodes in the 8dimensional grid of integration, and Δξ is the step of the velocity grid. The coefficient rν is deter mined by the energy conservation law. Its explicit expression will be given later. Note that, in (1.1), we passed from target distance b to the variable S = b2, since experience has shown that the statistical computational error is smaller when the integral is represented in this form. interp
The symbol ( f 'f 1' ) ν
stands for an approximation of the product of exact distribution functions in terms of the aftercollision velocities ξ 'αν , ξ 'βν . In the general case, ξ 'αν , ξ 'βν do not coincide with velocity interp nodes. Therefore, ( f 'f 1' ) ν is determined by interpolating the distribution function values from the nodes nearest to the aftercollision velocities. The standard choice is power interpolation. Additionally, λν, λν + sν and μν, μν – sν are used as interpolating nodes to reduce the amount of computations (of course, other nodes can be used in the general case): interp
( f 'f 1' ) ν
( ν ) ( ν ) 1 – rν
= ( f λν f μ ν )
(ν)
(ν)
rν
( f λν + sν f μ ν – sν ) .
An advantage of this interpolation is that the integral for the Maxwellian distribution automatically vanishes. The following assumptions are used in this paper. interp 1. The value of ( f 'f 1' ) ν is exactly found. Of course, this is not the case in practice, but the interpo lation of distribution functions in terms of the aftercollision velocities is simple and can always be solved to high accuracy. 2. The number of grid nodes satisfies Nν ∞. As a result, the computation is free of statistical errors, which can always be analyzed separately. Additionally, for convenience and notational brevity, the units of velocities and target distances are chosen so that Vsph = 1 and bmax = 1. Define N = Nν/N0. One pair of nodes interpolating the νth contribution to the integral sum is naturally chosen to be the nearest nodes to the aftercollision velocities. In what follows, they are denoted by the index near(m), m = 1, 2. The other pair of nodes is denoted by add(m). Due to these nodes, the resulting collision integral is made conservative. Thus, although they lead to an additional local deviation of the computed integral from its exact counterpart (which is hereafter called the projection error), its total accuracy is improved. The ultimate goal is to minimize this deviation. With the notation introduced, integral (1.1) rewritten as Nν
1 I γ = [ δ αν, γ + δ βν, γ – ( 1 – a ν ) ( δ near (1 ), γ + δ near ( 2), γ ) – a ν ( δ add (1 ), γ + δ add ( 2), γ ) ]Ω ν , ν ν ν ν 4 2N ν = 1
∑
(1.2)
where aν is the contribution of the additional nodes, which is related to the coefficients in the old notation depending on the energy of these nodes: aν = rν or aν = 1 – rν (in the old notation, rν is the contribution of pairs of maximalenergy nodes, which can be the nearest or additional nodes, respectively). Integral (1.2) is divided into two parts: ( non )
Iγ = Iγ
+ δI γ ,
(1.3)
where ( non ) Iγ
Nν
1 = [ δ αν, γ + δ βν, γ – δ near ( 1), γ – δ near (2 ), γ ]Ω ν ν ν 4 2N ν = 1
∑
(1.4)
is the nonconservative integral obtained without using additional nodes and Nν
1 a ν ( δ near ( 1 ), γ + δ near ( 2), γ – δ add ( 1 ), γ – δ add ( 2 ), γ )Ω ν δI γ = ν ν ν ν 4 2N ν = 1
∑
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
(1.5)
Vol. 52
No. 4
2012
ON THE ACCURACY OF THE PROJECTION COMPUTATION
617
is the projection error caused by the use of additional nodes. Expression (1.4) is not exactly equal to the (0) true collision integral I γ ≡ I ( ξ γ ) , and the total error of the integral is produced by both parts in (1.3). We examine each of them separately. 2. NONCONSERVATIVE INTEGRAL Consider the first two terms in (1.4). Under the assumptions made above, it is easy to see that they are equal to each other and (1) Iγ
Nν
3
N 0 2π 1
Δξ 1 δ αν, γ Ω ν = ≡ lim N ν → ∞ 2N 2π 2 ρ = 1 ν=1
∑
∑ ∫ ∫ ( f ( ξ'' )f ( ξ'
ργ )
ργ
– f ( ξ γ )f ( ξ ρ ) ) ξ γ – ξ ρ dS dε,
(2.1)
0 0
where ξ 'ργ , ξ ''ργ are the velocities produced by a collision of molecules moving at the analyzed velocity ξγ and the velocity ξρ over which the sum is computed (discrete analogue of the integration velocity ξ1 in the exact integral). Indeed, since Nν is infinite, for each pair of collision velocities ξγ, ξρ, there is a continuous set of azimuthal angles ε and target distances S, and the sum over them becomes an integral. Note that the difference distribution function under study is the grid projection of the exact function onto the grid: f γ ≡ f ( ξ γ ). (Recall that it is in this way that we determine the order of accuracy of the difference operator, namely, the grid projection of the exact solution is substituted into it and the deviation of the result from the projection of the exact operator onto the grid is analyzed.) It can be shown that the mixed integrosum (2.1) is equal up to highorder accuracy to the exact inte gral at the point ξγ. Consider an individual term of the sum. It approximates a piece of the exact integral: 1 2π 2
ξ ρ + Δξ/2 2π 1
∫ ∫ ∫ ( f ( ξ' )f ( ξ' ) – f ( ξ )f ( ξ ) ) ξ γ
1
1
γ
– ξ 1 dS dε dξ 1 .
(2.2)
ξ ρ – Δξ/2 0 0
Let us find the error of this approximation. The following notation is introduced for the multiple inner integral with respect to the azimuthal angle and the target distance: 2π 1
1 Φ ( ξ γ, ξ 1 ) = 2π 2
∫ ∫ ( f ( ξ' )f ( ξ' ) – f ( ξ )f ( ξ ) ) ξ γ
1
1
γ
– ξ 1 dS dε .
(2.3)
0 0
Applying the Taylor expansion formula yields ξ ρ + Δξ/2
∫
ξ ρ – Δξ/2
5
3 7 Δξ Φ ( ξ γ, ξ 1 ) dξ 1 = Φ ( ξ γ, ξ ρ )Δξ + Δ 1 Φ ( ξ γ, ξ ρ ) + O ( Δξ ) 24
2
= Φ ( ξ γ, ξ ρ )Δξ + Δξ 24 3
ξ ρ + Δξ/2
∫
7
Δ 1 Φ ( ξ γ, ξ 1 ) dξ 1 + O ( Δξ ),
ξ ρ – Δξ/2
where Δ1Φ is the Laplacian of Φ with respect to ξ1. Combining the results, we obtain (1) Iγ
=
(0) Iγ
2∞
4 Δξ – Δ 1 Φ ( ξ γ, ξ 1 ) dξ 1 + O ( Δξ ). 24
∫
(2.4)
–∞
It is easy to see that the second term tends to zero. Indeed, by the Gauss theorem,
∫ Δ Φ ( ξ , ξ ) dξ ≡ ∫ div ( ∇ Φ ( ξ , ξ ) ) dξ 1
Γ
γ
1
1
1
Γ
γ
1
1
=
∫ ∇ Φ ( ξ , ξ ) dA, 1
γ
1
A
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
618
ANIKIN
where A is the surface of an integration domain Γ. Sending its size to infinity and taking into account that Φ ( ξ γ, ξ 1 ) decays exponentially, we find that the desired integral is zero. Thus, the first two terms in (1.4) are equal to the exact integral up to O(Δξ4), which is omitted from our analysis: (1)
Iγ
(2)
= Iγ
(0)
= Iγ .
