François Dubois
email: francois.dubois@math.u-psud.fr

Pierre Lallemand
email: pierre.lallemand1@free.fr

Quartic Parameters for Acoustic Applications of Lattice Boltzmann Scheme

Keywords:

Using the Taylor expansion method, we show that it is possible to improve the lattice Boltzmann method for acoustic applications. We derive a formal expansion of the eigenvalues of the discrete approximation and fit the parameters of the scheme to enforce fourth order accuracy. The corresponding discrete equations are solved with the help of symbolic manipulation. The solutions are obtained in the case of D3Q27 lattice Boltzmann scheme. Various numerical tests support the coherence of this approach.

1) Introduction

• The lattice Boltzmann equation methodology is a general framework for approximating problems that are modelled by partial differential equations under conservative form arising in physics. It has been first proposed for fluid dynamics in the context of cellular automata (see e.g. Hardy et al [START_REF] Hardy | Time Evolution of a Two-Dimensional Classical Lattice System[END_REF], Wolfram [START_REF] Wolfram | Cellular Automaton fluids: basic theory[END_REF], d'Humières et al [START_REF] Humières | Lattice gas models for 3D-hydrodynamics[END_REF]). The classical lattice Boltzmann scheme (McNamara-Zanetti [START_REF] Mc Namara | Use of Boltzmann equation to simulate lattice gas automata[END_REF], Higuera et al [START_REF] Higuera | Lattice gas dynamics with enhanced collisions[END_REF], Qian et al [START_REF] Qian | Lattice BGK for Navier-Stokes equation[END_REF], d'Humières [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF]) has been first developed for nonlinear fluid problems. It has also the capability to approximate thermal flows. (Chen et al [START_REF] Chen | Thermal lattice Bhatnagar-Gross-Krook model without nonlinear deviations in macrodynamic equations[END_REF]), magnetohydrodynamics (Dellar [START_REF] Dellar | Lattice kinetic schemes for magnetohydrodynamics[END_REF]) or coupling between fluid and structures (see e.g. Kwon [START_REF] Kwon | Coupling of Lattice Boltzmann and Finite Element Methods for Fluid-Structure Interaction Application[END_REF]).

• We have proposed in [START_REF] Dubois | Une introduction au schéma de Boltzmann sur réseau[END_REF][START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF] the Taylor expansion method to analyze formally the d'Humières lattice Boltzmann scheme [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF] when the mesh size tends to zero. We have then replaced the (much more) formal Chapman-Enskog expansion methodology used in [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF] by a simple Taylor expansion relative to the grid spacing. In this way we can obtain the modified equations of the scheme (in the sense of Lerat-Peyret [START_REF] Lerat | Noncentered Schemes and Shock Propagation Problems[END_REF] and Warming-Hyett [START_REF] Warming | The modified equation approach to the stability and accuracy analysis of finite difference methods[END_REF]; see also [START_REF] Griffiths | On the scope of the method of modified equations[END_REF][START_REF] Carpentier | On the derivation of the modified equation for the analysis of linear numerical methods[END_REF][START_REF] Villatoro | On the method of modified equations. V: Asymptotic analysis of and direct-correction and asymptotic successive-correction techniques for the implicit midpoint method[END_REF]) at an arbitrary order in the general nonlinear case. We made more explicit in [START_REF] Dubois | Third order equivalent equation of lattice Boltzmann scheme[END_REF] the result at third order accuracy in all generality. We have observed at this occasion that a serious difficulty with the lattice Boltzmann scheme lies in the fact that the equivalent mass conservation equation contains an a priori non-null third order term. We have also proposed an algorithm to derive the modified equation in the case of linearized problems [START_REF] Dubois | Towards higher order lattice Boltzmann schemes[END_REF]. We have applied this methodology to derive the so-called "quartic parameters" to enhance the accuracy of the lattice Boltzmann scheme to simulate shear waves [START_REF] Dubois | Towards higher order lattice Boltzmann schemes[END_REF]. An application of this result is used by Leriche et al [START_REF] Leriche | Stokes eigenmodes in cubic domain: primitive variable and Lattice Boltzmann formulations[END_REF] for computing with spectral method and lattice Boltzmann scheme the eigenmodes of the Stokes problem in a cube. This Taylor expansion method can also be used for a detailed analysis of boundary conditions. In [START_REF] Dubois | On a superconvergent lattice Boltzmann boundary scheme[END_REF] we have shown that the "magic boundary parameters" of Ginzburg and Adler [START_REF] Ginzburg | Boundary flow condition analysis for three-dimensional lattice Boltzmann model[END_REF] depend in fact upon the detailed choice of the parameters of the lattice Boltzmann scheme.

• In this contribution, we consider linearized athermal acoustics in two and three space dimensions. In Section 2, we recall the essential properties of the d'Humières scheme. Then in Section 3, we use the method of formal expansion to expand the discrete eigenmodes of the acoustic system of partial differential equations as the mesh size tends to zero. In Section 4, we increase the accuracy of the lattice Boltzmann scheme with the development of "quartic" parameters and develop also a weaker approach by enforcing isotropy at fourth order accuracy. Applications in two and three space dimensions (D2Q13 and D3Q27 schemes) are presented in Sections 5 and 6 respectively. We detail our version of the D3Q27 scheme and the explicit formulae for the determination of the quartic parameters.

2) d'Humières lattice Boltzmann scheme

• In the framework proposed by d'Humières [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF], the lattice Boltzmann scheme uses a regular lattice L with typical mesh size ∆x and for each node x in L, a discrete set of (J + 1) velocities V is given. In this contribution, the set V does not depend on the vertex x. We introduce a velocity scale λ and the time step ∆t is obtained according to the so-called "acoustic scaling" :

(1) ∆t = ∆x λ

For x ∈ L and v j ∈ V, the point x + v j ∆t is supposed to be a new vertex of the lattice. The unknown of this lattice Boltzmann scheme is the particle distribution f j (x, t) for x ∈ L, 0 ≤ j ≤ J and discrete values of time t. The numerical scheme proceeds into two major steps.

