Hélène Barucq
email: helene.barucq@inria.fr

Nathan Rouxelin
email: nathan.rouxelin@inria.fr

Sébastien Tordeux
email: sebastien.tordeux@inria.fr

Low-order Prandtl-Glauert-Lorentz based Absorbing Boundary Conditions for solving the convected Helmholtz equation with Discontinuous Galerkin methods

Keywords: convected Helmholtz equation, Absorbing Boundary Conditions, ABC, Prandtl Glauert Lorentz transformation

We construct Absorbing Boundary Conditions (ABCs) for the convected Helmholtz equation that are easy to implement in a Hybridizable Discontinuous Galerkin (HDG) formulation. The construction is based upon the Prandtl-Glauert-Lorentz map which transforms the convected Helmholtz equation into the regular Helmholtz equation. The new ABCs are thus issued from classical Bayliss-Gunztburger-Turkel ABCs and are valid for carrier flows that vary inside the computational domain but become uniform far away from the source. They lead to accurate numerical results for low and intermediate Mach numbers using a HDG formulation with the acoustic potential and the total flux as unknowns.

Introduction

The study of acoustic waves in the presence of a flow is involved in many applications. The work presented here is motivated by the objective to develop an advanced computational framework to simulate acoustic waves propagating at shallow depths in the sun and in its atmosphere. In [START_REF] Rouxelin | Mixed condensed numerical methods for convected wave propagation[END_REF], the time harmonic Galbrun equations have been studied first and numerical experiments illustrate that in the presence of a flow, the simulations can be unstable if continuous finite elements are used. From this came the idea to develop a computational library based on less regular approximation spaces in order to better respect the regularity properties of the solution of the Galbrun equations. Our choice was to implement a Discontinuous Galerkin (DG) method and we opted for the development of Hybridizable DG (HDG) formulations in order to work with discrete systems of reasonable size. A direct DG approximation leads indeed to systems whose size is largely superior to the one obtained with a continuous finite element approximation, even more so if we use high orders of approximation. The advantage of HDG formulations is that they are constructed as mixed DG methods to which static condensation can be applied. The global system is thus defined to compute an auxiliary unknown defined only on the skeleton of the mesh and the global solution is reconstructed at low cost by solving in parallel local problems of very small size. For a nice introduction to HDG methods, we refer to [START_REF] Kirby | To CG or to HDG: A Comparative Study[END_REF]. As a first step, we consider here the convected Helmholtz equation and as in helioseismology the waves propagate in an infinite medium, we address the question of limiting externally the computational domain in the purpose of using finite element methods. Domain truncation is important for computational wave dynamics, and the two most popular truncation techniques are:

• Perfectly Matched Layers (PMLs), introduced in [START_REF] Berenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF]; they surround the domain of interest and involve a damping parameter introduced in the equations in such a way that (i) waves are perfectly transmitted across the interface between the real medium and the artificial layer and (ii) the waves are sufficiently absorbed in the layer to avoid the possible back propagation of waves from the layer;

• Absorbing Boundary Conditions (ABCs), introduced in [START_REF] Engquist | Absorbing Boundary Conditions for the Numerical Simulation of Waves[END_REF]; they select waves outgoing to the computational domain limited externally with a boundary meant to be transparent in an ideal world. Exact ABCs are often expressed in terms of nonlocal operators and finding local approximation of those operators is a problem that remains difficult and yet crucial to perform large-scale simulations with reasonable computational costs.

Regarding PMLs, it has been shown that standard PMLs might be unstable because of the possible existence of back-propagating modes. In [START_REF] Bécache | Perfectly matched layers for the convected helmholtz equation[END_REF], stabilized PMLs have been derived for the time-harmonic convected Helmholtz equation in a duct with a uniform flow, adapting ideas formerly developed in [START_REF] Hagstrom | Absorbing layers and radiation boundary conditions for jet flow simulations[END_REF] and [START_REF] Fang | A stable, perfectly matched layer for linearized euler equations in unsplit physical variables[END_REF] for time-dependent problems. In a very recent work [START_REF] Marchner | Stable perfectly matched layers with lorentz transformation for the convected helmholtz equation[END_REF], Marchner et al. have proposed a stabilized PML for the convected Helmholtz equation following the approach in [START_REF] Dubois | Lorentz transform and staggered finite differences for advective acoustics[END_REF] of applying the Prandtl-Glauert-Lorentz transformation (PGL) to remove the external flow to end up with the standard Helmholtz equation for which stable PMLs exist. It is worth noting that in [START_REF] Dubois | Lorentz transform and staggered finite differences for advective acoustics[END_REF], the PML is constructed for the advective acoustic equation when the velocity of the fluid is parallel to a particular direction while [START_REF] Marchner | Stable perfectly matched layers with lorentz transformation for the convected helmholtz equation[END_REF] only assumes the flow is uniform and subsonic. In this paper, we focus on the construction of ABCs for the convected Helmholtz equation which can be implemented easily in the different HDG formulations that we have introduced in a previous work [START_REF] Barucq | HDG and HDG+ methods for harmonic wave problems with convection[END_REF]. To the best of our knowledge, this has not yet been done. Actually, high-order ABCs for the convected Helmholtz equation have been constructed and implemented in finite difference schemes and continuous finite element formulation [START_REF] Hagstrom | High-Order Radiation Boundary Conditions for the Convective Wave Equation in Exterior Domains[END_REF][START_REF] Bécache | High-order Absorbing Boundary Conditions for anisotropic and convective wave equations[END_REF]. In [START_REF] Hagstrom | High-Order Radiation Boundary Conditions for the Convective Wave Equation in Exterior Domains[END_REF], the construction is based upon a multipole expansion of outgoing solutions and high-order conditions are obtained from the combination of loworder derivatives thanks to auxiliary unknowns. The resulting conditions are implemented in a finite difference scheme, assuming the flow is uniform outside a bounded region and aligns with a coordinate axis. In [START_REF] Bécache | High-order Absorbing Boundary Conditions for anisotropic and convective wave equations[END_REF], the construction is extended to anisotropic and subsonically convective media in the case of rectangular artificial boundary, numerical experiments with finite elements being reserved to the anisotropic case. Following the original idea of [START_REF] Hagstrom | High-Order Radiation Boundary Conditions for the Convective Wave Equation in Exterior Domains[END_REF], complete absorbing (or radiation) boundary conditions have been proposed in [START_REF] Kim | Analysis of the convected Helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions[END_REF] for the case of the convected Helmholtz equation with a uniform flow in a waveguide. Very recently, radiation boundary conditions have been constructed by applying a micro-local factorization of the convected Helmholtz operator when the flow is aligned with a coordinate axis [START_REF] Marchner | Construction and Numerical Assessment of Local Absorbing Boundary Conditions for Heterogeneous Time-Harmonic Acoustic Problems[END_REF]. The construction of an exact ABC is also feasible from a theoretical point of view but it involves a non-local operator whose discretization induces an additional computational load. Some attempts have been carried out, for instance in [CES14, BCD + 14] where the non-local operator is approximated with an integral equation coupled with a volume finite element solution. Another approach consists in using infinite elements (or wave-envelope elements) as it is proposed in [START_REF] Astley | Infinite elements for transient flow acoustics | 7th AIAA/CEAS Aeroacoustics Conference and Exhibit[END_REF] for the convected Helmholtz equation. To deal with the unbounded nature of the domain, the solution is computed as a truncated multipole expansion outside a convex bounded domain which contains the sources. Then the solution is computed in conjunction with finite elements inside the bounded domain. It turns out that the coupling with DG approximations inside the bounded region containing the sources is still a challenging problem that deserves a work in itself.