Consider the third term in (1.4) (for the fourth one, the line of reasoning is similar). Due to the Kro necker delta in the expression, from the set of all integration nodes, we single out a subset of nodes for which the aftercollision velocities ξ 'αν belong to the cell γ. In other words, δ near ( 1), γ = Δ γ ( ξ 'αν ) , where ν
ξ ∈ [ ξ γ – Δξ/2, ξ γ + Δξ/2 ],
⎧ 1, Δγ ( ξ ) = ⎨ ⎩ 0,
ξ ∉ [ ξ γ – Δξ/2, ξ γ + Δξ/2 ].
By analogy with (2.1), it is easy to see that (3) Iγ
N0
3
N 0 2π 1
Δξ = – 2π 2 κ = 1 ρ = 1
∑ ∑ ∫ ∫ Δ ( ξ'
κρ ) ( f ( ξ '' κρ )f ( ξ 'κρ )
γ
– f ( ξ κ )f ( ξ ρ ) )g κρ dS dε,
(2.5)
0 0
where gκρ = ξ κ – ξ ρ . Note that (2.5) approximates the integral ∞ ∞ 2π 1
1 – 3 Δ γ ( ξ' ) ( f ( ξ '1 )f ( ξ' ) – f ( ξ 1 )f ( ξ ) )g dS dε dξ 1 dξ. 2π 2Δξ –∞ –∞ 0 0
∫ ∫ ∫∫
Let us show that the approximation error has a high order of smallness. Define the function 3
N0
N 0 2π 1
Δξ ˜I (γ3 ) ( η ) = – 2π 2 κ = 1 ρ = 1
∑ ∑ ∫ ∫ Δ ( ξ˜ ' γ
˜'
˜ ''
κρ ) ( f ( ξ κρ )f ( ξ κρ )
– f ( ξ κ + η )f ( ξ ρ ) ) ξ κ + η – ξ ρ dS dε,
0 0
where ξ˜ 'κρ , ξ˜ ''κρ are the velocities after a collision with velocities ξ κ + η and ξρ. (3)
Repeating the argument presented for (2.1) (Taylor expansion of ˜I γ ( η ) and the vanishing of the sec ond derivatives after integration), we obtain (3) Iγ ∞
Δξ/2
1 ˜I (γ3 ) ( η ) dη + O ( Δξ 4 ) = 3 Δξ –Δξ/2
∫
(2.6)
N 0 2π 1
4 1 Δξ 3 = – Δ γ ( ξ 'ρ ) ( f ( ξ 'ρ )f ( ξ ''ρ ) – f ( ξ )f ( ξ ρ ) ) ξ – ξ ρ dS dε dξ + O ( Δξ ), 3 2π 2Δξ –∞ ρ=1 0 0
∫
∑ ∫∫
where ξ 'ρ , ξ ''ρ are the velocities after a collision with the velocities ξ and ξρ. (3) Note that, since the integral in (2.5) involves the discontinuous function Δ γ ( ξ ) , the function ˜I γ ( η ) or its derivatives can also have discontinuities, which prevents the Taylor expansion. Consider this issue in more detail. For fixed κ and ρ, the action of Δγ is reduced to cutting out a piece of the sphere swept by the after collision velocities ξ 'κρ in integration with respect to ε and S if the sphere hits the neighborhood γ of the (3) node (see Fig. 1). The collection of such pieces forms the function ˜I γ ( η ) . A variation in η leads to a dis
placement of these spheres (which is linearly connected with the value of the variation), and discontinu (3) ities in ˜I γ ( η ) or its derivatives can occur only when new spheres appear in the cell or old spheres disap (3) pear from it. Obviously, ˜I γ ( η ) itself is continuous, since the size of the piece grows continuously from zero as the sphere penetrates into the cell. Its first derivatives are continuous as well, since the size (area)
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
ON THE ACCURACY OF THE PROJECTION COMPUTATION
619
ξ''κρ ξρ
ξγ
ξκ ξ'κρ Fig. 1.
of the piece grows proportionally to the squared penetration depth in the sphere. However, the second derivative is discontinuous. This difficulty can be overcome as follows. In (2.6) the strongly discontinuous function Δγ is replaced ˜ γ identical to the former in the entire cell, except for a narrow domain with a width of order by a function Δ
δξ on the cell surface, where the function gradually decays. Accordingly, an O(δξ2/Δξ2) error is introduced ˜ γ is an even function about into the computed integral (if the decay law is chosen so that the derivative Δ the cell boundary). Let us estimate the size of δξ required for completely smoothing the jumps in the higher derivatives. (3) (3) –5 The function ˜I γ ( η ) consists of O( Δξ ) spherical pieces—eventually, ˜I γ ( η ) approximates the collision –3
–2
integral, which consists of N 0 = O ( Δξ ) similar spheres, each consisting of O( Δξ ) similar pieces. If (3) η varies by a value of order Δξ, the collection of pieces forming ˜I γ ( η ) is updated, since the displace
ment of the spheres is linearly connected with the variation in η . Thus, on average, jumps in the deriva tives occur with a frequency of δη ~ O(Δξ6). As a consequence, by using δξ ~ δη ~ O(Δξ6), the higher derivatives can be made smooth. The error in the computed integral is then O(δξ10), i.e., negligibly small. By analogy, passing in (2.6) from the sum over ξρ to an integral with respect to ξ1 (this time without discontinuities) and omitting the O(Δξ4) terms, we obtain (3) Iγ
∞ ∞ 2π 1
1 = – 3 Δ γ ( ξ' ) ( f ( ξ '1 )f ( ξ' ) – f ( ξ 1 )f ( ξ ) )g dS dε dξ 1 dξ. 2π 2Δξ –∞ –∞ 0 0
∫ ∫ ∫∫
(2.7)
By using one of the basic symmetries of the collision integral, (2.7) can be rewritten as (3) Iγ
∞ ∞ 2π 1
ξ γ + Δξ/2
∫ ∫ ∫∫
∫
1 1 = Δ γ ( ξ ) ( f ( ξ '1 )f ( ξ' ) – f ( ξ 1 )f ( ξ ) )g dS dε dξ 1 dξ = 3 3 2π 2Δξ –∞ –∞ 0 0 Δξ ξ
γ
– Δξ/2
2
I ( ξ ) dξ = I γ + Δξ ΔI γ . 24 (0)
(0)
Combining all the results, we finally have ( non )
Iγ
2
= I γ + Δξ ΔI γ . 48 (0)
(0)
(2.8)
Thus, the nonconservative integral has an error of the second order of smallness with respect to the mesh size. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
620
ANIKIN p ξγ−sp Γγ−s p
~ g
ξγ
Γγp
ξ1– S p ~ g
Sp
ξ1
Fig. 2.