• The first step describes the relaxation f -→ f * of particle distribution f towards a locally defined equilibrium. It is local in space and nonlinear in general. In this paper devoted to acoustics we consider only linear contributions. It is defined with the help of a fixed invertible matrix M. The moments m k are defined through a simple linear relation

(2) m k = J j=0 M kj f j , 0 ≤ k ≤ J .
Note that this very simple linear hypothesis is not satisfied in the scheme proposed by Geier et al [START_REF] Geier | Cascaded digital lattice Boltzmann automata for high Reynolds number flow[END_REF]. The first d + 1 moments (where d is the space dimension, equal to d = 2 or d = 3 in the present applications) are the total density ρ and the momentum q α :

(3)

           m 0 = ρ ≡ J j=0 f j m α = q α ≡ J j=0 v α j f j , 1 ≤ α ≤ d .
We denote by W the vector (in R N ) composed of the density and the components of the momentum. These moments are supposed to be at equilibrium:

(4) m * i = m i ≡ m eq i ≡ W i , 0 ≤ i ≤ d .
The relaxation evolution m -→ m * due to linearized collisions is local and trivial for the vector W of conserved quantities, as remarked in [START_REF] Chen | Thermal lattice Bhatnagar-Gross-Krook model without nonlinear deviations in macrodynamic equations[END_REF]. For the other moments, a distribution of equilibrium moments G(•) is given as a (linear for the present study of acoustic waves) function of the vector W :

(5)

m eq k = G k (W ) , d < k ≤ J .
For k ≥ N, the evolution m -→ m * is supposed to be uncoupled:

(6) m * k = m k + s k (m eq k -m k ) , k > d.
It is parametrized by the so-called "relaxation rate parameters" s k . For stability reasons (see e.g. Lallemand and Luo [START_REF] Lallemand | Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF]), we have the inequalities

(7) 0 < s k < 2 , k > d.
When the components m * k (x, t) are computed for each x ∈ L at discrete time t, the distribution f * j (x, t) after relaxation is reconstructed by inversion of relation ( 2):

(8) f * j = J ℓ=0 M -1 jℓ m * ℓ , 0 ≤ j ≤ J .
• The second step is just a free advective evolution without collision through characteristics:

(9)

f j (x, t + ∆t) = f * j (x -v j ∆t, t) , 0 ≤ j ≤ J , x ∈ L .
• The asymptotic analysis of cellular automata provides evidence supporting asymptotic partial differential equations with transport coefficients related to the induced parameter defined by the so-called Hénon's relation [START_REF] Hénon | Viscosity of a Lattice Gas[END_REF] (10)

σ k ≡ 1 s k - 1 2 .
The lattice Boltzmann scheme is classically considered as second order accurate (see e.g. [START_REF] Lallemand | Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF]). We describe in Section 6 the D3Q27 (d = 3, J = 26) numerical scheme that we used for acoustic application.

3) Formal development of eigenmodes

• With the method described in [START_REF] Dubois | Towards higher order lattice Boltzmann schemes[END_REF], it is possible to derive the equivalent system of (d + 1) equations associated to a lattice Boltzmann scheme. The system of linearized conservation equations issued from this analysis can be written under the form

(11) A(∆x, ∂) • W = 0
which uses a compact notation stating that A(∆x, ∂) is a (d + 1) × (d + 1) (4 × 4 for three-dimensional applications, 3 × 3 for two-dimensional models) matrix of differential operators of high order acting on the conserved variables W .

We seek the eigenmodes of the operator A(∆x, ∂), id est the eigenvalues λ j (∆x, ∂) and the eigenvectors r j (∆x, ∂) such that [START_REF] Dubois | On a superconvergent lattice Boltzmann boundary scheme[END_REF] A(∆x, ∂)

• r j (∆x, ∂) = λ j (∆x, ∂) r j (∆x, ∂) , 0 ≤ j ≤ d.
We introduce the diagonal matrix Λ(∆x, ∂) composed of the eigenvalues λ j (∆x, ∂) and the square matrix R(∆x, ∂) composed of the eigenvectors. Then relation [START_REF] Dubois | On a superconvergent lattice Boltzmann boundary scheme[END_REF] can be written under the following synthetic form:

(13) A(∆x, ∂) • R(∆x, ∂) = R(∆x, ∂) • Λ(∆x, ∂) .
• Moreover, the operator A(∆x, ∂) in equation ( 11) is a formal series in the variable ∆x that is truncated at order 4 in this contribution:

(14) A(∆x, ∂) ≡ A 0 (∂) + ∆x A 1 (∂) + ∆x 2 A 2 (∂) + ∆x 3 A 3 (∂) + O ∆x 4 .
We can apply in this case the so-called stationary perturbation theory for linear operators (see e.g. [START_REF] Cohen-Tannoudji | Quantum Mechanics, two volumes[END_REF] for an elementary introduction). First for ∆x = 0, the operator A 0 (∂) is exactly the perfect linear acoustic model. For d = 3 (the case d = 2 is a simple consequence of this particular analysis and we omit the details), we have

(15) A 0 (∂) =         ∂ t ∂ x ∂ y ∂ z c 2 0 ∂ x ∂ t 0 0 c 2 0 ∂ y 0 ∂ t 0 c 2 0 ∂ z 0 0 ∂ t         .
Introducing the Laplacian operator ∆ ≡ ∂ 2 x + ∂ 2 y + ∂ 2 z , a system R 0 (∂) of reference eigenvectors can be given under the form

(16) R 0 (∂) =         c 0 √ ∆ -c 0 √ ∆ 0 0 c 2 0 ∂ x c 2 0 ∂ x ∂ y -∂ 2 xz c 2 0 ∂ y c 2 0 ∂ y -∂ x -∂ 2 yz c 2 0 ∂ z c 2 0 ∂ z 0 ∂ 2 x + ∂ 2 y         .
It is elementary to observe the relation

(17) A 0 (∂) • R 0 (∂) = R 0 (∂) • Λ 0 (∂)
and the associated matrix Λ 0 (∂) of order zero is simply given by ( 18)

Λ 0 (∂) =      ∂ t + c 0 √ ∆ 0 0 0 0 ∂ t -c 0 √ ∆ 0 0 0 0 ∂ t 0 0 0 0 ∂ t      .
We observe that even if the eigenvalue ∂ t is degenerated, i.e. when the eigenvalues are multiple, the family of eigenvectors proposed in ( 16) is complete and defines a basis. This allows us to show the existence of two acoustic waves and two shear waves (see e.g. [START_REF] Landau | Fluid Mechanics[END_REF]).