Herein, we construct a family of local ABCs based on the Prandtl-Glauert-Lorentz transformation for the convected Helmholtz equation. These conditions are meant to be integrated in HDG discretizations. Hence they are constructed for the first-order formulation of the convected Helmholtz equation. They are constructed from the ABCs formerly derived for the standard Helmholtz equation in [START_REF] Bayliss | Boundary conditions for the numerical solution of elliptic equations in exterior regions[END_REF] using the Prandtl-Glauert-Lorentz transformation. Without saying so, it is worth mentioning that this idea has already been exploited in [START_REF] Bayliss | Far field boundary conditions for compressible flows[END_REF] for a convective wave equation with a constant free stream flow. That the flow is constant probably explains that the authors did not introduce the Prandtl-Glauert-Lorentz transformation as the change of variables to go from the convective wave equation to the acoustic wave equation was quite natural. Finally, it is important to note that the Prandtl-Glauert-Lorentz transformation is only used to compute the coefficients on the artificial boundary. Hence the new ABCs are valid for a flow that varies inside the domain but becomes uniform at infinity. Numerical experiments based upon the solution of a HDG formulation (see [START_REF] Barucq | HDG and HDG+ methods for harmonic wave problems with convection[END_REF]) illustrate the efficiency of the new ABCs for low and intermediate Mach numbers. This paper is organized as follows:

• In Section 1, we recall some results about the convected Helmholtz equation and precisely state the assumptions under which the new ABCs are derived.

• In Section 2, we use the Prandtl-Glauert-Lorentz transformation to construct the new ABCs.

• In Section 3, we present numerical results to illustrate the performance of the new ABCs. Only low-order conditions are investigated as the integration of high-order conditions in HDG formulations is still challenging.

Model problem and geometric settings

In this section, we introduce the convected Helmholtz equation and recall some of its properties. We also precisely state the assumptions under which the new Absorbing Boundary Condition are constructed.

Equation and carrier flow

We consider the following convected Helmholtz equation

ρ 0 -ω 2 p -2iωv 0 • ∇p + v 0 • ∇[v 0 • ∇p] -div ρ 0 c 2 0 ∇p = s, ( 1
)
where p is the acoustic potential, v 0 is the velocity of the carrier flow, c 0 is the adiabatic sound speed, ρ 0 is the density of the flow and s is the acoustic source.

In this paper, we use the following convention for time-harmonic solutions

p(x, t) = p(x, ω)e -iωt .
To make the notations lighter, the e -iωt factor is omitted, and, in this paper vectors are assumed to be column vectors. The physical perturbations of pressure p and of velocity v can be recovered from the potential using the following relations p = -ρ 0 c 0 (-iω + v 0 • ∇)p, and v = -c 0 ∇p.

As usual in acoustics, this model is obtained thanks to a linearization process and is therefore only valid for small perturbations.

As there are several ways to write the convected Helmholtz equation, see [START_REF] Pierce | Wave equation for sound in fluids with unsteady inhomogeneous flow[END_REF], we have chosen the simple form (1) as a model problem for this paper. The construction of the ABCs presented here can then be adapted to the other forms of the equation without major difficulties. In particular, the results for the velocity potential formulation

(-iω + v 0 • ∇) 1 c 2 0 (-iω + v 0 • ∇) ϕ - 1 ρ 0 div (ρ 0 ∇ϕ) = s, (2)
will also be stated.

In this paper we will make the following usual assumptions on the carrier flow:

Assumption 1 (Incompressible flow). The density of the flow ρ 0 is uniform. Upon nondimensionalization we can always choose ρ 0 ≡ 1. The velocity v 0 satisfies the following incompressibility equation div (v 0 ) = 0.

Remark 1.1. Taking other values of ρ 0 into account only amounts to small changes in the construction of the ABCs. Notice that if ρ 0 is allowed to vary inside the computational domain but becomes equal to one close to the boundary, then Assumption 1 should be replaced with the usual mass conservation equation

div (ρ 0 v 0 ) = 0.
Assumption 2 (Subsonic flow). The carrier flow is subsonic, ie.

inf c 2 0 -|v 0 | 2 > 0.
We introduce the matrix

K 0 := c 2 0 Id -v 0 v T 0 , ( 3
)
where Id denotes the identity matrix. Then, using Assumption 1, (1) can be rewritten as

-ω 2 p -div (K 0 ∇p + 2iωpv 0 ) = s. ( 4
)
The quantity σ := -K 0 ∇p -2iωpv 0 (5) is called the total flux and allows to rewrite the convected Helmholtz equation as a firstorder system conducive to get a mixed DG formulation. Actually, it is showed in [START_REF] Barucq | HDG and HDG+ methods for harmonic wave problems with convection[END_REF] that the HDG method for solving the following first-order system σ + K 0 ∇p + 2iωpv 0 = 0, -ω 2 p + div (σ) = s, has some attractive properties when compared to other HDG methods for the convected Helmholtz equation. In particular it provides an optimal order of convergence and does not depend on the choice of a penalization parameter, as an optimal penalization parameter can be computed a priori. The aim of this paper is therefore to construct ABCs that takes both effects of wave propagation and convection into account and that are easy to implement in a HDG solver.

Geometric assumptions for the background flow

As the Absorbing Boundary Conditions will be constructed using the Prandtl-Glauert-Lorentz transformation, the background flow must be locally uniform around the artificial boundary. We therefore make the following assumption.

Assumption 3 (Uniform flow outside a compact).

There exists a compact K such that v 0 and c 0 are constant outside K.

Assumption 3 is a reasonable assumption: in practice we can always assume that the background flow becomes uniform if the artificial boundary is located far away from the acoustic source. It is also worth noticing that a similar assumption is usually used to derive ABCs for the standard Helmholtz equation. A sketch of this configuration is given in Figure 1.

K ∂K v 0 (x, y) v 0 = v ∞ e x

Prandtl-Glauert-Lorentz transformation and approximate ABCs

In this section we introduce the Prandtl-Glauert-Lorentz transformation on which the construction of the new ABCs is based. We recall that the Prandtl-Glauert-Lorentz transformation maps the convected Helmholtz equation into the standard one. We then show how the boundary conditions are transformed under this change of coordinates. Finally we construct new ABCs for the convected Helmholtz equation in 2D.

Prandtl-Glauert-Lorentz transformation

We define the Lorentz factor

α := √ 1 -M 2 ,
where M is the Mach-number defined as

M 0 := 1 c 0 v 0 , and M 2 := M T 0 M 0 .
Following [HPN19, MBAG21, Cha00], we use the following frequency-domain transformation

ω = ω α ; x = Ax := Id + 1 α(1 + α) M 0 M T 0 x. (6)
Depending on the sources, it can be called Lorentz transformation, Prandtl-Glauert transformation or sometimes Prandtl-Glauert-Lorentz transformation. The term Lorentz transformation is used as this transformation is close to the Lorentz transformation arising in special relativity.