3. VALUE OF THE PROJECTION ERROR The coefficient a in (1.2) is found assuming that the total energy of the system is conserved under the contribution to the integral made by an individual collision ν: 2
2
2
2
2
2
ξ αν + ξ βν – ( 1 – a ν ) ( ξ near ( 1 ) + ξ near ( 2 ) ) – a ν ( ξ add (1 ) + ξ add ( 2) ) = 0, ν
ν
ν
ν
so that the total integral is conservative with respect to energy. Moreover, pairs of additional nodes are cho sen so that the integral is conservative with respect to momentum: ξ αν + ξ βν = ξ near ( 1 ) + ξ near ( 2 ) = ξ add ( 1) + ξ add ( 2) = 2ξ cν . ν
ν
ν
ν
Combining this with ξ 'αν – ξ cν = ξ αν – ξ cν , after elementary algebra, we obtain g˜ ν δξ ν a ν = , ˜ ( g ν + S ν – δξ ν )S ν
(3.1)
where g˜ ν = ξ near ( 1 ) – ξ 'βν ,
δξ ν = ξ 'αν – ξ near ( 1 ) ,
ν
ν
S ν = ξ add ( 1) – ξ near ( 1) . ν
ν
Consider the first term in (1.5) (the second terms is similar): (1) δI γ
Nν
Nν
1 1 a ν δ near (1 ), γ Ω ν = a ν Δ γ ( ξ 'αν )Ω ν . = ν 2N ν = 1 2N ν = 1
∑
∑
Proceeding as in the case of (2.5) with Δγ replaced by aΔγ, we obtain (1) δI γ
ξ γ + Δξ/2 ∞
1 = – 3 Δξ ξ
∫ ∫ a ( ξ , ξ, ξ )Φ ( ξ, ξ ) dξ dξ , γ
γ
1
1
(3.2)
1
– Δξ/2 – ∞
where Φ ( ξ, ξ 1 ) is given by (2.3) and a(ξγ, ξ, ξ1) is defined by (3.1) with ξ 'αν and ξ 'βν replaced by the respec tive variables ξ and ξ1 and ξ near (1 ) is replaced by the discrete parameter ξγ. ν
p
The third term in (1.5) is more complicated. Suppose that δξ hits the domain Γ γ and the chosen addi tional node is γ + sp, which lags behind γ by the vector S p (see Fig. 2). The index p ranges from 1 to 26, COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
ON THE ACCURACY OF THE PROJECTION COMPUTATION
621 p
which is the number of neighboring nodes for the chosen one. The shape and sizes of Γ γ depend, on the one hand, on the constraint imposed on the values of a for stability reasons: 0≤a≤1
(3.3)
which sweeps the subdomains of the cell allowed for choosing the node S p. On the other hand, they depend on some optimizing considerations, according to which a single additional node is chosen from several allowed ones. Under the natural assumption that this optimization is independent of the distribu tion function and is determined only by the position δξ of the aftercollision velocity and by the admissible p coefficients a at this point, it follows from (3.1) that the shape and size of Γ γ are functions of ξ γ – ξ 1 and p
p
of the node index p. Moreover, for the same g˜ , Γ γ – sp is the domain Γ γ displaced through the vector –S p (see Fig. 2). Rewriting (3.2) according to these considerations, we obtain (1) δI γ
∞ ⎛ 1 ⎜ = – 3 ⎜ Δξ –∞ ⎝
⎞ a ( ξ γ, ξ, ξ 1 )Φ ( ξ, ξ 1 ) dξ⎟ dξ 1 . ⎟ p ⎠ Γ
∫ ∑∫ p
(3.4)
γ
The third term consists of contributions for which γ is an additional node for its neighbors. Combining this with (3.4) yields (3) δI γ
∞ ⎛ 1 ⎜ = 3 ⎜ Δξ –∞ ⎝
∫ ∑∫ p
⎞ a ( ξ γ, – S , ξ, ξ 1 )Φ ( ξ, ξ 1 ) dξ⎟ dξ 1 . ⎟ ⎠ p
p Γγ – s p
(3.5)
Here, we took into account that γ is the pth node for ξ γ – sp = ξγ – S p (see Fig. 2). It follows from (3.1) that a satisfies a ( ξ γ, ξ, ξ 1 ) = a ( ξ γ + η, ξ + η, ξ 1 + η ), where η is an arbitrary discrete vector. Accordingly, (3.5) is rearranged into (3) δI γ
∞ ⎛ 1 ⎜ = 3 ⎜ Δξ –∞ ⎝
⎞ p p a ( ξ γ, ξ + S , ξ 1 + S )Φ ( ξ, ξ 1 ) dξ⎟ dξ 1 . ⎟ ⎠
∫ ∑∫ p
p
Γγ – s
p
p Making the substitutions ξ˜ = ξ + S p and ξ˜ 1 = ξ1 + S p and taking into account the equivalence of Γ γ – sp p
and Γ γ , we rewrite the previous equation as (3)
δI γ
∞ ⎛ 1 ⎜ = 3 ⎜ Δξ –∞ ⎝
⎞ p p a ( ξ γ, ξ, ξ 1 )Φ ( ξ – S , ξ 1 – S ) dξ⎟ dξ 1 ⎟ p ⎠ Γ
∫ ∑∫ p
(3.6)
γ
(here and below, the tilde is dropped for notational brevity). Combining (3.4) and (3.6) gives the following expression for the complete projection error: (3) δI γ
∞ ⎛ 1 = 3 ⎜ ⎜ 2Δξ –∞ ⎝
⎞ p p a ( ξ γ, ξ, ξ 1 ) { Φ ( ξ – S , ξ 1 – S ) – Φ ( ξ, ξ 1 ) } dξ⎟ dξ 1 . ⎟ ⎠
∫ ∑∫ p
p
Γγ – s
(3.7)
p
(2)
Here, we took into account the coefficient 1/4, which was omitted from (1.5), as well as δI γ COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
(4)
and δI γ .
622
ANIKIN
Using Taylor expansions with respect to ξc = ξ + ξ1 and g = ξ – ξ1, we rearrange the expression in curly brackets in (3.7) into 2
∂Φ ( ξ γ + δξ, ξ 1 ) p 1 ∂ Φ ( ξ γ + δξ, ξ 1 ) p p – S i + S i S j . 2 ∂ξ ci ∂ξ ci ∂ξ cj (Here and below, for notational brevity, we use the Einstein summation convention for repeated indices.) Expanding the resulting expression in δξ and neglecting the O(Δξ3) terms gives 2
2
∂ Φ ( ξ γ, ξ 1 ) p p ∂Φ ( ξ γ, ξ 1 ) p ∂ Φ ( ξ γ, ξ 1 ) p – S i – S i δξ j + 1 Si Sj . 2 ∂ξ ci ∂ξ cj ∂ξ ci ∂ξ j ∂ξ ci Defining 1 A ij = 3 Δξ
∑ S S ∫ a ( ξ , ξ, ξ ) dξ, p p i j
p
γ
1
p Γγ
1 C ij = 3 Δξ
1 B i = 3 Δξ
∑ S ∫ a ( ξ , ξ, ξ ) dξ, p i
p
γ
1
p Γγ
(3.8)
∑ S ∫ a ( ξ , ξ, ξ )δξ dξ p i
p
γ
1
j
p Γγ
we rewrite (3.7) as ∞
2
∂ ( B i Φ ( ξ γ, ξ 1 ) ) ( ξ γ, ξ 1 )⎞ 1 ∂ ( A ij Φ ( ξ γ, ξ 1 ) )⎞ ∂ ⎛ C ∂Φ δI γ = 1 ⎛ – – + dξ 1 . ij ⎠ 2 ⎠ 2 ⎝ ∂ξ ci ⎝ ∂ξ ci ∂ξ j ∂ξ ci ∂ξ cj
∫
(3.9)
–∞
Here, we took into account that values (3.8) depend only ξγ – ξ1 and, hence, are treated as constants when differentiated with respect to ξc. Finally, after using the relation ∂ ∂ ∂ = + ∂ξ i ∂ξ 1i ∂ξ ci and the fact that all the total derivatives with respect to ξ1 vanish after integration with infinite limits (the Gauss theorem, the exponential decay of the distribution function), the final expression for the artificial error is written as ∞