• The parameter ∆x is supposed to be infinitesimal and we introduce a formal expansion of the eigenvalues with diagonal matrices Λ j (∂):

(19) Λ(∆x, ∂) ≡ Λ 0 (∂) + ∆x Λ 1 (∂) + ∆x 2 Λ 2 (∂) + ∆x 3 Λ 3 (∂) + O ∆x 4
and relative perturbations Q j (∂) of the eigenvectors:

(20) R(∆x, ∂) ≡ R 0 (∂) • Id + ∆x Q 1 (∂) + ∆x 2 Q 2 (∂) + ∆x 3 Q 3 (∂) + O(∆x 4 ) .
We adopt in this work the scaling condition for eigenvectors proposed e.g. by Cohen-Tannoudji et al [START_REF] Cohen-Tannoudji | Quantum Mechanics, two volumes[END_REF]. The component of r j (∆x, ∂) relative to the corresponding unperturbed eigenvector is exactly equal to unity. In other words, all the diagonal terms of the Q j (∂) matrices are null:

(21) Q j (∂) ℓℓ = 0 , 0 ≤ ℓ ≤ d , j ≥ 1 .
We insert the expansions ( 20) and ( 21) into the eigenmode condition [START_REF] Geier | Cascaded digital lattice Boltzmann automata for high Reynolds number flow[END_REF]. We introduce the expression A j (∂) for the perturbation relative to the unperturbed eigenbasis, id est

(22) A j (∂) ≡ R 0 (∂) -1 • A j (∂) • R 0 (∂) .
Then we identify each term relative to ∆x and we obtain in this manner:

(23) Λ 1 (∂) ≡ A 1 (∂) + Λ 0 (∂) • Q 1 (∂) -Q 1 (∂) • Λ 0 (∂) (24) Λ 2 (∂) ≡ A 2 (∂) + A 1 (∂) • Q 1 (∂) -Q 1 (∂) • Λ 1 (∂) + Λ 0 (∂) • Q 2 (∂) -Q 2 (∂) • Λ 0 (∂) (25) Λ 3 (∂) ≡      A 3 (∂) + A 2 (∂) • Q 1 (∂) -Q 1 (∂) • Λ 2 (∂) + A 1 (∂) • Q 2 (∂) -Q 2 (∂) • Λ 1 (∂) +Λ 0 (∂) • Q 3 (∂) -Q 3 (∂) • Λ 0 (∂)
.

Explicit forms of the eigenvalues at each order may be found without difficulty using symbolic manipulation.

• We introduce a reference density ρ 0 , a reference sound velocity c 0 , the shear viscosity µ and the bulk viscosity ζ. It is well known (see e.g. Landau-Lifshitz [START_REF] Landau | Fluid Mechanics[END_REF]) that the four compressible isothermal acoustic linearized equations take the form:

(26)                    ∂ t ρ + ρ 0 div u = 0 ρ 0 ∂ t u + c 2 0 ∂ x ρ -µ ∆u -ζ + µ 3 ∂ x div u = 0 ρ 0 ∂ t v + c 2 0 ∂ y ρ -µ ∆v -ζ + µ 3 ∂ y div u = 0 ρ 0 ∂ t w + c 2 0 ∂ z ρ -µ ∆w -ζ + µ 3 ∂ z div u = 0
with div u ≡ ∂ x u + ∂ y v + ∂ z w and we recognize matrix ( 15) when µ = ζ = 0. This system has two families of eigenvalues.

• On the one hand, the acoustic waves λ ± are given by ( 27)

λ ± = ∂ t -γ ∆ ± c 0 √ ∆ 1 + γ 2 c 2 0 ∆
and are parametrized by the attenuation of the sound waves γ given according to

(28) γ = 1 2 ρ 0 ζ + d + 1 d µ .
We can expand the previous acoustic wave eigenvalues in powers of γ when the attenuation is sufficiently small compared to the frequency of the sound waves. Then we have the expansion

(29) λ ± = ∂ t ± c 0 √ ∆ -γ ∆ ± 1 2 γ 2 c 0 ∆ 3/2 + O(γ 4 ) .
The previous (relatively unusual) notations with matrices of operators are exactly the one used when implementing without difficulty (due to linearity) our approach with a symbolic manipulation software. As is well known, the pseudo-differential operators like √ ∆ and ∆ 3/2 are defined by their action on Fourier transforms (we refer e.g. to the book of Hörmander [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators[END_REF]). It is also possible to introduce a Fourier decomposition on harmonic waves of the type exp i (ω tk•x) . Then we have the usual change of notation: 3 , etc. The corresponding dispersion relation associated with the annulation of the eigenvalues λ ± given in [START_REF] Qian | Lattice BGK for Navier-Stokes equation[END_REF] allows the introduction of the parameter Γ ℓ according to [START_REF] Qian | Simulating thermohydrodynamics with lattice BGK models[END_REF] -

∂ t ≡ i ω, ∇ ≡ -i k, ∆ ≡ -|k| 2 , ∆ 3/2 ≡ -i |k|
i ω ≡ Γ ℓ = γ |k| 2 ± i c 0 |k| 1 - γ 2 2 c 2 0 |k| 2 + O γ 4 |k| 5 .
• On the other hand, the shear wave λ 0 define a multiple eigenmode that is simply given by ( 31)

λ 0 = ∂ t -ν ∆ with ν ≡ µ ρ 0 .
With the spectral notation, relation [START_REF] Rubinstein | Theory of the lattice Boltzmann equation: Symmetry properties of discrete velocity sets[END_REF] can be written with the help of the attenuation Γ t of the shear waves under the form ( 32)

-i ω ≡ Γ t = ν |k| 2 .
As previously, a family of two independent eigenvectors with two different "polarizations" corresponds to this shear eigenvalue. Moreover, the perturbation ∆x 14) is diagonal on the basis of unperturbed eigenvectors proposed at relation [START_REF] Griffiths | On the scope of the method of modified equations[END_REF]. Due to well-established arguments (see e.g. the textbook of Quantum Mechanics of Cohen-Tannoudji et al [START_REF] Cohen-Tannoudji | Quantum Mechanics, two volumes[END_REF] or e.g. the sections "multiple roots" and "degenerate roots" in the book of Hinch [START_REF] Hinch | Perturbation methods[END_REF]), the expansion (20) ( 21) is completely valid even in this (relatively) complicated degenerate case.

A 1 (∂)+∆x 2 A 2 (∂)+ ∆x 3 A 3 (∂) + • • • of relation (

4) Increasing the accuracy of shear and acoustic waves

• When we use a lattice Boltzmann scheme, the viscosity µ admits a representation of the type (c.f. the D3Q27 scheme):

(33) µ = 1 3 λ ∆x σ x
and similarly the bulk viscosity ζ is given for the D3Q27 scheme (to fix the ideas) according to

(34) ζ = λ ∆x σ e 5 9 -c 2 0
with the notations of Section 6. We can identify the developments [START_REF] Hinch | Perturbation methods[END_REF] on the one hand and (29)(31) on the other hand because, due to (33), ( 34) and ( 28), the viscosities γ and µ are proportional to ∆x. The usual implementation of lattice Boltzmann scheme consists in adjusting the eigenvalues Λ 0 (∂) and Λ 1 (∂) in order to fit the viscosities.