Using the Sherman-Morrisson formula [START_REF] Sherman | Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix[END_REF], it is easy to show that

A -1 = Id - 1 1 + α M 0 M T 0 .
We can therefore write the inverse Prandtl-Glauert-Lorentz transformation

ω = α ω ; x = A -1 x := Id - 1 1 + α M 0 M T 0 x.
A direct computation, see [Rou21, Lemma 4.2.2], allows to prove the Lemma 1. Let p be a solution of (1). If c 0 and v 0 are constant, then p defined by

p( x, ω) = α exp i ω α 2 c 0 M 0 • x p(x, ω) (7)
satisfies the standard Helmholtz equation -ω 2 p -c 2 0 ∆ p = s, where ∆ is the Laplace operator in Prandtl-Glauert-Lorentz coordinates and

s( x, ω) = α exp iω M 0 • x α 2 c 0 s(x, ω).
The Prandtl-Glauert-Lorentz transformation will be used to construct ABCs for the convected Helmholtz equation. The validity of this approach is ensured by the following lemma whose proof is postponed to Appendix A.

Lemma 2. The Prandtl-Glauert-Lorentz transformation maps outgoing solutions of the convected Helmholtz equation to outgoing solutions of the Helmholtz equation.

Transformation of the boundary condition

We now introduce a boundary Σ, the so-called artificial boundary, which limits a region including the support of the acoustic source and thus defines a bounded computational domain O. The ABC is then set on the artificial boundary Σ. We also denote Σ the artificial boundary in Lorentz coordinates. For simplicity, we will choose a circle centered on the origin for Σ, Σ will therefore be an ellipse. An example is depicted on Figure 2. More precisely, the following lemma holds.

-2 -1 0 1 2 -2 -1 0 1 2 Σ Σ
Σ = x | x| 2 = R 2 = x = R (cos(t)e x + sin(t)e y ) t ∈ [0, 2π) ,
then the artificial boundary in physical coordinates satisfies

Σ = x |Ax| = R 2 = x = R (cos(t)A 1 + sin(t)A 2 ) t ∈ [0, 2π) ,
where A 1 and A 2 are the first and second columns of A -1 .

Remark 2.1. At first glance, the other choice (taking a circle in physical coordinates and an ellipse in Lorentz coordinates) may seem more natural. However ABCs for the Helmholtz equation on an ellipsoidal boundary are more complicated to implement. If they are derived using an analytical point of view, those ABCs are expressed using special functions whose numerical evaluation may be unstable, see [START_REF] Barucq | Non-reflecting boundary condition on ellipsoidal boundary[END_REF][START_REF] Saint-Guirons | Construction et Analyse de Conditions Absorbantes de Type Diritchlet-to-Neumann Pour Des Frontières Ellipsoïdales[END_REF]. On the other hand, if a geometric point of view is used, the resulting ABCs involve curvature operators that are difficult to approximate, see [START_REF] Bayliss | Boundary conditions for the numerical solution of elliptic equations in exterior regions[END_REF][START_REF] Antoine | Bayliss-Turkel-like Radiation Condition on Surfaces of Arbitrary Shape[END_REF].

We will now state and prove the main result of this paper.

Theorem 4. Let p be the solution of

-ω 2 p -div (K 0 ∇p + 2iωpv 0 ) = s, in O, (8a) σ • n + Zp = 0, on Σ, ( 8b
)
where K 0 is defined by (3) and scales like c 2 0 , and σ is the total flux defined by (5). Then if v 0 and c 0 are constant, and if p is defined by (7), then p is a solution to

-ω 2 p -c 2 0 ∆ p = s, in O, (9a)
∂ n p + Z p = 0, on Σ, ( 9b
)
if the boundary operators satisfy

Z(x, ω) = -c 2 0 |A -T n| Z( x, ω) + iωv 0 • n. ( 10
)
Furthermore the problems (8a)-( 8b) and (9a)-(9b) are both well-posed.

Remark 2.2. We can notice that Z and Z do not have the same physical dimension. This is not a consequence of the Prandtl-Glauert-Lorentz transformation, but it comes from the choice of unknown for the convected Helmholtz equation. Indeed, ABCs based on σ • n + Zp for the convected Helmholtz equation should be interpreted as -c 2 0 ∂ n p + Zp, whereas most ABCs for the standard Helmholtz equation are written as ∂ n p + Zp. This explains both the minus sign and the difference in dimensions between the coefficient. We recall that we have chosen to use σ because it is convenient for constructing mixed finite element methods, see [START_REF] Barucq | HDG and HDG+ methods for harmonic wave problems with convection[END_REF].

Remark 2.3. Notice that even if Theorem 4 is stated for a uniform flow, the new Absorbing Boundary Conditions can be used in the more general case described by Assumption 3. Indeed as the Prandtl-Glauert-Lorentz transformation will only be used to construct the boundary operator, the carrier flow only needs to be locally uniform close to the artificial boundary.

Remark 2.4. ABCs for the standard Helmholtz equation satisfy Im Z < 0, using (10) we can see that ABCs for the convected Helmholtz equation satisfy ImZ -ωv 0 • n > 0. This can be understood as σ • n should be interpreted as -∂ n p. If the signs are reversed, then the boundary conditions select incoming waves instead of outgoing ones.

Remark 2.5. Equation (10) can also be written as

Z(x, ω) = -c 2 0 |A -T n| Z( x, ω) -i ω c 0 M 0 • n ,
however we have chosen to use the formulation (10) as it is closer the implementation choices that were performed in [START_REF] Barucq | HDG and HDG+ methods for harmonic wave problems with convection[END_REF].

Proof (of Theorem 4). Let us now focus on the boundary conditions (8b) and (9b). This is adapted from [HPN19, Sec. 3].

We assume that Σ and Σ have the following cartesian equations Σ : {Φ(x) = 0} and Σ : Φ( x) = 0 .

Step 1: Because of the chain rule, we have the following relationship between their normal directions

∇Φ = ∂Φ ∂x = ∂ x ∂x ∂ Φ ∂ x = A T ∇ Φ
Let n and n be the unit normal vectors to Σ and Σ respectively. Using that ∇Φ = |∇Φ|n, we have

n = 1 | ∇ Φ| ∇ Φ = 1 |A -T ∇Φ| A -T ∇Φ = |∇Φ| |A -T ∇Φ| A -T n, which implies that n = µA -T n, (11)
where µ = |A -T n| -1 is a scaling factor that ensures that | n| = 1.