⎞ 1 ∂ 2 ( { A ij – 2C ij }Φ ( ξ γ, ξ 1 ) )⎞ ⎫ 1 ⎛ ∂ ⎛ ⎧ ∂C δI γ = ⎜ ⎜ ⎨ ij – B i ⎬Φ ( ξ γ, ξ 1 )⎟ + ⎟ dξ 1 . 2 ⎝ ∂ξ i ⎝ ⎩ ∂ξ j ∂ξ i ∂ξ j ⎠ 2 ⎠ ⎭ –∞
∫
(3.10)
It follows from (3.8) that Aij and Cij are O(Δξ2) quantities. Let us show that Bi also has the second order of smallness with respect to the velocity mesh size. Having used formula (3.1), we expand the denominator of the coefficient a in small terms: gδξ gδξ a = p = p + O ( Δξ ). p gS ( g + S – δξ )S Neglecting the linear term, we see that a is an even function of δξ: a ( – δξ ) = a ( δξ ) , since the inversion of δξ leads to the inversion of the set of additional nodes S p admissible at the given point: in the projection method, an additional node is chosen from seven nodes that, together with γ, form a cube around δξ. As a result, the function p
a ( δξ )S i
p
is odd and Bi = 0. In fact, Bi = O(Δξ2), since we have neglected the O(Δξ) S i terms. Thus, we have obtained an important result: the projection error and, hence, the total error in the com puted collision integral has the second order of smallness with respect to the mesh size. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
ON THE ACCURACY OF THE PROJECTION COMPUTATION
623
4. AVERAGING OF THE ERROR When comparing the exact and numerical integrals, we use the norm
∑ y Δξ , 2 γ
y =
3
γ
which is a conventional criterion for the proximity of grid functions in the theory of numerical methods. In the general case, projection error (3.10) is extremely difficult to analyze. However, for a spherically symmetric collision integral I = I ( ξ ), which depends only on the magnitude of the velocity, the analysis can be substantially simplified. The norm of the total error in the computed collision integral is δI
sum
2
= Δξ ΔI + δI = 48
∑ γ
2
2
⎛ Δξ ΔI ⎞ + 2 ⎝ 48 γ⎠
∑ γ
2
Δξ ΔI γ δI γ + 48
∑ δI . 2 γ
γ
The second term in the radicand approximates the integral
∑ γ
∞
∞
∫
∫
1 4π ξ 2 ΔI ( ξ )δI ( ξ ) dξ ≈ ΔI γ δI γ ≈ 3 ΔI ( ξ )δI ( ξ ) dξ = 3 Δξ –∞ Δξ 0
∑ ΔI δI , γ
γ
γ
where 1 δI ( ξ ) = 4π
π 2π
∫ ∫ δI ( ξ ) sin θ dθ dϕ
(4.1)
0 0
and ξ, θ, and ϕ are spherical coordinates. Let us find an expression for δI ( ξ ) . Substituting (3.10) into (4.1) and using the fact that an arbitrary vector a satisfies the equality π 2π
π 2π
⎧1 ∂
∂
1
1
∂a ϕ ⎫
( ξ a ) + ( sin θa ) + ⎬ sin θ dθ dϕ ∫ ∫ div ( a ) sin θ dθ dϕ = ∫ ∫ ⎨⎩ ξ ∂ξ ξ sin θ ∂θ ξ sin θ ∂ϕ ⎭ 2
0 0
2
ξ
θ
0 0
1 ∂⎛ 2 = 2 ⎜ ξ ξ ∂ξ ⎝
π 2π
⎞
∫ ∫ a sin θ dθ dϕ⎟⎠ , ξ
a ξ = an ξ ,
0 0
ξ nξ = , ξ
we obtain 2 1 ∂ ⎛ ξ δI ( ξ ) = ⎜ 2 2ξ ∂ξ ⎝ 4π
π 2π
π 2π
∫∫
∫∫
0
2 ⎞ ⎞ ∂D 1 ∂ ⎛ξ E i n ξi sin θ dθ dϕ⎟ + 2 ⎜ ij n ξi sin θ dθ dϕ⎟ , ∂ξ j ⎠ 4ξ ∂ξ ⎝ 4π ⎠ 0 0 0
(4.2)
where ∞
Ei =
⎧ ∂C ij ( g ) ⎫ ⎨ – B i ( g ) ⎬Φ ( ξ, ξ 1 ) dξ 1 , ∂ξ j ⎩ ⎭ –∞
∫
∞
D ij =
∫ { A ( g ) – 2C ( g ) }Φ ( ξ, ξ ) dξ . ij
ij
1
1
–∞
Representing the integrand in the second term in (4.2) as ∂D ij ∂ ( D ij n ξi ) ∂ ( D ij n ξi ) D ii D ij ∂ ξ n ξi = – D ij ⎛ i⎞ = – + n ξi n ξj ⎝ ⎠ ∂ξ j ξ ∂ξ j ξ ξ ∂ξ j ∂ξ j and again applying the angular averaging of divergence, we rewrite (4.2) in the form π 2π
2 ⎞ 1 ∂ ⎛ ξ 1 ∂ ∂ 2 ˜ ) + ξD ˜ – ξD⎞ , δI ( ξ ) = n sin θ d θ d ϕ E ⎜ ⎟ + 2 ⎛⎝ ( ξ D i ξi 2 ∂ξ 4π ⎠ ∂ξ ∂ξ ⎝ ⎠ 4ξ 2ξ 0 0
∫∫
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
(4.3)
2012
624
ANIKIN ng
ng H
H
n⊥
n⊥
Fig. 3.
where 1 ˜ = D 4π
π 2π
∫∫
D ij n ξi n ξj sin θ dθ dϕ,
1 D = 4π
0 0
π 2π
∫∫D
ii sin θ dθ dϕ.
0 0
Consider the first term in (4.3). It is easy to see that, in the case of spherical symmetry, Φ ( ξ, ξ 1 ) = Φ ( ξ, ξ 1, ξξ 1 ). In the integral with respect to ξ1, passing to spherical coordinates ξ1, θ1, and ε1, where the angle θ1 is mea sured from the vector ξ, we rewrite the integral expression in the first term in (4.2) as 1 4π
π 2π ∞ π 2π
⎛ ⎜ ⎝
⎞ ⎧ ∂C ij ( g ) ⎫ 2 ⎨ – B i ( g ) ⎬Φ ( ξ, ξ 1, ξξ 1 cos θ 1 )ξ 1 sin θ 1 dξ 1 dθ 1 dε 1⎟ n ξi sin θ dθ dε ⎠ ⎩ ∂ξ j ⎭ 0
∫ ∫ ∫∫ ∫ 0 0
00
(4.4)
∞ π 2π
=
∫ ∫ ∫ Κ ( g, θ , ε )Φ ( ξ, ξ , ξξ cos θ )ξ 1
1
1
1
1
2 1 sin θ 1 dξ 1 dθ 1 dε 1 ,
00 0
where 1 Κ ( g, θ 1, ε 1 ) = 4π
π 2π
∫ ∫ H ( g )n i
ξi sin θ dθ dϕ,
0 0
∂C H i = ij – B i . ∂ξ j
(4.5)
Note that, in integration in (4.5), the vector ξ1 also varies in such a manner that θ1, ε1 = const. In other words, when averaged with respect to ξ, the vector g turns in the same way as ξ, so that their relative posi tion remains unchanged. We introduce a basis consisting of ng = g/g and unit vectors n⊥ and n ⊥' perpendicular to ng and each other. Decomposing H over this basis gives H = H g n g + H ⊥ n ⊥ + H ⊥' n ⊥' . Based on the structure of formulas (3.1) and (3.8) and symmetry considerations, we see that, if the vector g is aligned with a coordinate axis, then H is also aligned with it; i.e., H⊥ = H ⊥' = 0. From this and symmetry considerations, we conclude that, if two Cartesian coordinates of H simultaneously reverse their signs, then the components H⊥, H ⊥' also reverse their signs, while Hg = Hng remains unchanged (see Fig. 3). Since nξng, nξn⊥, and n ξ n '⊥ in the integral in (4.5) are constant, it follows that ng nξ Κ ( g, θ 1, ε 1 ) = 4π
π 2π
∫ ∫ Hn sin θ dθ dϕ.