• When we identify the developments ( 19) and (31) at the order 4 for the shear waves, we recover the "quartic parameters" obtained previously [START_REF] Dubois | Towards higher order lattice Boltzmann schemes[END_REF], in particular for the D2Q9 [START_REF] Lallemand | Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF] and D3Q19 [START_REF] Tölke | Lattice Boltzmann Simulations of Binary Fluid Flow through Porous Media[END_REF] schemes. In these cases, the shear viscosity µ follows the "quartic" condition

(35) µ = 1 √ 108 λ ∆x .
If we add to the previous study the fitting of the acoustic waves [START_REF] Qian | Lattice BGK for Navier-Stokes equation[END_REF], there is no solution of the equations for the two previous schemes D2Q9 and D3Q19. We have then considered the previous conditions for extended stencils such as the D2Q13 [START_REF] Qian | Simulating thermohydrodynamics with lattice BGK models[END_REF][START_REF] Weimar | Nonlinear reactions advected by a flow[END_REF] and D3Q27 (see e.g. [START_REF] Mei | Lattice Boltzmann method for 3-D flows with curved boundary[END_REF]) lattice Boltzmann schemes. The results for quartic coefficients are displayed in Section 6 when fitting all the waves ( 29) and ( 31) of the linear problem.

• In order to take advantage of the flexibility of the d'Humières scheme, we have developed an "isotropic" condition, not as strong as the quartic condition. With this isotropic condition, we do not impose anymore the conditions that the numerical waves [START_REF] Hinch | Perturbation methods[END_REF] fit exactly the physical ones ( 29) and ( 31) but suggest that the numerical waves are isotropic.

In other words, the operator matrices Λ 2 (∂) (for the order 3) and Λ 3 (∂) (for the order 4) are differential operators that are constrained in order to depend only on the radial variable r (r 2 ≡ x 2 +y 2 +z 2 ) and on the operator ∂/∂r . When this condition is imposed, it is possible to fit isotropic waves for the four previous schemes. In Table 1 and 2, we display the number of (a priori non-independent) discrete equations that must be satisfied by the parameters of the lattice Boltzmann schemes when we impose isotropy for the shear waves, isotropy for the acoustic waves, exact fitting of the shear waves and exact fitting of the acoustic waves. All these discrete equations have been solved exactly with the help of symbolic manipulation. To fix the ideas, we give in Section 6 the parametrization of the quartic shear and acoustic waves for the D3Q27 lattice Boltzmann scheme (solution of 18 discrete equations, according to Note that the same numbers apply to both lattices.

5) Linearized acoustics with the D2Q13 scheme

• Two numerical "experiments" have been performed: emission from the center (Figures 1, 2 and 3) and emission from the boundary of the circle (see Figures 1,[START_REF] Chen | Thermal lattice Bhatnagar-Gross-Krook model without nonlinear deviations in macrodynamic equations[END_REF][START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF][START_REF] Humières | Lattice gas models for 3D-hydrodynamics[END_REF]. In both cases, a first order anti-bounce-back boundary condition (see e.g. [START_REF] Bouzidi | Momentum transfer of a Boltzmann lattice fluid with boundaries[END_REF]) is implemented to impose a given density (pressure) on the boundary: either a constant when sound is emitted at the center, or a sinusoidal time dependent value for the emission by the boundary. In each case, we measure the value of the density at a fixed position in space vs time (see Figure 6) or we analyze the pressure field at a given time (see Figures 2 to 5). Obviously the maximum duration of the simulation with a radius R and sound velocity c 0 is R/c 0 when the source is at the center and 2R/c 0 when it is on the circular boundary. These experiments are sensitive to anisotropic behaviour of both the velocity and of the attenuation as can be seen in the figures.

harmonic source harmonic source

6) D3Q27 athermal linearized Acoustics

• The D3Q27 Lattice Boltzmann scheme, described with details e.g. in Mei et al. [START_REF] Mei | Lattice Boltzmann method for 3-D flows with curved boundary[END_REF] is simply obtained by taking a tensorial product for the set V of discrete velocities: [START_REF] Wolfram | Cellular Automaton fluids: basic theory[END_REF] v j = λ ε x , ε y , ε z , ε x , ε y , ε z = -1, 0, +1 , 0 ≤ j ≤ 26 .

Due to the large number of moments, we detail in this subsection the way the matrix M parametrizing the transformation (2) is obtained. First, velocities v α j for 0 ≤ j ≤ J ≡ 26 and 1 ≤ α ≤ 3 are given according to relation [START_REF] Wolfram | Cellular Automaton fluids: basic theory[END_REF]. The first four conserved moments ρ and q α are determined according to [START_REF] Cohen-Tannoudji | Quantum Mechanics, two volumes[END_REF]. The generation of other moments is done according to the tensorial nature of the variety of moments that can be constructed, as analyzed by Rubinstein and Luo [START_REF] Rubinstein | Theory of the lattice Boltzmann equation: Symmetry properties of discrete velocity sets[END_REF]: scalar fields are naturally coupled with one another, idem for vector fields, and so on. So components of kinetic energy are introduced:

(37) m 4 = e ≡ 26 j=0 3
β=1 v β j 2 f j
The entire set of second order tensors is completed according to (38)

           m 5 = XX ≡ 26 j=0 2 (v 1 j ) 2 -(v 2 j ) 2 -(v 3 j ) 2 f j m 6 = W W ≡ 26 j=0 (v 2 j ) 2 -(v 3 j ) 2 f j (39)                        m 7 = XY ≡ 26 j=0 v 1 j v 2 j f j m 8 = Y Z ≡ 26 j=0 v 2 j v 3 j f j m 9 = ZX ≡ 26 j=0 v 3 j v 1 j f j
Flux of the energy and of the square of energy:

(40)

           m 9+α = ϕ α ≡ 3 26 j=0 3 β=1 v β j 2 v α j f j m 12+α = ψ α ≡ 9 2 26 j=0 3 β=1 v β j 2 2 v α j f j , 1 ≤ α ≤ 3
Square and cube of the energy:

(41) (42)

           m 16 = ε ≡ 3 2 26 j=0 3 β=1 v β j 2 2 f j m 17 = e 3 ≡ 9
           m 18 = XX e ≡ 3 26 j=0 2 (v 1 j ) 2 -(v 2 j ) 2 -(v 3 j ) 2 3 β=1 v β j 2 f j m 19 = W W e ≡ 3 26 j=0 (v 2 j ) 2 -(v 3 j ) 2 3 β=1 v β j 2 f j
Extra-diagonal second order moments of the energy: two different components times energy and permutations (43)

                       m 20 = XY e ≡ 3 26 j=0 v 1 j v 2 j 3 β=1 v β j 2 f j m 21 = Y Z e ≡ 3 26 j=0 v 2 j v 3 j 3 β=1 v β j 2 f j m 22 = ZX e ≡ 3 26 j=0 v 3 j v 1 j 3 β=1 v β j 2 f j j=0 M ij M kj = 0 for i = k .
Note that the coefficients in (37), ( 38), (39), ( 40), ( 41), ( 42), ( 43), ( 44) and ( 45) have been chosen such that the coefficients of matrix M are all integers.