Step 2:

We have

A -1 M 0 = Id - 1 1 + α M 0 M T 0 M 0 = 1 - |M 0 | 2 1 + α M 0 = αM 0 .
Starting from [HPN19, Eq. ( 11)], multiplying by A -1 on the left and by A -T on the right, we have

c 2 0 M 0 M T 0 -c 2 0 Id = -c 2 0 A -1 A -T ⇐⇒ A -1 A -T = Id -M 0 M T 0 . ( 12
)
Step 3:

We recall that p( x, ω) = α exp iω α 2 c 0 M 0 • x p(x, ω). ( 13
)
Using the chain rule, we have

∇p = A -T ∇p, and ∇ α exp iω α 2 c 0 M 0 • x = iω α 2 c 0 α exp iω α 2 c 0 M 0 • x A -T M 0 ,
and therefore

∂ n p = n • ∇ p
using the definition (13) of p we have

∂ n p = α exp iω α 2 c 0 M 0 • x n T A -T ∇p + iω α 2 c 0 pA -T M 0 ,
then, (11) yields

∂ n p = α exp iω α 2 c 0 M 0 • x µn T A -1 A -T ∇p + iω αc 0 pM 0 , using (12)
leads to

∂ n p = α exp iω α 2 c 0 M 0 • x µn T (Id -M 0 M T 0 )∇p + i ω c 0 pM 0 ,
we then divide by c 2 0 and use the definition (5) of σ to obtain

∂ n p = α exp iω α 2 c 0 M 0 • x µ c 2 0 n T   K 0 ∇p + 2iωpv 0 =-σ -iωpv 0    ,
finally, the ABC (8b) gives

∂ n p = α exp iω α 2 c 0 M 0 • x µ c 2 0 (Zp -iωpv 0 • n) .
We obtain (9b) using (10) and

p(x, ω) = 1 α exp -iω α 2 c 0 M 0 • x p( x, ω).
Step 5: The well-posedness of the standard Helmholtz equation with ABCs is a wellknown result that can be found eg. in [START_REF] Duprat | Conditions Aux Limites Absorbantes Enrichies Pour l'équation Des Ondes Acoustiques et l'équation d'Helmholtz[END_REF]Sec. 4.3]. Using Lemma 2, we can see that the convected Helmholtz equation with ABCs inherits the well-posedness of the standard Helmholtz equation through the Prandtl-Glauert-Lorentz transformation.

Up to this point, we have worked with the total flux σ which is a convenient quantity to write a mixed finite formulation of the problem. However when working with continuous primal finite-element methods, it is possible to obtain a formulation whose volumetric part is hermitian, which would lead to an alternative boundary flux [HPN17, HPN19, LMG + 20]. This flux is

g := -c 2 0 ∇p + (-iωp + v 0 • ∇p)v 0 = -K 0 ∇p -iωpv 0 .
This alternative flux is related to σ by

g = σ + iωpv 0 ,
and we can define the following ABC

g • n + Z g p = 0.
Corollary 1. The boundary coefficient Z g satisfies

Z g (x, ω) = -|A -T n|c 2 0 Z( x, ω).
The construction of the ABCs can also be adapted to the velocity formulation of the convected Helmholtz equation (2) which is recalled below for a uniform carrier flow with ρ 0 ≡ 1

1 c 2 0 (-iω + v 0 • ∇) 2 ϕ -div (∇ϕ) = s.
For this formulation, the total flux can be defined as

σ ϕ = Id -M 0 M T 0 ∇ϕ -2i ω c 0 ϕM 0 , ( 14
)
and we have

Corollary 2. If Z ϕ satisfies Z ϕ = -|A -T n| Z + i ω c 0 M 0 • n, then σ ϕ • n + Z ϕ ϕ = 0,
is an ABC for (14).

New ABCs for the convected Helmholtz equation in 2D

In this section, we derive the practical values used for the ABCs for the convected Helmholtz equation from Absorbing Boundary Conditions formerly derived for the Helmholtz equation in [START_REF] Bayliss | Boundary conditions for the numerical solution of elliptic equations in exterior regions[END_REF].

Low-order ABCs

In this section, we recall some Absorbing Boundary Conditions on a circle for the Helmholtz equation. The ABCs of this section are specific to the 2D setting, however the extension to 3D is straightforward as the result of Theorem 4 holds for any dimension. Notice that even if Theorem 4 is stated for a uniform flow, the new Absorbing Boundary Conditions can be used in the more general case described by Assumption 3. Indeed as the Prandtl-Glauert-Lorentz transformation will only be used to construct the ABCs, the carrier flow only needs to be locally uniform close to the artificial boundary, which we can always assume if the boundary is located far away from the source. The ABCs for the standard Helmholtz equation will define the Z operator applied on Σ. The value of Z will then be obtained using the transformation of the time-harmonic solutions described in Lemma 1 and the transformation of the boundary coefficients described in Theorem 4.

We will consider the low-order Bayliss-Guntzburger-Turkel Absorbing Boundary Conditions, described in [START_REF] Bayliss | Boundary conditions for the numerical solution of elliptic equations in exterior regions[END_REF], in the Lorentz variable.

We assume that the artificial boundary in Lorentz coordinates Σ is a circle of radius R.

Proposition 1. Low-order ABCs for the convected Helmholtz equation are:

• ABC of order 0:

σ • n + Zp = 0, with Z := i c 0 α |A -T n| + v 0 • n ω, ( ABC0
)
• ABC of order 1:

σ • n + Zp = 0, with Z := -c 2 0 2R |A -T n| + i c 0 α |A -T n| + v 0 • n ω. (ABC1)
Proof. ABC of order 0. According to [START_REF] Bayliss | Boundary conditions for the numerical solution of elliptic equations in exterior regions[END_REF], the zeroth order ABC the standard Helmholtz equation reads

∂ n -i ω c 0 p = 0.
The boundary coefficients are therefore defined by

Z := -i ω c 0 and Z := i c 0 α |A -T n| + v 0 • n ω.
ABC of order 1. According to [START_REF] Bayliss | Boundary conditions for the numerical solution of elliptic equations in exterior regions[END_REF], the first order ABC for the standard Helmholtz equation reads

∂ n -i ω c 0 + 1 2R p = 0.
The boundary coefficients are therefore defined by

Z := -i ω c 0 + 1 2R and Z := -c 2 0 2R |A -T n| + i c 0 α |A -T n| + v 0 • n ω.
Remark 2.6. In [START_REF] Bayliss | Far field boundary conditions for compressible flows[END_REF], the author derived far-field boundary conditions for the steady-state Euler's equations by applying a change of coordinates which corresponds to the Prandtl-Glauert-Lorentz transformation. If they had used their idea to obtain an ABC for the convected Helmholtz equation, this would have led to (ABC1).

It is interesting to dispose of another ABC constructed by adopting another point of view. For that purpose, we will derive an ABC that selects outgoing plane waves that are locally orthogonal to the artificial boundary Σ.

Proposition 2. The ABC that selects the outgoing plane waves that are locally orthogonal to the boundary is

σ • n + Zp = 0, with Z := i(c 0 + v 0 • n)ω. (ABC-PW)
Proof. We begin by computing the plane waves for the convected Helmholtz equation (1). We write p as

p P W = p 0 e iκ•x
where p 0 ∈ C and where κ is the wave vector. We can rewrite (1) as

p 0 -ω 2 + 2ωv 0 • κ + κ T K 0 κ = 0,
and therefore

-ρ 0 ω 2 + 2ωρ 0 v 0 • κ + ρ 0 c 2 0 |κ| 2 -ρ 0 (v 0 • κ) 2 = 0. ( 15
)
As we only have one equation for two unknowns, we make the following assumptions

κ := κ cos θ sin θ , where θ ∈ [0, 2π
) is assumed to be known,

v 0 := M c 0 cos θ 0 sin θ 0 , where θ 0 ∈ [0, 2π).
With those assumptions, (15) becomes

-ω 2 + 2ωκM c 0 (cos θ 0 cos θ + sin θ 0 sin θ) + c 2 0 κ 2 -M 2 c 2 0 κ 2 (cos θ 0 cos θ + sin θ 0 sin θ) 2 = 0,
or equivalently

-ω 2 + 2ωκM c 0 cos(θ -θ 0 ) + c 2 0 κ 2 1 -M 2 cos 2 (θ -θ 0 ) = 0.
Solving this last equation for κ leads to the two following solutions