(4.6)
g
0 0
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
ON THE ACCURACY OF THE PROJECTION COMPUTATION
625
P(g)/Δξ2 0.25 0.20 0.15
0.10 0.05
0
1
2
3
4
5 Δξ
6
7
8
9
10
Fig. 4.
Let us find Hng. Using (3.1) and (3.8) and neglecting the O(Δξ3) terms produces 1 g i C ij = 3 Δξ
ξ γ + Δξ/2
∑ p
p gi Si
gi gδξ p δξ j dξ = 3 p g S ( + – δξ )S Δξ p ξ
∫ Γ
∫
γ
– Δξ/2
gj 2 δξ i δξ j dξ = Δξ . 12
Applying this expression and taking into account ∂gi/∂ξj = δij and δii = 3, we have 2 ∂ ( g i C ij ) ∂C ∂g g i ij = – C ij i = Δξ – C ii . ∂ξ j ∂ξ j 4 ∂ξ j
On the other hand, 1 g i B i = 3 Δξ
∑ p
p gδξ 1 g i S i p dξ = 3 p Δξ ( g + S – δξ )S p
p p ⎛ ( S – δξ )S ⎞ gδξ ⎜ 1 – ⎟ dξ p ⎝ ⎠ gS p
∫
∑∫
Γ
Γ
1 = – 3 Δξ
∑ ∫ a(S p
Γ
p
p
p
– δξ )S dξ = –A ii + C ii .
p
Combining the results yields n g n ξ ⎛ Δξ 2 n g n ξ ⎛ Δξ 2 Κ ( g, θ 1, ε 1 ) = + A ii ( g ) – 2C ii ( g )⎞ dn ξ = + P ( g )⎞ , ⎠ ⎠ 4πg ⎝ 4 g ⎝ 4
(4.7)
1 { A ( g ) – 2C ( g ) } dn . P ( g ) = ii ii g 4π
(4.8)
∫
where
∫
In the above derivation, we used the fact that complete averaging over nξ is equivalent to that over ng if θ1, ε1 = const. It follows from (3.1) and (3.8) that 2
P ( g ) = ( P 0 + O ( Δξ/g ) )Δξ , which is confirmed by a numerical experiment. Figure 4 shows the graph of P(g) computed numerically using formulas (3.1), (3.8), and (4.8) with an additional node chosen so that the coefficient a has the COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
626
ANIKIN
smallest value from all admissible ones for the given point δξ (choice no. 3 in Section 5). The function P(g)/Δξ2 rapidly tends to its limiting value P0 = 0.174 with a significant deviation from the latter occurring only at a point of several Δξ. Neglecting this deviation and substituting (4.7) into (4.4), we find the first term in (4.3), namely, 2
( 1 + 4P 0 )Δξ ∂ 2 ˜ ( ξ J ), 2 ∂ξ 8ξ where J˜ ( ξ ) =
∫ ( f ( ξ' )f ( ξ' ) – f ( ξ )f ( ξ ) )n n dS dε dξ . 1
g ξ
1
(4.9)
1
Let us simplify the second term in (4.3). Following the above line of reasoning gives 2
D = Δξ P 0 ΔI ( ξ ). ˜ = D /3. Indeed (temporarily abandoning the Einstein summation convention), On the other hand, D
∑D n
= D ξξ ,
ij ξi n ξj
i, j
where Dξξ is the corresponding component of the tensor D in spherical coordinates. After complete angu lar averaging, the components Dξξ, Dθθ, and Dϕϕ are equivalent and 1 1 D ξξ = D θθ = D ϕϕ = ( D ξξ + D θθ + D ϕϕ ) = 3 3
∑D
ii .
i
In the last equality, we used the fact that the trace of a tensor is an invariant. Combining the results, we rewrite (4.3) as P0 ⎫ 2 ⎧ 1 + 4P 0 ∂ 2 δI ( ξ ) = Δξ ⎨ ( ξ J˜ ) + ΔI ⎬. 2 ∂ξ 12 ⎭ ⎩ 8ξ
(4.10)
The norm of the error is rewritten as δI
sum
=
∑ γ
2
2
⎛ Δξ ΔI + δI γ⎞ + ⎝ 48 γ ⎠
∑ δI – ∑ δI 2 γ
γ
2 γ.
γ
Substituting (4.10) and, in a rough approximation, neglecting
∑ δI – ∑ δI 2 γ
γ
2 γ,
γ
we derive the final expression for the norm of the total error: δI
sum
2
∂ ( ξ 2 J˜ ) . 6 = ( 1 + 4P 0 ) Δξ ΔI + 2 48 ξ ∂ξ
(4.11)
Thus, the optimization of the projection method is reduced to the minimization of P0, which, in turn, is reduced to decreasing the integrand in (4.8): 1 A ii ( g ) – 2C ii ( g ) = 3 Δξ
ξ γ + Δξ/2
∑ ∫
p
p
aS i ( S i – 2δξ i ) dξ.
p ξ – Δξ/2 γ p
(Here, S p is a function of the integration variable ξ, depending on which of the domains of Γ γ it belongs to.) This implies an important practical conclusion, namely, if an additional node is chosen so that the quantity p
p
aS i ( S i – 2δξ i ), COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
ON THE ACCURACY OF THE PROJECTION COMPUTATION
627
has the smallest value at the given point δξ, then the error of the computed spherically symmetric collision integral (more exactly, its norm) has the smallest possible value. 5. NUMERICAL EXPERIMENT When computing collision integral (1.2), we discard the integration nodes giving aftercollision veloc ities going beyond the velocity grid and the sum in (1.2) extends over the remaining nodes. (Note that the factor 1/ N multiplying the sum contains the number of all integration nodes, since the discarded nodes technically remain in the sum, but Ων at them is set equal to zero.) Applied to the computation of the exact collision integral, the procedure for eliminating collisions for which the aftercollision velocities lie beyond the sphere ξcut does not change the basic properties of the integral (namely, the conservation laws and the Htheorem [4]) and adds an error to macroscopic param eters that decay exponentially with ξcut. In what follows, by the exact integral used as a reference for com parison with its numerical counterpart, we mean the integral computed in this manner. In the first numerical experiment, the collision integral was found for the spherically symmetric distri bution function 2 2 ⎧ ξ ξ ⎞⎫ 1 ⎛ – f = C ⎨ exp ⎛ – ⎞ + . exp 3/2 ⎝ 2⎠ ⎝ 2T⎠ ⎬ T ⎩ ⎭
(5.1)
∫
The constant C was determined from the concentration normalization condition n = fdξ = 1. All the computations were performed for T = 0.5. The velocity mesh size was determined by Nξ: Δξ = 2ξ cut /N ξ . The cutoff velocity was specified as ξcut = 4.8. In all the experiments, we used a collection of grids with N ξ = 30, 36, 40, 46, 50, 54, 60. The intermolecular interaction was defined by the hardsphere model and bmax = 1. Integral (1.2) was compared with the exact one calculated using the Monte Carlo method (with the use of pseudorandom Korobov grids) on the basis of the classical formula. sum
In the case of finite integration grids, the regular error δI γ
is supplemented with the random deviation:
sum sum stat δI˜ γ = δI γ + δI γ .