• The equilibrium values of the moments define the equilibrium distribution. The first moments ρ and q α are in equilibrium and have respectively a scalar and a vectorial structure. The equilibrium values of the other moments respect this structure. We have (47)

                              
e eq = θ λ 2 ρ XX eq = W W eq = XY eq = Y Z eq = ZX eq = 0 ϕ eq α = c 1 λ 2 q α ψ eq α = c 2 λ 4 q α ε eq = β λ 4 ρ e eq 3 = ξ λ 6 ρ XX eq e = W W eq e = XY eq e = Y Z eq e = ZX eq e = 0 τ eq α = c 3 λ 2 q α XY Z eq = 0 .

We have the following relation between the parameter θ and the sound velocity c 0 :

(48) θ = 3 c 2 0 -2 .
The relaxation parameters for these moments are denoted respectively by the following values (49)

                                  
e eq : s e XX eq , W W eq , XY eq , Y Z eq , ZX eq : s x ϕ eq α : s ϕ ψ eq α : s ψ ε eq : s ε e eq 3 : s ξ XX eq e , W W eq e : s γ XY eq e , Y Z eq e , ZX eq e : s χ τ eq α : s τ XY Z eq : s ω .

With these relaxation parameters, the shear viscosity is isotropic when

(50) c 1 = -2 c 3 = 0
• The quartic parameters for athermal acoustics are solution of the 13 + 2 + 3 = 18 discrete equations obtained at Table 2. They are displayed in this subsection.

(51)

     β = - 1 2 (σ x -σ e ) -9 c 2 0 σ x -18 c 2 0 σ e + 27 c 4 0 σ x + 180 c 2 0 σ x σ 2 e +144 c 2 0 σ 3 x -8 σ x + 8 σ e -324 c 4 0 σ x σ 2 e (52) c 2 = -42 σ 2 x + 5 2 (53) σ ϕ = 1 12 σ x
                                     σ ε = - 1 48 N ε D ε N ε = -76 σ 2 x + 27 c 2 0 -27 c 4 0 -180 c 2 0 σ 2 e -468 c 2 0 σ 2 x + 324 c 4 0 σ 2 e + 7776 c 4 0 σ 4 x -93312 c 4 0 σ 4 x σ 2 e -4 σ 2 e + 80 σ x σ e -336 σ 3 x σ e -1344 σ 2 e σ 2 x + 240 σ 4 x -10800 c 2 0 σ 3 x σ e -46656 c 4 0 σ 3 x σ 3 e + 20736 σ 3 x σ 3 e c 2 0 + 62208 c 2 0 σ 5 x σ e -324 c 4 0 σ x σ e + 3888 c 4 0 σ x σ 3 e + 864 c 2 0 σ 2 x σ 2 e -4752 c 2 0 σ x σ 3 e +51840 c 2 0 σ 4 e σ 2 x + 3888 c 4 0 σ 2 e σ 2 x -46656 c 4 0 σ 4 e σ 2 x + 324 c 2 0 σ e σ x -3456 σ 6 x + 2880 σ 3 x σ 3 e + 20736 c 2 0 σ 6 x + 8064 σ 4 x σ 2 e + 6912 σ 5 x σ e -864 c 2 0 σ 4 x + 1440 σ x σ 3 e -14400 σ 4 e σ 2 x + 324 c 4 0 σ 2 x + 31104 c 2 0 σ 4 x σ 2 e D ε = σ x (σ x -σ e ) -84 σ 3 x + 432 c 2 0 σ 3 x + 84 σ 2 x σ e + 540 c 2 0 σ x σ 2 e -23 σ x -972 c 4 0 σ x σ 2 e -27 c 2 0 σ x + 81 c 4 0 σ x + 23 σ e -54 c 2 0 σ e
We observe that the presence of the factor (σ x -σ e ) in the denominator of the algebraic expression (54) of σ ε makes this expression incompatible with a BGK or TRT [START_REF] Ginzburg | Study of Simple Hydrodynamic Solutions with the Two-Relaxation-Times Lattice Boltzmann Scheme[END_REF] type hypothesis. The same remark holds for relations (55) and (56) relative to the parameters σ γ and σ χ respectively.

(55)

               σ γ = 1 336 σ x (-1 + 12 σ 2 x ) (σ x -σ e ) 2 N γ N γ = 1968 σ 4 x -144 c 2 0 σ 2 x -15552 c 2 0 σ 4 x -624 σ 2 e σ 2
x -1344 σ 3

x σ e +4608 σ 5

x σ e + 8064 σ (59)

               σ χ = 1 96 σ x (-1 + 84 σ 2 x ) (σ x -σ e ) 2 N χ N χ = 192 σ 4 x -828 c 2 0 σ 2 x + 972 c 4 0 σ 2 x + 6480 c 2 0 σ 2 x σ 2 e -
                         σ τ = 1 12
N τ D τ N τ = 76 σ 2 x -
                                 σ ω = 1 12 N ω D ω N ω = -816 σ 4 x -828 c 2 0 σ 2 x + 972 c 4 0 σ 2 x + 6480 c 2 0 σ 2 x σ 2 e -
             s e = 0.95057034220532319391634980988593155893536121673004 s x = 1.8552875695732838589981447124304267161410018552876 c 0 = 0.623538 c 2 = 2.436118000 ξ = 1
and the "quartic" relaxation parameters:

(60)

                               β = 0.50345521670787922851706691010021052631578947368420 s φ = 0.37925445705024311183144246353322528363047001620745 s ψ = 1.3 s ε = 0.34253657030513141274711235609982461596733955718034 s ξ = 1.2 s γ = 1.9945477114942149093456286590460711496091258797386 s χ = 1.2940799466197218037166471960307116873345869400515 s τ = 1.9451616927239606019667013153498030552793202566039 s ω = 0.25131560984615404581501005329813711811837082814760 .
Of course, we used the previous numbers with the usual truncation allowed by the 32 bit or 64 bit floating point arithmetic. We observe that for these particular parameters, due to the link (33) between the shear viscosity µ and the parameter σ x , to relation [START_REF] Warming | The modified equation approach to the stability and accuracy analysis of finite difference methods[END_REF] between the bulk viscosity ζ and the parameter σ e and to formula [START_REF] Mei | Lattice Boltzmann method for 3-D flows with curved boundary[END_REF] Periodic wave with the lattice Boltzmann scheme D3Q27. Error for shear eigenvalue Γ t defined in [START_REF] Tölke | Lattice Boltzmann Simulations of Binary Fluid Flow through Porous Media[END_REF]. For quartic parameters, we have Γ t -ν |k| 2 = O(|k| 6 ). Periodic wave with the lattice Boltzmann scheme D3Q27. Error for the imaginary part of the acoustic eigenvalue Γ ℓ defined in [START_REF] Qian | Simulating thermohydrodynamics with lattice BGK models[END_REF]. For quartic parameters, we have fifth order accuracy:

Im Γ ℓ -c 0 |k| 1 -γ 2 |k| 2 /(2 c 2 0 ) = O(|k| 5
) with γ detailed at relation [START_REF] Mei | Lattice Boltzmann method for 3-D flows with curved boundary[END_REF]. The numerical experiments have been done for a three-dimensional converging acoustic wave. A pulsating sphere of radius R = 46.08 is embedded in a 95 3 cube. The simulations used only 64 bit or even 32 bit data with CUDA: (using a Nvidia 9800GT card, one full update of D3Q27 takes approximately 11 nanoseconds per site). The "linear anti-bounceback" numerical boundary conditions (in the sense given in Bouzidi et al. [START_REF] Bouzidi | Momentum transfer of a Boltzmann lattice fluid with boundaries[END_REF]) are used to impose a boundary density on the sphere oscillating with a period T = 10. Since the sound velocity is around 0.62 (see (59)), there are around six mesh points by wave length. In Figure 10, the scatter plot of density vs radius after 82 iterations is displayed. The results are of excellent quality with the use of quartic parameters. Nevertheless, since the attenuation of the sound wave is relatively important for the above parameters (γ ≈ 0.0547 as recalled in (61)), the range of propagation is relatively limited (five wave lengths in our case, see Figure 10). For the less stringent "isotropic" case, i.e. to fix the ideas (62)

         σ φ = 1 6 σ x β = 4 -9c 2 0 c 0 = 1 √ 3
we can find situations of "TRT type" satisfying (63) σ φ = σ ψ = σ τ = σ ω σ e = σ x = σ ǫ = σ ξ = σ γ = σ χ with much lower attenuation of sound. An example is given in figure 10 with parameters of the scheme following relations (62) and (63) with σ x = 0.006, that is shear viscosity 0.002 and sound attenuation 0.002. These results show that choosing the goal in terms of accuracy or small attenuation will influence the choice of the parameters in the lattice Boltzmann simulation. . Sound wave emitted by a sphere. Numerical results with the D3Q27 lattice Boltzmann scheme with "TRT" isotropic parameters described at relations (62) (63) and with quartic parameters given by relations (59) and (60). Six points by wave length.

7) Conclusion

• We have proposed to use the Taylor expansion method in conjunction with the use of symbolic manipulation to increase the accuracy of the lattice Boltzmann scheme in the multiple time relaxation approach proposed by D. d'Humières for linear acoustic waves. The result is the use of the previous scheme with very particular "quartic" parameters. We have obtained a family of such parameters for the D3Q27 numerical scheme. Numerical experiments confirm the predictions of the theoretical analysis. The catch is to make sure that situations are stable. This problem cannot be solved in practice with developments around small wave numbers. In the absence of efficient techniques to find stable situations with small viscosities and/or sound attenuation, we tried at random a number of sets of the free parameters and have used the best of them for the explicit results shown above. We note that the less stringent "isotropic" condition is compatible with small viscosities which may be quite useful for practical simulations. Comparison of various stencils is under way and will be presented elsewhere.

Figure 1 .

 1 Figure 1. Diverging (left) and converging (right) acoustic test cases.

Figure 2 .

 2 Figure 2. D2Q13 Acoustic propagation, diverging acoustic test case, time period = 8 (7 points per wave length, c 2 0 = 0.80): usual (left), isotropic (center) and quartic (right).

Figure 3 .

 3 Figure 3. D2Q13 diverging acoustic test case. Measurements after 200 time steps; usual parameters (left) and quartic parameters (right).

Figure 4 .

 4 Figure 4. D2Q13 Converging acoustic propagation, time period = 8 (7 points per wave length, c 2 0 = 0.80): usual (left), isotropic (center) and quartic (right).

Figure 5 .

 5 Figure 5. D2Q13 Converging acoustic propagation, time period = 10 (9 points per wave length, c 2 0 = 0.80): usual (left), isotropic (center) and quartic (right).

Figure 6 .

 6 Figure 6. D2Q13 Converging acoustic propagation, time period = 8 (left) and time period = 10 (right).

  Product of XX and WW by the energy:

François

  error of shear waves logarithm of the inverse of the wave vector usual : order 4 quartic : order 6

Figure 7 .

 7 Figure 7.Periodic wave with the lattice Boltzmann scheme D3Q27. Error for shear eigenvalue Γ t defined in[START_REF] Tölke | Lattice Boltzmann Simulations of Binary Fluid Flow through Porous Media[END_REF]. For quartic parameters, we have Γ t -ν |k| 2 = O(|k| 6 ).

  error of sound velocity logarithm of the inverse of the wave vector usual : order 3 quartic : order 5

Figure 8 .

 8 Figure 8.Periodic wave with the lattice Boltzmann scheme D3Q27. Error for the imaginary part of the acoustic eigenvalue Γ ℓ defined in[START_REF] Qian | Simulating thermohydrodynamics with lattice BGK models[END_REF]. For quartic parameters, we have fifth order accuracy:Im Γ ℓ -c 0 |k| 1 -γ 2 |k| 2 /(2 c 2 0 ) = O(|k| 5) with γ detailed at relation[START_REF] Mei | Lattice Boltzmann method for 3-D flows with curved boundary[END_REF].

6 Figure 9 .

 69 Figure 9. Periodic wave with the lattice Boltzmann scheme D3Q27. Error for the real part of acoustic eigenvalue Γ ℓ . For quartic parameters, we have Re Γ ℓ -γ |k| 2 = O(|k| 6 ) .

Figure 10

 10 Figure 10. Sound wave emitted by a sphere. Numerical results with the D3Q27 lattice Boltzmann scheme with "TRT" isotropic parameters described at relations (62) (63) and with quartic parameters given by relations (59) and (60). Six points by wave length.

Table 2

 2

			).
		isotropic shear	isotropic acoustic	exact shear	exact acoustic
	order 3	0	1	0	2
	order 4	1	1	2	2

Table 1 .