κ -= -ω c 0 (1 -M cos(θ -θ 0 )) and κ + = ω c 0 (1 + M cos(θ -θ 0 )) .
Notice that κ can be rewritten as

κ = κ ± M c 0 cos(θ -θ 0 )v 0 + sin(θ -θ 0 )v ⊥ 0 .
The plane-wave solutions of (4) are therefore

p P W = p 0 e iκ ± •x , with κ ± = ±ω c 0 (1 ± M cos(θ -θ 0 )) ,
and the total flux σ P W therefore satisfies

σ P W = -i c 0 κ ± M (1 -M 2 ) cos(θ -θ 0 )v 0 + sin(θ -θ 0 )v ⊥ 0 p 0 e iκ•x .
As we want to select plane-waves that are locally orthogonal to the artificial boundary Σ, we have

κ ± |κ ± | = n = cos θ sin θ , leading to v 0 • n = M c 0 cos(θ -θ 0 ), and v ⊥ 0 • n = M c 0 sin(θ -θ 0 ). We then have σ • n = -iωc 0 (1 ± M cos(θ -θ 0 )) p = -iω (c 0 ± v 0 • n) p.
We can now define the boundary coefficients Z as

Z = -σ P W • n p P W = iω(c 0 + v 0 • n),
which leads to what we call (ABC-PW)

σ • n + iω(c 0 + v 0 • n)p = 0.
We have chosen the ABC with the + sign as a consequence of the convention e -iωt that we use for time-harmonic solutions .

Remark 2.7. The two ABCs (ABC0) and (ABC-PW) are different. Indeed for the standard Helmholtz equation, the zeroth order Bayliss-Guntzburger-Turkel ABC is exact for plane waves that are locally orthogonal to the boundary. This property is not preserved when we convert the Helmholtz equation into its convected counterpart via the Prandtl-Glauert-Lorentz transformation. To illustrate why this fact, let us consider the simple case where v 0 = M c 0 e x . In this case, a plane wave solution to the convected Helmholtz equation (1) with incident angle θ is given by

p(x, ω) = exp iω c 0 (1 + M cos θ) (x cos θ + y sin θ) ,
using Lemma 1, we have

p( x, ω) = α exp iω α 2 c 0 M x exp iω c 0 (1 + M cos θ) (x cos θ + y sin θ) , which leads to p( x, ω) = α exp iω αc 0 M α + α cos θ 1 + M cos θ x + iω αc 0 α cos θ 1 + M cos θ y .
Using that x = A -1 x, we get x = α x and y = y, and we finally obtain that

p( x, ω) = α exp iω αc 0 M + cos θ 1 + M cos θ x + α sin θ 1 + M cos θ y = α exp i ω c 0 cos θ x + sin θ y ,
where θ = arctan α sin θ M + cos θ .

When M = 0, we have θ = θ, and we can therefore conclude that the incident angle of a plane wave changes under the Prandtl-Glauert-Lorentz transformation. As (ABC0) is derived for a plane wave that is locally orthogonal to the boundary in PGL coordinates, the resulting plane wave is not normal to the boundary in physical coordinates. This explains the difference with (ABC-PW). This is natural since the acoustic intensity vector is not orthogonal to the wavefront in the presence of a flow.

Extension to higher order ABCs

In this section, we will illustrate how the Prandtl-Glauert-Lorentz transformation can be used to derived higher-order ABCs, even if those ABCs do not naturally fit in the framework of DG methods. For the sake of simplicity, we only consider the case v 0 = M c 0 e y . The general case can then be obtained by rotation. We introduce the elliptical coordinates (µ, ν) defined by

x = c cosh µ cos ν, y = c sinh µ sin ν, and the artificial boundary Σ is described by setting a constant value for µ. On the other hand, the artificial boundary in PGL-coordinates is represented as

x = R cos θ, y = R sin θ,
for some constant R. Using that x = A -1 x, we have

c cosh µ = R, c sinh µ = Rα, ν = θ.
It is now straightforward to obtain a second-order ABC for the convected Helmholtz equation. For the standard Helmholtz equation, a complete second-order ABC on a circular boundary is given by

∂ n p + B p + D∂ 2 θ θ p = 0, where
B := - iω c 0 + 1 2R + 1 8i ω c 0 R(1 + ic 0 ωR ) , D := 1 2 iω c 0 R(1 + ic 0 ωR ) 1 R 2 ,
see [ABB99, Eq. ( 28)]. We can therefore define Z as Z := B + D∂ 2 θ θ , using Theorem 4 we finally obtain the following ABC for the convected Helmholtz equation

σ • n + Zp = 0, where Zp(x, ω) = -c 2 0 |A -T n| Bp + D ∂ 2 νν p + 2iωM c α 2 c 0 sinh µ cos ν∂ ν p + iωM c α 2 c 0 sinh µ 2iωM c α 2 c 0 sinh µ cos 2 ν + i sin ν p - iωM αc 0 sin νp .
This resulting ABC is different from the ones derived in [HHT03, BGH10], as they take the elliptical shape of the boundary into account while the conditions in [HHT03, BGH10] have been derived for flat boundaries only. This ABC will not be considered in this paper because its implementation into the underlying HDG formulation is a challenging issue due to the discretization of the first and second order surface derivatives. This question will be addressed in a further work.

Numerical experiments

The numerical experiments of this section have been performed using the HDG-σ method of [START_REF] Barucq | HDG and HDG+ methods for harmonic wave problems with convection[END_REF] which has been implemented in the open-source software package hawen, see [START_REF] Faucher | hawen': Time-harmonic wave modeling and inversion using hybridizable discontinuous Galerkin discretization[END_REF].

We solve the following first-order in space and second-order in frequency formulation:

σ + K 0 ∇p + 2iωpv 0 = 0, -ω 2 p + div (σ) = s.
As the total flux σ is an unknown of the method, it is very natural to work with the ABCs of Proposition 1. The experiments that are presented in this paper have been carried out with a nodal basis with interpolation degree k = 5.

Experiments with a uniform flow

We consider a point-source in a uniform flow

M 0 := M cos θ 0 sin θ 0 , s = δ 0 , c 0 = 1, ω = 6π.
Following [START_REF] Marchner | Stable perfectly matched layers with lorentz transformation for the convected helmholtz equation[END_REF], we define the number of degrees of freedom per shortest wavelength as

d λ := 2πk ωh (1 -M ).
In practice we choose h = 0.05, and for the highest Mach number M = 0.8 we have d λ 6.7, which is similar to the parameters used in [START_REF] Marchner | Stable perfectly matched layers with lorentz transformation for the convected helmholtz equation[END_REF]. The Dirac source-term located at x 0 is discretized as

K δ x 0 Φ K i dx = Φ K i (x 0 ),
on element K and for all basis functions Φ K i . To handle the singularity due to this term, the mesh is refined around x 0 by halving h. In the numerical experiments presented below, we have taken x 0 = 0.

A reference solution of the Helmholtz equation reads:

p ref ( x, ω) = i 4 H (1) 0 ( ω r),
and from Lemma 1, we deduce that a reference solution to the convected Helmholtz equation is:

p ref (x, ω) = i 4α H (1) 0 ω α |Ax| exp - iω α 2 c 0 M 0 • x . ( 16
)
To begin with, we will use θ 0 = π 4 as it leads to a circular artificial boundary in both physical and PGL coordinates. Numerical experiments illustrating the influence of the flow angle θ 0 will be given in the end of this section.