The norm of the total error is approximately calculated as sum δI˜ ≈
δI
sum 2
2
+ σ ,
(5.2)
stat 2
where σγ = ( δI γ ) is the variance of the random error in the computed integral Iγ. Indeed, consider the velocity nodes lying in a relatively narrow spherical layer. Since the problem is spherically symmetric, the stat collection of random error values at these nodes can be interpreted as a statistical ensemble of values δI γ stat
at a single node. Since δI γ
= 0, we then obtain (5.2).
To eliminate the statistical error, we used large integration grids with N ~ 10–14.5 million integration nodes per velocity node. To reduce the amount of computations, the computational domain was reduced to a single cubic quadrant of the sphere (out of available eight). Inside the quadrant, we used the spherical symmetry I γx, γy, γz = I γy, γx, γz = I γx, γz, γy = I γz, γy, γx . As a result, the effective cardinality of the integration grids was increased up to N ~ 80–116 million. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
628
ANIKIN ξy 2
4 7
5 γ
1 ξx
3 6 ξz Fig. 5.
The calculation of (1.2) at a single velocity node makes use of 4N integration nodes on average. The theory of Monte Carlo methods [5] implies that the rms deviation of the resulting integral from its limiting ∞) is value (as N σ γ = Κ γ / 4N,
(5.3)
where 3
4πξ cut 2 2 Κ γ = 2πb max Ω dξ 1 dS dϕ, 3
1 Ω = ( f ( ξ' ) ( ξ '1 )f – f ( ξ γ )f ( ξ 1 ) )g. 2 2π
∫
(5.4)
Accordingly, the relative random error is σ k s stat = = , I 4N
K . k = I
The integral in (5.4) was numerically evaluated, and we eventually obtained k = 4.45 for distribution func tion (5.1) and the indicated parameters ξcut and bmax. For the given values of N , formula (5.3) gives the errors sstat = (2–2.5) × 104, which agrees well with the standard estimate of σγ, which was obtained by dividing the integration grid into m uniform subgrids (in the experiment, m = 16) and computing σγ =
1 m(m – 1)
m
∑
k ( Iγ
2
– Iγ ) ,
k=1
1 I γ = m
m
∑I , k γ
k=1
k
where I γ is the numerical collision integral computed on the kth subgrid. The resulting errors were sstat = (2.2–2.7) × 10–4. The relative regular error s reg = δI
sum
/ I
varied from 0.02–0.03 for the coarsest velocity grids to 0.005 for the grid with Nξ = 60 (see Fig. 6) and was higher than the statistical error (taking into account that they are added according to the quadratic law). To improve the accuracy of sreg, the root in (5.2) was represented by its expansion up to the first term. Accordingly, 2
s stat s reg = ˜s reg – , 2s˜ reg
sum δI˜ ˜s reg = . I
(5.5)
Generally speaking, the computation error in the exact integral also has to be taken into account in (5.5). However, due to the spherical symmetry, the integral was computed to high accuracy. In the (0) experiment, the relative computation error of I γ amounted to ~(2–3) × 10–5, i.e., was negligibly small. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
ON THE ACCURACY OF THE PROJECTION COMPUTATION
629
logsreg −1.4 −1.5 −1.6
Method 1 Method 2 Method 3 Method 4
−1.7 −1.8 −1.9 −2.0 −2.1 −2.2 −2.3 −0.80 −0.75 −0.70 −0.65 −0.60 −0.55 −0.50 log(Δξ) Fig. 6.
Figure 6 shows sreg as a function of the mesh size on logarithmic scales. Four strategies were used for choosing additional nodes. 1. Minimization of P(g). Seven nodes to choose from (which, together with γ, form a cube around the aftercollision velocity, see Fig. 5) are assigned the weights a 1 ( Δξ – 2 δξ x ), a 2 ( Δξ – 2 δξ y ), a 3 ( Δξ – 2 δξ x ), 2a 4 ( Δξ – δξ x – δξ y ),
2a 5 ( Δξ – δξ y – δξ z ),
2a 6 ( Δξ – δξ x – δξ z ),
(5.6)
3a 7 ( Δξ – 2 { δξ x + δξ y + δξ z }/3 ) according to the indexing used in Fig. 5. The node with the minimal weight from the subset of nodes sat isfying constraint (3.3) on the coefficients a is chosen to be additional. According to (4.11), this choice gives the smallest error. In Fig. 6 the error is shown by circles. The slope of the line is 1.974. 2. Minimization of the distance between the aftercollision velocity and the additional node. The com puted results are marked with squares. The slope is equal to 1.978. 3. Minimization of the admissible coefficients a. The error is depicted by stars. The slope is equal to 1.925. 4. The fourth case demonstrates an inappropriate choice of additional nodes. The node to be chosen was the furthest from γ. Nodes no. 7 maximally separated from γ were chosen in most cases. The slope is equal to 1.953. In addition to the exact integral, we calculated its Laplacian, the divergence of integral (4.9), and the ratio of norms 6 ∂ 2 ΔI + 2 ( ξ J˜ ) / I = 6.77. ξ ∂ξ ˜ 0 that, in the limit Substituting this into (4.11) and using sreg for the smallest velocity step yields some P Δξ 0, must converge to its theoretical value (which, of course, is also obtained numerically using def ˜ 0 and P for each method of choosing addi inition (4.8) and formulas (3.1) and (3.8)). Table 1 compares P 0 tional nodes. Table 1
P0 ˜ P 0
Choice no. 1
Choice no. 2
Choice no. 3
Choice no. 4
0.0856
0.0988
0.1743
0.3633
0.0855
0.1035
0.1668
0.3727
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
630
ANIKIN logsreg −1.3 −1.4 −1.5 −1.6 −1.7 −1.8 −1.9 −2.0 −2.1 −2.2 −2.3
Method 1 Method 2 Method 3 Method 4
−0.80 −0.75 −0.70 −0.65 −0.60 −0.55 −0.50 log(Δξ) Fig. 7.
Thus, the numerical results agree well with the theoretical conclusions. This confirms the validity of the derivation and the assumptions made. Note an interesting and important fact that the contribution of the projection error to the total error was smaller than the contribution of the error in the nonconservative part of the integral, in particular, for optimal strategies of choosing additional nodes. Indeed, δI
sum
/ δI
non
6 ∂ ( ξ 2 ˜J ) / ΔI ≈ 0.877 ( 1 + 4P ). = ( 1 + 4P 0 ) ΔI + 0 2 ∂ξ ξ
Using P0 = 0.0856 (the value for optimal choice no. 1), we obtain δI
sum
/ δI
non
= 1.177
(the value obtained by directly computing I(non) is 1.183); i.e., the norm of the total error is only 17% larger than the norm of the error in the nonconservative integral. The optimality of choice no. 1 follows directly from the structure of (4.11), which was derived assuming that the collision integral is spherically symmetric. Is choice no. 1 optimal in the case of the general dis tribution function? In my view, the answer is in the affirmative. However, before saying general words jus tifying this assumption, we present the results produced in three numerical experiments with distribution functions that have only polar symmetry in the velocity space. In numerical experiment no. 2, the collision integral was computed for the Maxwellian distribution with an anisotropic temperature: ⎧ ξx + ξy ξz ⎞ ⎫ f = C ⎨ exp ⎛ – – ⎬, ⎝ 2⎠⎭ 2T ⎩ 2
2
2
T = 0.5.