 1 Fitting "isotropic" and "exact" shear and acoustic waves at various orders of accuracy. Number of equations for the D2Q9 and D2Q13 lattice Boltzmann schemes. Note that the same numbers apply to both lattices.

		isotropic shear	isotropic acoustic	exact shear	exact acoustic
	order 3	0	1	0	2
	order 4	10	2	13	3

Table 2 .

 2 Fitting "isotropic" and "exact" shear and acoustic waves at various orders of accuracy. Number of equations for the D3Q19 and D3Q27 lattice Boltzmann schemes.

  The results of simple periodic waves are displayed in Figures7, 8 and 9. The numerical results confirm that parameters proposed in (59) and (60) allow one to get fourth order (relative) accuracy.

	•
			making explicit
	the attenuation γ of sound waves, we have
		 	µ = 0.013 λ ∆x
	(61)		ζ = 0.0920492 λ ∆x γ = 0.0546913 λ ∆x .

Acknowledgments

The referees conveyed to the authors very interesting remarks. Some of them have been incorporated into the present edition of the article.

Third order pseudovector τ α (44)

with a natural permutation convention and the third order totally antisymmetric tensor XY Z:

(45)

A first (non-orthogonal) matrix M is obtained by using (3) (37) (38) (39) (40) (41) (42) (43) (44) and (45) as linear combinations of the f j 's :

(46)

Then matrix M is orthogonalized from relations (3), (

, ( 44) and ( 45) with a Gram-Schmidt orthogonalization algorithm:

The coefficients g iℓ are computed recursively in order to enforce orthogonality:


Gerald Reiner
email: gerald.reiner@vetmed.uni-giessen.de

Christina Fresen

Sebastian Bronnert

Ingo Haack

Hermann Willems

: S Bronnert

: H Willems

Prevalence of infection in hunted wild

Keywords: Haemophilus parasuis, Epidemiology, Wild boar

published or not. The documents may come

Introduction

Haemophilus parasuis infection is considered one of the economically most important diseases in swine because of the high costs of antibiotic treatments and loss of animals [START_REF] Oliveira | Development of a PCR test to diagnose Haemophilus parasuis infection[END_REF]. The gram-negative, non-haemolytic, nicotinamide adenine dinucleotide-dependent bacterium is the etiological agent of Glässer's disease, a syndrome characterized by fibrinous polyserositis, arthritis, and meningitis. The bacterium is an early colonizer of pigs and is commonly isolated from the upper respiratory tract of healthy individuals [START_REF] Oliveira | Development of a PCR test to diagnose Haemophilus parasuis infection[END_REF][START_REF] Oliveira | Haemophilus parasuis: new trends on diagnosis, epidemiology and control[END_REF][START_REF] Rapp-Gabrielson | Haemophilus parasuis[END_REF]. Regarding the lower respiratory tract, H. parasuis has been isolated from domestic pigs with pneumonia, but the organism is not commonly isolated from normal lungs (Little 1970;[START_REF] Moller | Optimization of the detection of NAD dependent Pasteurellaceae from the respiratory tract of slaughterhouse pigs[END_REF]. Isolation of H. parasuis from pigs with pneumonia has increased substantially during the past few years [START_REF] Rapp-Gabrielson | Haemophilus parasuis[END_REF].

There is little information available regarding H. parasuis infection in wild boars (Sus scrofa). Glässer's disease has never been reported [START_REF] Olvera | First isolation of Haemophilus parasuis and other NAD-dependent Pasteurellaceae of swine from European wild boars[END_REF]), and in one of two studies that have attempted to detect H. parasuis in wild boar by serology, no antibodies were detected in 78 animals from southcentral Spain [START_REF] Vicente | Antibodies to selected viral and bacterial pathogens in European wild boars from southcentral Spain[END_REF]. In the second study, 18% of wild boars in Slovenia were seropositive [START_REF] Vengust | A serological survey of selected pathogens in wild boar in Slovenia[END_REF]. In an attempt to isolate H. parasuis from the respiratory tract of wild boars in Spain, [START_REF] Olvera | First isolation of Haemophilus parasuis and other NAD-dependent Pasteurellaceae of swine from European wild boars[END_REF] cultured two strains of H. parasuis from nasal swaps of 4.8% of boars sampled.

Diagnosis of H. parasuis infections is mostly based on clinical signs, presence of lesions, and isolation of the organism. H. parasuis, however, is a fastidious organism, which hampers its isolation from clinical samples [START_REF] Rapp-Gabrielson | Haemophilus parasuis[END_REF]).

In the present study, we used a PCR assay to evaluate the prevalence and distribution of H. parasuis infection in wild boars in Germany. To determine if there is a difference between the prevalence of infection between the upper and lower respiratory tract of wild boars, we analysed the tonsils [START_REF] Oliveira | Development of a PCR test to diagnose Haemophilus parasuis infection[END_REF][START_REF] Rapp-Gabrielson | Haemophilus parasuis[END_REF] and lung tissue (Little 1970;[START_REF] Moller | Optimization of the detection of NAD dependent Pasteurellaceae from the respiratory tract of slaughterhouse pigs[END_REF]) from each boar separately.

Materials and methods

The total area of Germany covers 357,000 km 2 . Actual numbers of wild boars are unknown, however, during each hunting season from 2004 to 2007, an average of 400,000 wild boars were shot [START_REF] Djv | ? meta_id=250 Little TWA (1970) Haemophilus infection in pigs[END_REF]. Samples for the present study were collected between latitudes 48.1°and 54.5°N and between longitudes 6.1°and 14.0°E.

Tonsils and lung samples were collected from 531 boars during 52 hunts. The hunts took place in 14 of the 16 federal states of Germany. Samples were collected from boars hunted from November 2004 to March 2007. Samples were taken randomly. Age was determined based on eruption of teeth [START_REF] Stubbe | Vom Frischling zum Hauptschwein[END_REF]. The sex ratio of the sampled wild boars was 44%:56% (males/females). The average age was 11 months (range, 1 to 84 months) and the average carcass weight was 35 kg (range, 5 to 120 kg).

Samples of approximately 5 cm 3 were taken 0.5 to 2 h after death using new sterile disposable gloves and scalpel blades to collect each tonsil and lung sample. Tonsils were excised from the dorsal part of the pharynx. Lung samples were removed from the right dorsal lobe. They were stored in individual freezer bags at-20°C until further usage.

DNA was extracted separately from lung and tonsil samples with the Qiagen Viral RNA Mini kit (Qiagen, Hilden, Germany) as described elsewhere [START_REF] Reiner | Prevalence of Actinobacillus pleuropneumoniae infection in hunted wild boars (Sus scrofa) in Germany[END_REF]).