As the parameter R, which represents the size of the domain in PGL coordinates, is not really meaningful, we introduce the Helmholtz number

He := ωαR 2π(1 + M ) ,
which describes the number of largest wavelengths inside the domain. In the above formula, ω/(2π(1 + M )) denotes the wavenumber of the waves propagating with the flow and, αR is the size of the domain in physical coordinates (as |Ax| |x|/α).

Validation of the ABCs

For a numerical solution p h , we define the relative error as

E O := K,i |p h -p ref | 2 (x K i ) K,i |p ref | 2 (x K i )
, where (x K i ) i are the degrees of freedom in element K ∈ T h . The triangulation T h is composed of triangles K. As the solution p ref is singular at x = 0, the error will be computed on O\B(0, ρ) where B(0, ρ) is the open ball centered on x = 0 with radius ρ = 2h. 1 for a low Mach number M = 0.4. In this case, we can see that the three ABCs perform well, with a clear advantage in favor of the two PGL-based ABCs ((ABC0) and (ABC1)). As expected, (ABC1) performs better than (ABC0) and for M = 0.4, we can see that a relative error of ∼ 0.1% is obtained even with He = 1.

Low Mach number. The relative error E O for various values of He is given in Table

R

He (ABC0) (ABC1) (ABC-PW) 0.5 1 3.47% 0.12% 4.8% 1.0 2 1.51% 0.13% 3.72% 1.5 3 0.89% 0.12% 3.33% 2.0 4 0.70% 0.12% 3.12% Intermediate Mach number. In Table 2, the relative error E O is computed for several values of R for M = 0.6. In this case (ABC-PW) performs badly, leading to an error level of ∼ 10%, whereas the PGL-based ABCs give good numerical results with an error level of ∼ 1%. This is also illustrated in Figure 3. Once again (ABC1) performs better than (ABC0). Indeed (ABC1) leads to an error level below 1% even for very small values of He whereas a larger He is required to obtain a similar error level with (ABC0). R He (ABC0) (ABC1) (ABC-PW) 0.5 0.75 2.14% 0.67% 8.3% 1.0 1.5 1.21% 0.62% 7.31% 1.5 2.25 0.98% 0.66% 8.02% 2.0 3.0 0.83% 0.64% 7.1% In Figure 4, the local error is plotted for (ABC1) and (ABC-PW) for M = 0.6 and R = 2. For (ABC-PW) there is a clear pattern: the ABC performs better in the top-right and bottom-left parts of the domain. This can be understood as this ABC is constructed to select outgoing plane-waves that are locally orthogonal to the boundary. By looking at Figure 3, we can see that p is almost orthogonal to the boundary in the top-right and bottom-left parts of the domain where (ABC-PW) performs well. However due to the presence of convection, p is clearly not orthogonal to the boundary in the top-left and bottom-right parts of the domain, where ABC-PW exhibits its worst behavior. For (ABC1) the error seems better distributed inside the domain, and we can clearly see that the error level is one order of magnitude smaller than the error level of (ABC-PW).

To further illustrate this idea, we also consider the local error on the artificial boundary Σ

E Σ := |(p h -p ref )| Σ | .
This error is depicted in Figure 5 for (ABC1) and (ABC-PW). For (ABC-PW), the same effect as in Figure 4 can be seen. For (ABC1) however, we notice that most of the error is located in bottom part of the domain. The effects of the flow angle θ 0 on this error distribution will be discussed in Table 4 at the end of this section. We would like to point out that the error levels for (ABC1) are one order of magnitude lower than those of (ABC-PW) as expected. Large Mach number. For a large Mach number, the use of the PGL-based ABCs leads to a higher error. It therefore seems natural to consider larger domain, but as it can be seen in Table 3, even for He = 10 the error does not get below 1.84% with (ABC1). This level of error occurs even if the Helmholtz number He is higher than in the cases of low and intermediate Mach numbers. It also seems that in this case, using (ABC1) instead of (ABC0) does not improve the quality of the solution. In Figure 6, the numerical solutions p h obtained with (ABC1) and (ABC-PW) are depicted.

R He (ABC0) (ABC1) (ABC-PW) 3 3 2.41% 2.39% 15.29% 10 10 1.85% 1.84% 14.87% In Figure 7 we have depicted the numerical solution p h and the local error E loc for ABC1 with He = 10. Even if the relative error E O is of 1.84%, by looking at the plot of E loc it seems possible to trust the solution far enough from the artificial boundary. A domain of this size satisfies the convergence criterion R > (1 -M 2 ) -1 suggested in [START_REF] Hagstrom | High-Order Radiation Boundary Conditions for the Convective Wave Equation in Exterior Domains[END_REF], as the error level is still quite important we can probably conclude that the difficulties in dealing with high Mach numbers are an intrinsic feature of low-order ABCs. A similar conclusion can be drawn from [HHT03, FIG. 5].

To further illustrate the influence of the Mach number, the relative error E O has been plotted as a function of M with fixed Helmholtz number He = 3 in Figure 8. In this figure, we can see that the error stays below 1% for Mach numbers up to 0.6. Starting at M = 0.7, the errors become quite large and more advanced domain truncation techniques should be preferred, as for instance the use of high-order ABCs.

Finally we would like to point out that considering a larger domain leads to a much more expensive problem from a computational point of view. The numerical experiments were run on a miriel node of the plafrim cluster 1 equipped with 2 dodeca-core Haswell Intel Xeon E5-2680 v3 with a clock rate of 2.5 GHz and 128 Go of memory. The case of the small domain with He = 3 ran in 1min, whereas for the test with He = 10, it took 1h15min to run.

Comparison with PMLs. For the convected Helmholtz equation, usual PMLs are known to be unstable. In [START_REF] Bécache | Perfectly matched layers for the convected helmholtz equation[END_REF] stable PMLs for the convected Helmholtz equation were derived, but only for propagation in a waveguide. This work was extended to arbitrary geometries in [START_REF] Marchner | Stable perfectly matched layers with lorentz transformation for the convected helmholtz equation[END_REF], this formulation is also based on the Prandtl-Glauert-Lorentz transformation.

For high Mach numbers, the so-called Lorentz PMLs of [START_REF] Marchner | Stable perfectly matched layers with lorentz transformation for the convected helmholtz equation[END_REF] seem to perform better than the ABCs of this paper. Indeed, the authors were able to obtain an error level of 1.2% using a smaller domain for M = 0.8. However this formulation seems difficult to implement in the context of HDG methods and, as stated in [START_REF] Marchner | Stable perfectly matched layers with lorentz transformation for the convected helmholtz equation[END_REF], their efficiency for non-uniform flows remains unclear. However, as it will be illustrated in Subsection 3.2, the PGL-based ABCs can be used for cases where the background flow changes inside the domain.

Comparison with other low-order ABCs. In [Gab03, Chap. 3] and [Bér08, Chap. 5], ABCs for Galbrun's equation are studied. Galbrun's equation is an aeroacoustic model which is more realistic than the convected Helmholtz equation as it allows for more general background flows. However the ABCs derived in those dissertations were only illustrated with uniform flows for which the two models are equivalent. This is why as the reference solution used in [Bér08, p. 129 -last equation] is the same as the one we described in (16). Even if various conditions are described, the best performing one is somewhat similar to (ABC-PW), and the authors only obtained an error level of 4.5% for M = 0.3 and ω = 6π. It seems interesting to extend the PGL-based ABCs to Galbrun's equation. This generalization is difficult, except in the case of a potential flow, where Galbrun's equation reduces to the convected Helmholtz equation.