(5.7)
The integration grids were used with N ~ 33–49 million. Due to the polar symmetry I γx, γy, γz = I γy, γx, γz , the cardinality of the grids increased to N ~ 66–98 million. The statistical error was larger than in the first case, sstat = (4.6–6.1) × 10–4, but still sufficiently small. In experiment nos. 3 and 4, the bimodal distribution function 2 2 ⎧ ξ – U) ⎞ ( ξ + U )⎞ ⎫ f = C ⎨ exp ⎛ – ( + exp ⎛ – ⎬. ⎝ ⎝ 2 ⎠ 2 ⎠⎭ ⎩
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
(5.8)
Vol. 52
No. 4
2012
ON THE ACCURACY OF THE PROJECTION COMPUTATION
631
logsreg −1.6 Method 1 Method 2 −1.7 Method 3 Method 4 −1.8 −1.9 −2.0 −2.1 −2.2 −2.3 −2.4 −2.5 −2.6 −0.80 −0.75 −0.70 −0.65 −0.60 −0.55 −0.50 log(Δξ) Fig. 8.
logsreg 1.8 −1.9
Method 1 Method 2 Method 3 Method 4
−2.0 −2.1 −2.2 −2.3 −2.4 −2.5 −2.6 −0.80 −0.75 −0.70 −0.65 −0.60 −0.55 −0.50 log(Δξ) Fig. 9.
was used in the computations. In both cases, the vector U had the identical magnitude U = 0.75, but its direction relative to the velocity grid was different. In experiment no. 3, the vector was aligned with the ξz axis: U = ( 0, 0, U ). In experiment no. 4, it was rotated by an angle of 45° relative to the axes in the (ξy, ξz) plane: U U U = ⎛ 0, , ⎞ . ⎝ 2 2⎠ In experiment no. 3, the final cardinalities of the integration grids were N ~ 132–196 million with sstat = (2.1–3) × 10–4. In experiment no. 4, the grid cardinality was less by half and, accordingly, sstat = (3–4.2) × 10–4. In all three cases, the parameters for the integral computation (the cutoff velocity, the intermolecular potential, and the velocity mesh sizes) were the same as in the first experiment. Figures 7–9 show the errors sreg for experiment nos. 2–4, respectively. All the important results obtained in the first experiment are preserved (at least, at the qualitative level) for the distribution func tions with polar symmetry. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
632
ANIKIN
Table 2 Experiment
Choice no. 1
Choice no. 2
Choice no. 3
Choice no. 4
No. 2 No. 3 No. 4
1.953 1.958 1.945
1.944 1.943 1.961
1.992 1.984 1.923
2.002 2.014 1.954
Experiment
Choice no. 1
Choice no. 2
Choice no. 3
Choice no. 4
No. 2 No. 3 No. 4
1.488 1.560 1.477
1.643 1.690 1.525
1.562 1.532 1.954
3.465 3.336 2.580
Table 3
The projection method is secondorder accurate. Table 2 gives the slopes of the graphs for each method of choosing additional nodes. The Laplacians of the collision integrals were also calculated: ⎧ 5.615 ΔI = ⎨ I ⎩ 3.302
for
( 5.7 ),
for
( 5.8 ).
Accordingly, the computational errors in the nonconservative integrals were estimated. Table 3 presents sum non their ratio to the total error δI . / δI Thus, in the case of successful choices, the basic contribution to the total error is also made by I non. Choice no. 4 is again unsuccessful, and the discrepancy between it and the optimal strategies becomes even larger. Inspection of Table 3 shows that, irrespective of the distribution function, choice no. 1 is superior or nearly superior (in the third experiment, choice no. 1 is only 2% inferior to choice no. 3 in terms of accuracy). 6. APPLICABILITY OF CHOICE NO. 1 IN THE GENERAL CASE It is reasonable to assume that the optimality of choice no. 1, which was rigorously justified in the case of spherical symmetry, is not random in the examples of distribution functions with polar symmetry. Below are some arguments justifying the superiority of this choice to the others in the general case. In view of the symmetry of the problem, coefficients (3.8) are periodic function of the direction of g with the period being the solid angle made, for example, by nodes 2, 5, and 7 (see Fig. 5). Due to the spher ical symmetry of the distribution function, the periodic part of the coefficients vanishes after angular aver aging, and the optimization of the projection method is reduced to the minimization of P(g). Of course, for an arbitrary distribution function, the periodic part of the coefficients can correlate with local features of the distribution function, so that choice no. 1 becomes unsuccessful. However, since the coefficients exhibit fast periodic oscillations (the complete solid angle contains 48 periods), the probability of a signif icant correlation is sufficiently low. As a result, by analogy with the usual onedimensional integral with a rapidly oscillating function that tends to zero, the basic contribution to the error is again made by the con stant part of the coefficients. This issue can be viewed from a different angle. Usually, the distribution function is spatially dependent and, at a qualitative level, the error in the computed integral can be spatially averaged so as to reduce the possible local nonoptimality of choice no. 1. For example, a distribution function that structurally and qualitatively resembles the above bimodal distribution arises in a strong shock wave or in the related problem of a supersonic gas flow impinging on an obstacle, for example, on a wing edge. Then U in (5.8) is the vector difference between the velocities of the flow impinging on and reflected from the wing: U = V – V'. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
ON THE ACCURACY OF THE PROJECTION COMPUTATION
633
However, near the edge (which is the most interesting and important domain for analysis), the directions of the reflected gas velocity V' (and the directions of U) lie within a large solid angle. As a result, choice no. 1, according to the third and fourth experiments, even though inferior to the other choices in certain directions, surpasses them overall. Thus, the theoretical results and the numerical experiments suggest that the method for choosing addi tional nodes based on the minimization of P(g) is optimal in the general case. 7. STATISTICAL ERROR As was indicated above, the relative statistical error of the computed integral at a particular node is ~( 4N )–1/2. Even for a velocity grid with Nξ = 30 consisting of N0 = 14328 velocity nodes on an integration grid of cardinality Nν = 106 nodes (for which significant computational resources are usually required), the error is on the order of 0.06, which several times as much as the corresponding regular error in the numerical experiments performed. As the velocity step decreases, this discrepancy becomes even larger, since the statistical error grows as sstat ~ O(Δξ–3/2), while the regular error decreases as sreg ~ O(Δξ2). Is it reasonable to focus on the regular error when the statistical error occurring in practice is so large? The matter is that, generally speaking, the distribution function itself is of no interest, since it carries excessively detailed information. The goal is always some macroscopic parameters obtained by integrating the distribution function over the velocity space with a certain weight. In this case, important is the accu racy of quantities obtained by averaging the collision integral over large volumes of velocity space rather than the accuracy of the collision integral itself. Moreover, in contrast to the regular systematic error, the random deviations of the computed integral at individual nodes somehow cancel each other and the sta tistical error of a macroscopic parameter is determined by Nν for the entire integration grid. Let us illus trate what was said by solving the onedimensional heat conduction problem. The stationary onedimensional Boltzmann equation is ∂f = I ( f, f ). ξ x ∂x
(7.1)
In the presence of a weak temperature gradient, the collision integral is given by (see [6]) 2 1 ∂T I ( ξ ) = f M ( ξ ) ξ x ⎛ ξ – 5 ⎞ , ⎝ T ∂x 2⎠
(7.2)
where fM(ξ) is a local Maxwellian distribution. Assume that the temperatures of the boundaries differ slightly. Therefore, fM(ξ) and the temperature gradient are nearly independent of x, and the integral can be treated as spatially homogeneous. From this, the solution of Eq. (6.1) is the function (L is the distance between the boundaries) x ⎧ f ( 0, ξ ) + I ( ξ ), ξ x > 0, ξx ⎪ f ( x, ξ ) = ⎨ – x I ( ξ ), ξ < 0. ⎪ f ( L, ξ ) – L x ⎩ ξx
(7.3)
The approximation to the integral is found with an error depending chaotically on the time step: Iγ ( tj ) = Iγ + δj Iγ . Consider the case of ξx > 0. Solution (6.3) can be rewritten as x/ξ x
f ( x, ξ ) = f ( 0, ξ ) +
x/ ( ξ x τ )
∫ I ( ξ ) dt = f ( 0, ξ ) + τ ∑
I ( ξ ).