H. parasuis-specific DNA was amplified as described by [START_REF] Reiner | Prevalence of Actinobacillus pleuropneumoniae infection in hunted wild boars (Sus scrofa) in Germany[END_REF], but with hairpin-shaped primers HPS-forward (5′-TGA TGA GGA AGG GTG GTG T-3′) and HPS-reverse (5′-GGC TTC GTC ACC CTC TGT-3′) [START_REF] Oliveira | Development of a PCR test to diagnose Haemophilus parasuis infection[END_REF]) and an annealing temperature of 63°C. The sensitivity of this PCR is 10 2 CFU/mL [START_REF] Olvera | First isolation of Haemophilus parasuis and other NAD-dependent Pasteurellaceae of swine from European wild boars[END_REF]. The PCR covers all H. parasuis serotypes [START_REF] Oliveira | Development of a PCR test to diagnose Haemophilus parasuis infection[END_REF]) and excludes other Pasteurellaceae or less-related bacteria, except Actinobacillus indolicus. In order to exclude amplification of A. indolicus, we used a Taq DNA polymerase, which makes it possible to increase the annealing temperature from 59°C [START_REF] Oliveira | Development of a PCR test to diagnose Haemophilus parasuis infection[END_REF]) to 63°C and to decrease the extension time from 2 to 1 min. Success of this approach was tested with up to 10 8 DNA copies of A. indolicus isolate 46KC2 (German Collection of Microorganisms and Cell Cultures, DSMZ) per PCR reaction, which did not provide any signal.

Statistics were calculated with the Statistical Package for Social Sciences, version 15.0 (SPSS, Inc., Chicago, IL, USA). Differences in tissue prevalence and effects of hunting ground, season, and sex were calculated with Chi 2 -test. Effects of age and body weight were calculated by the Spearman correlation analysis. The threshold for statistical significance was set to <0.05.

Results

The overall prevalence of H. parasuis in wild boars (tissue independent) in Germany was 74.2%. H. parasuis was detected in 69.1% of the tonsils and 40.4% of the lungs. H. parasuis was detected in the tonsils but not in the lungs in 33.8%, in the lungs but not in the tonsils in 5.1%, and in both tissues in 35.5% of the sampled wild boars. Figure 1 shows the spatiotemporal distribution of hunting grounds and the respective prevalence of H. parasuis. Over the complete time period, there were significant differences in prevalence between states, with overall lowest values in Berlin, Saarland, Hesse, and Brandenburg and highest values in Schleswig-Holstein, Bavaria, Baden-Wurtemburg, and Mecklenburg-West Pomerania (Fig. 1).

Whether wild boars were infected with H. parasuis or not was not significantly influenced by sex, age, or body weight.

Discussion

The present study provides prevalence of H. parasuis in wild boars in Germany. H. parasuis DNA was detected in 74.2% of the sampled individuals by PCR. Seroprevalences reported in recent studies from Spain [START_REF] Vicente | Antibodies to selected viral and bacterial pathogens in European wild boars from southcentral Spain[END_REF] and Slovenia [START_REF] Vengust | A serological survey of selected pathogens in wild boar in Slovenia[END_REF] were low (0 to 18%) and [START_REF] Olvera | First isolation of Haemophilus parasuis and other NAD-dependent Pasteurellaceae of swine from European wild boars[END_REF] detected H. parasuis in 4.8% of wild boars from Spain by culture and PCR. Although these studies represent different methods, prevalences are consistently lower than those identified in the present study. The difference may thus be based more on the hunting location than on the methodology.

H. parasuis is regularly isolated from the upper respiratory tract of healthy pigs [START_REF] Oliveira | Development of a PCR test to diagnose Haemophilus parasuis infection[END_REF][START_REF] Oliveira | Haemophilus parasuis: new trends on diagnosis, epidemiology and control[END_REF][START_REF] Rapp-Gabrielson | Haemophilus parasuis[END_REF], indicating that pigs can be a reservoir for this pathogen without the necessity to develop disease. Thus, the usefulness of the PCR with the nasal or tonsillar swabs in clinical diagnostics has been questioned [START_REF] Olvera | First isolation of Haemophilus parasuis and other NAD-dependent Pasteurellaceae of swine from European wild boars[END_REF]. With regard to the lower respiratory tract, H. parasuis is more frequently isolated from cases with pneumonia than from normal lungs (Little 1970; [START_REF] Moller | Optimization of the detection of NAD dependent Pasteurellaceae from the respiratory tract of slaughterhouse pigs[END_REF]). Significant differences in virulence between H. parasuis are well documented and susceptibility of pigs to H. parasuis-associated disease depends on the individual resistance and immunity as well as on distinct risk factors. Thus, whether an H. parasuis infection stays in the upper or reaches the lower respiratory tract, depends on both, factors of the agent and of the host that have not been addressed by the present study. Considerable differences in the prevalence of Actinobacillus pleuropneumoniae between the lungs (6.4%) and the tonsils (34.8%) argue for the carrier status of wild boars regarding a further member of the Pasteurellaceae [START_REF] Reiner | Prevalence of Actinobacillus pleuropneumoniae infection in hunted wild boars (Sus scrofa) in Germany[END_REF]. We have, therefore, isolated DNA from the tonsils and lung tissue separately. With the improved PCR assay, prevalences of the H. parasuis DNA turned out to be high in both tissues. However, in spite of the high prevalence of H. parasuis in the lung tissue, the sampled wild boars did not show any gross pathological signs of disease.

In conclusion, wild boars have to be interpreted as a reservoir for H. parasuis. While cross contamination between samples can be excluded because of the sampling technique, these high lung prevalences without signs of disease might be explained by either low virulence of the strains or high resistance of the wild boar to H. parasuis or both. The high prevalence of H. parasuis in wild boars in Germany might be interpreted as a high risk for domestic pig infection as the spread of pathogens from wild boars to domestic pigs is not uncommon. On the other hand, wild boar populations might be under risk to be infected with H. parasuis from domestic pig herd emissions. Further research should aim at the identification of H. parasuis strains involved and their respective virulence and at comparing them with strains isolated from domestic pigs. This will provide further insight into the degree and impact of exchange of H. parasuis between both species.

Fig. 1

 1 Fig. 1 Spatiotemporal distribution of wild boars infected with H. parasuis in Germany. Numbers show prevalence by hunting ground (H. parasuis DNA in the tonsils left and lung tissue right, respectively). White circles hunting season 2004/2005, light grey circles hunting season 2005/2006, dark grey circles hunting season

Acknowledgements We thank the German Research Foundation (DFG; Deutsche Forschungsgemeinschaft) for the financial support. We thank Mrs. Stoll and Mrs. Hopf for perfect assistance.