Comparison with high-order ABCs. In [START_REF] Hagstrom | High-Order Radiation Boundary Conditions for the Convective Wave Equation in Exterior Domains[END_REF], high-order ABCs were constructed for the time-domain convective wave equation. Obviously the comparison is not really relevant but for M = 0.4, we see on Figures 3 and4 of this paper that the error level of (ABC1) is lower than the error level of ABC with l = 1 and similar to the error level of ABC with l = 5 in [START_REF] Hagstrom | High-Order Radiation Boundary Conditions for the Convective Wave Equation in Exterior Domains[END_REF]. For M = 0.8, we can see on FIG. 5 that the low-order ABC of [HHT03] has a similar or higher error level than (ABC1), however the higher-order ABCs lead to better numerical results. This observation turns in favor of using high-order conditions for high Mach numbers.

Influence of the flow angle

In this section, we focus on the influence of the flow angle θ 0 on the performance of the ABCs. In Table 4 the errors obtained when using (ABC1) are given for various values of θ 0 . We can clearly see that E O is smaller when the flow is aligned with one of the axis (for θ 0 = 0 or θ 0 = π/2). This can be understood as the change of spatial coordinates A -1 , which is used to construct the ABCs, is contracting: the domain is smaller in physical coordinates than in PGL ones. When θ 0 = 0 or π/2, this contraction only occurs in one direction, whereas it occurs in both directions for the other values of θ 0 . The error is therefore smaller when the flow is aligned with one of the axis. The maximal error appears to be achieved for θ 0 = π/4 which is the angle with the highest contraction, as both directions are contracted in the same way.

Flow angle θ

0 Error E O 0 1.32 • 10 -3 % π/3 0.29% π/4 0.51% π/6 0.30% π/2 1.31 • 10 -3 % Table 4: Domain error E O for M = 0.
6 with (ABC1) and R = 2.5 corresponding to He = 3.75

As the error seems minimal for a flow aligned with one of the axis, performing a rotation should therefore be considered before using those ABCs. However as the relative error stays below 1%, results obtained for other values of θ 0 are still acceptable.

Some of the cases of Table 4 are depicted in Figure 9. On those figures, the elliptical shape of the domain can clearly be seen when θ 0 = π/4, as well as the contraction effect due to the change of coordinates.

Experiments with multiple sources

To illustrate the ability of the ABCs to handle more complex cases, we consider multiple point-sources in a uniform flow. For the numerical simulations, we consider the following parameters associated to a large Helmholtz number He 9.4, which corresponds to

M = 0.6, θ 0 = π 4 , ω = 12.5π, R = 2. ( 17
)
The sources will be located at (±0.1, ±0.1) for the case with two point-sources and at (±0.1, ±0.1) and (±0.1, ∓0.1) for the case with four point-sources. In those cases, an analytic solution can be obtained by superposition of the analytic solution (16) for one point-source.

To evaluate the error, a ball of radius ρ = 4h is excluded to remove the points where the solution is singular.

In Figure 10 the results obtained with the above parameters and (ABC1) are depicted. We can clearly see the interference patterns between the sources as well as changes in the apparent frequency due to the Doppler effect. Those physical phenomena are well-reproduced without any visible spurious effect of the ABC. For the case with two point-sources, we obtained an error level of E O = 1.003% and E rel Σ = 3%, where

E rel Σ = K,i |p h -p ref | 2 (x K∩Σ i ) K,i |p ref | 2 (x K∩Σ i ) ,
is the relative error on the boundary. In [START_REF] Astley | Infinite elements for transient flow acoustics | 7th AIAA/CEAS Aeroacoustics Conference and Exhibit[END_REF], the authors used infinite elements for the convected elements. The parameters described in (17) are similar to the case depicted in [AH01, Fig. 5 & 6, case b) with kD = 10]. The performances of their numerical method are better than ours when Mesh A is used as they obtain an error level of E rel Σ = 0.2%2 . If Mesh B is used, the infinite elements and the PGL-based ABCs exhibit similar performances. It is worth noting that in [START_REF] Astley | Infinite elements for transient flow acoustics | 7th AIAA/CEAS Aeroacoustics Conference and Exhibit[END_REF], the error is computed when a flux issued from an analytic solution is prescribed at the beginning of the infinite elements, and the coupling between infinite elements and a numerical method in the interior domain is not done. The implementation of this coupling would involve an approximation of the Dirichlet-to-Neumann operator at the interface between infinite and finite elements, which should introduce some additional error. Moreover, it is also not clear whether and how this coupling could be adapted to the framework of HDG methods. The ABCs of this paper are constructed for an artificial boundary Σ centered on the source. The expression for the potential flow should therefore be translated as it is usually written for a circular obstacle located at (0, 0).

We assume that the obstacle has a radius R C and its center is located at x C = (x C , y C ). Let M ∞ be the Mach number of the flow at infinity, the potential flow is naturally expressed in the previous polar coordinates as:

v 0 := M ∞ 1 - R 2 C r 2 cos θe r -1 + R 2 C r 2 sin θe θ ,
leading to the following expression in cartesian coordinates

v 0 = M ∞ x -x C r 1 - R 2 C r 2 cos θ + y -y C r 1 + R 2 C r 2 sin θ e x + y -y C r 1 - R 2 C r 2 cos θ - x -x C r 1 + R 2 C r 2 sin θ e y .
The configuration for this case is depicted in Figure 11. In Figure 12, we plot p h obtained with the following parameters

x C = (±1, 0), R C = 0.5, M ∞ = 0.4, ω = 6π.
It seems there is no reflection at the artificial boundary Σ and the expected physical phenomena are visible. We can clearly see a change in apparent frequency due to the Doppler effect, phase-shifts due to a refraction-like effect and an interference pattern due to the reflection of the wave on the obstacle. For the upstream example, we can also see a silent zone behind the obstacle as expected. For the downstream case, creeping waves can be seen around the obstacle, and a constructive interference pattern is visible behind the obstacle.

Similar cases to the upstream one have been studied in [BCD + 14], the differences between the results obtained by the authors and our results may be explained by a different kind of source term and differences between 2D and 3D simulations. In particular, the pressure inside the silent cone is lower in our results.

Similar cases to the downstream one have been considered in [LMG + 20]. The results obtained by the authors are very similar to the ones of this paper, therefore validating the new ABCs that we have constructed.

Conclusion

In this paper, we have seen that the use of the Prandtl-Glauert-Lorentz transformation to construct ABCs for the convected Helmholtz equation leads to very good results for low and intermediate Mach numbers. For higher Mach numbers, a very large domain seems to be required and the ABCs constructed in this paper should be used carefully. Due to the presence of convection, the usual idea of selecting waves that are locally orthogonal to the boundary leads to bad results, and the PGL-based ABCs should be preferred. Among those ABCs, the use of (ABC1) instead of (ABC0) gives better results for no additional cost. Finally we would like to point out that the ABCs constructed in this paper are really easy to implement in an existing finite-element solver for the convected Helmholtz equation as the PGL transformation is only required to compute the coefficient Z.