j=1
0
Approximating this expression yields the solution x/ ( ξ γ τ )
x/ ( ξ γ τ ) x
fγ ( x ) = fγ ( 0 ) + τ
∑
j=1
x
Iγ ( tj ) = f γ ( x ) + τ
∑
δj Iγ
(7.4)
j=1
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
634
ANIKIN
up to O(τ2) (here, we omitted the notation for the unimportant discretization in the spatial variable). An expression similar to (6.4) is derived for ξx < 0. By using (6.4), the error in the computed energy flux can be written as x–L
3
Δξ δq ( x ) = 2
∑ γ
x
⎛ ⎞ ξγ τ ξγ τ Nξ x x 3 ⎜ N ξ /2 ⎟ 2 2 2 Δξ τ δj Iγ + ξ γx ξ γ δ j I γ⎟ . ξ γx ξ γ ξ γx ξ γ δf γ ( x ) = ⎜ 2 ⎜ ⎟ γ x = N ξ /2 + 1 γ y, γ z j–1 j–1 ⎝ γx = 1 γy, γz ⎠
∑∑
∑
∑ ∑
∑
(7.5)
The random variables appearing under the summation signs in (6.5) are independent, since they cor respond to different time steps or different velocity nodes. Accordingly, the rms deviation of the entire expression is determined by adding up the squared deviations of the terms. Since the average deviation of the collision integral is independent of time and is even about ξx = 0, we obtain 1 6 δq = Δξ Lτ 2
Nξ
∑ ∑ξ
2
4 2 γ x ξ γ δI γ .
(7.6)
γ x = N ξ /2 + 1 γ y, γ z
By way of estimation, we assume that the relative error is independent of γ and σ γ = 1/ 4N.
(7.7)
Substituting (6.2) and (6.7) into (6.6) and replacing the sum with the resulting integral yields ∞
∞ ∞
∫
∫∫
3 1 2 2 3 δq = σ C 0 Δξ Lτ ξ x dξ x 2 2
0
–∞ –∞
4 2 5 2 –ξ2 ξ ⎛ ξ – ⎞ e dξ y dξ z , ⎝ 2⎠
1 ∂T C 0 = . 3/2 ( 2π ) ∂x
(the temperature and concentration are assumed to weakly differ from unity everywhere in the domain). After simple but cumbersome rearrangements, we obtain ∞
2
–ξx 3 2 4 6 8 2 2 3 2 2 3 δq = 1 σ C 0 Δξ Lτ ξ x ⎛ 13 + 13 ξ x + 13 ξ x – ξ x + ξ x ⎞ e dξ x = 135 σ C 0 Δξ Lτ. ⎝ ⎠ 2 2 2 4 4 2
∫
(7.8)
0
By using the energy flux for a hardsphere gas [6] (in dimensionless form) 75 ∂T q = – 2π 64 ∂x and applying the equality 3
N0 πξ cut 3 σ δξ = Δξ = 4N ν 3N ν 2
3
the relative random error in the calculated heat flux is finally written as 2 Cq δq = s q ≡ , 1/2 q Nν
3
C q = 0.1285 Lτξ cut .
(7.9)
Thus, as expected from general considerations, sq depends only on the total number of integration nodes; therefore, sq is relatively small and is independent of the velocity mesh size (i.e., of the number of velocity nodes). Note that the random error is sensitive to an increase in ξcut, which, among other things, provides additional evidence in favor of the maximum possible restriction of the cutoff sphere. Figure 10 shows sq against the size of the integration grid on logarithmic scales as obtained in two numerical experiments. The parameters of the problem were specified as L = 50,
ξ cut = 4.8,
τ = 0.2.
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
ON THE ACCURACY OF THE PROJECTION COMPUTATION
635
logsq −1.6 Korobov MonteКаrlo
−1.8 −2.0 −2.2 −2.4 −2.6 −2.8 −3.0
−3.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 logNv Fig. 10.
In the first experiment (depicted by squares), randomly generated Monte Carlo grids were used for integration. The resulting line agrees well with the theoretical prediction: for the indicated parameters, formula (6.9) yields Cq = 4.27 and the numerical results well fit into the approximating line (solid) – 0.508
4.09N ν
,
In the second experiment, the collision integral was computed with the help of pseudorandom Korobov grids [7] (referred to by Korobov himself as optimal parallelepipedal nets). It is easy to see that the Korobov grids perform better than the Monte Carlo ones. Specifically, Fig. 2 shows that the former produce smaller – 0.7
errors that decay according to the faster law ~ N ν . Note that the Korobov grids yield a better overall pat tern of the solution due to the correlations between its parts, while the accuracy of the computed collision integral at an individual velocity node is nearly identical for both types of grids and, in both cases, – 0.5
is O( N ). The fact is that the Kronecker deltas in (1.2) “cut out” a very small part of the 8dimensional integration grid, so that correlations do not have enough time to manifest themselves. In this context, it –1
was stated in [1, 2] that the random error of the computed integral on Korobov grids is O( N ν ), which is not an accurate assertion. 8. CONCLUSIONS The accuracy of the projection method as applied to the computation of the collision integral was examined. It was shown that its basic unavoidable error associated with the spread of an individual con tribution to the integral sum over nodes neighboring the aftercollision velocity has the second order of smallness with respect to the mesh size. Therefore, the interpolation law chosen for the distribution func tion depending on the problem in question has to have equivalent or even better accuracy. Previously, a second pair of velocity nodes “interpolating” an individual contribution in the projection method was chosen according to intuitive considerations (velocities that are nearest to the aftercollision velocities and lead to the smallest spread coefficient, etc.). Based on the expression derived for the projec tion error, an optimal method for choosing additional nodes was found. As a result, the total accuracy of the computed collision integral was improved. Numerical experiments were performed that confirmed the optimality of the method proposed. Additionally, the statistical error in the computed integral was analyzed. The superiority of Korobov grids to traditional Monte Carlo meshes was shown by solving the heat conduction problem. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
636
ANIKIN
REFERENCES 1. F. G. Cheremisin, “A Conservative Method for Calculation of the Boltzmann Collision Integral,” Dokl. Phys. 42, 607–610 (1997). 2. F. G. Tcheremissine, “Solution to the Boltzmann Kinetic Equation for HighSpeed Flows,” Comput. Math. Math. Phys. 46, 315–329 (2006). 3. Y. Sone, Molecular Gas Dynamics (Birkhäuser, Boston, 2007). 4. F. Rogier and J. Schnider, “A Direct Method for Solving the Boltzmann Equation,” Transport Theory Stat. Phys. 23 (1–3) (1994). 5. I. M. Sobol’, Numerical Monte Carlo Methods (Nauka, Moscow, 1973). 6. L. D. Landau and E. M. Lifshitz, Physical Kinetics (Pergamon, Oxford, 1964; Fizmatlit, Moscow, 2001). 7. N. M. Korobov, Exponential Sums and Their Applications (Nauka, Moscow, 1989; SpringerVerlag, New York, 1992).
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 52
No. 4
2012