When compared to other domain-truncation methods available in the literature, the PGL-based ABCs introduced in this paper have better performances than other low-order ABCs introduced eg. in [START_REF] Gabard | Méthodes Numériques et Modèles de Sources Aéroacoustiques Fondées Sur l'équation de Galbrun[END_REF][START_REF] Bériot | Eléments Finis d'ordre Élévé Pour l'opérateur de Galbrun En Régime Harmonique[END_REF]. As it can be expected, more advanced techniques such as the infinite elements [START_REF] Astley | Infinite elements for transient flow acoustics | 7th AIAA/CEAS Aeroacoustics Conference and Exhibit[END_REF], stabilized PMLs [START_REF] Marchner | Stable perfectly matched layers with lorentz transformation for the convected helmholtz equation[END_REF] or high-order ABCs [HHT03] can lead to better numerical results. In particular, those methods could be efficient to deal with high Mach numbers. But the integration of more sophisticated truncation methods in HDG formulations turns out to be a challenging work that is beyond the scope of this paper. By showing that HDG formulations can be coupled with ABCs for the convected Helmholtz equation, we have achieved an important start in the construction of advanced numerical methods for solving convected time harmonic wave equations posed in large domains.

Perspectives. The extension of this work to 3D is straightforward as the result of Theorem 4 relating Z to Z holds for any dimension. Extension of this work to higher order boundary conditions should be possible, we have already provided the expression of a second-order ABC. This is really an interesting purpose, as it was shown in [START_REF] Hagstrom | High-Order Radiation Boundary Conditions for the Convective Wave Equation in Exterior Domains[END_REF]. High-order ABCs seem to efficiently deal with problems involving large Mach numbers. But implementation of high order ABCs in HDG formulations is a non-trivial process, therefore rendering low-order ABCs attractive.

Extension to more realistic aeroacoustic models seem also possible. The most straightforward model to consider for this extension is Goldstein's equation [START_REF] Mercier | Acoustic propagation in a vortical homentropic flow[END_REF] which consists in a coupling between a convected Helmholtz equation and a vectorial transport equation. The ABCs of this paper could be used for the convected Helmholtz equation and it is possible to obtain an exact outgoing condition for the transport equation. The extension to the Linearized Euler's Equations (LEE) [START_REF] Bogey | Computation of Flow Noise Using Source Terms in Linearized Euler's Equations[END_REF] or Galbrun's equation [BMM + 12] should also be feasible. Indeed, for a uniform flow those equations are equivalent to the convected Helmholtz equation, and it is therefore possible to construct ABCs for LEE from ABCs for the convected Helmholtz equation. A similar process was already carried out in [START_REF] Bayliss | Far field boundary conditions for compressible flows[END_REF] to compute steady-state solutions of the Euler's equations.

As we are now working in time-domain, we recall that the space-time Prandtl-Glauert-Lorentz transformation reads

t = αt + M 0 • x αc 0 ; x = Ax := Id + 1 α(1 + α) M 0 M T 0 x,
see [START_REF] Fang | On the use of a Prandtl-Glauert-Lorentz transformation for acoustic scattering by rigid bodies with a uniform flow[END_REF]. The solutions to the convected wave equation and to the standard wave equations are linked by the following lemma.

Lemma 5. If c 0 and v 0 are constant, the quantity p( x, t) := p(x, t), satisfies the following wave equation

∂ 2 t t p -c 2 0 ∆ p = s,
where ∆ is the Laplace operator with respect to x and ( x, t) is defined by (6).

Step 1: Green functions in time-domain. The Green function for the two-dimensional acoustic wave-equation in time-domain reads The Green function for the convected wave equation in time-domain has been computed in the case M 0 = M e x using the Cagniard-de Hoop method in [DJ06, Th. 6]. The case of a generic M 0 reduces to this one by rotation. In this paper, the parameter σ denotes the absorption parameter used in PMLs, so we take σ = 0 leading to A = B = 0.

G std (x, t) = H t -r c 0 2π t 2 -r 2 c 2 0 ,
With the assumption that M 0 = M e x , the Prandtl-Glauert-Lorentz transformation reads

t = αt + 1 αc 0 M x, x = x α , y = y, r 2 = x 2 + y 2 , θ = arctan y x . (A.1)
We define the following change of variables in space The Green function corresponding to the convected acoustic wave equation reads

G conv (x, t) = H(t -r c 0 (1 -M cos θ)) 2π √ 1 -M 2 (t + M r c 0 cos θ) 2 - r2 c 0 .
Using the intermediate change of variables (A.2a)-(A.2b) and expressing in terms of the Prandt-Glauert-Lorentz coordinates (A.1), we have

G conv (x, t) = H t + M c 0 x -r c 0 2πα (t + M c 0 x) 2 - r2 c 2 0 = H t -r c 0 2π t 2 -r 2 c 2 0 = G std ( x, t).
As G conv is mapped to G std through the Prandtl-Glauert-Lorentz transformation, we can therefore prove that this transform also maps outgoing solutions to outgoing solutions by using standard arguments of the limiting amplitude principle.

Step 2: Limiting amplitude principle for the standard Helmholtz equation. We recall that the following identities hold Using the integral representation we have

p std (x, t) = R 2 R G std (x 1 , t 1 )g(x -x 1 , t -t 1 )dt 1 dx 1 ,
using the integral identity (A.3) for the Hankel function, we get

= R 2 +∞ |x 1 | c 0 g(x -x 1 , t -t 1 ) 2π t 2 1 -|x 1 | 2 c 2 0 dt 1 dx 1 using that g(x, t) = s(x)e -iωt H(t), where s is a L 2 -function with compact support, = R 2 s(x -x 1 )e -iωt t |x 1 | c 0 e iωt 1 2π t 2 1 -|x 1 | 2 c 2 0 dt 1 dx 1 = R 2 s(x -x 1 )e -iωt i 4 H (1) 0 ω c 0 |x 1 | + r std dx 1 29 where r std (x 1 , t) = - +∞ t e iωt 1 2π t 2 1 -|x 1 | 2 c 2 0 dt 1 with t > |x 1 | c 0 , = R 2 s(x -x 1 )e -iωt i 4 H (1) 0 ω c 0 |x 1 | dx 1 + R std ,
and

R std (x, t) = e -iωt R 2 s(x-x 1 )r std (x 1 , t)dx 1 = -e -iωt R 2 s(x-x 1 ) +∞ t e iωt 1 2π t 2 1 -|x 1 | 2 c 2 0 dt 1 dx 1 .
We now prove that R std (x, t) ----→ t→+∞ 0. The first step is to show that the integral defining r std is actually finite-valued. We have We can therefore express r std as

r std (x 1 , t) := - +∞ t e iωt 1 2π t 2 1 -|x 1 | 2 c 2 0 dt 1 = -lim A→+∞         e iωt 1 2iπω t 2 1 -|x 1 | 2 c 2 0     A t + A t t 1 e iωt 1 2π t 2 1 -|x 1 | 2 c 2 0 3 2 dt 1     < + ∞,
r std (x 1 , t) = - e iωt 2iπω t 2 -|x 1 | 2 c 2 0 - +∞ t t 1 e iωt 1 2π t 2 1 -|x 1 | 2 c 2 0 3 2 dt 1
and R std as

R std (x, t) = R 2 s(x -x 1 ) 2π t 2 -|x 1 | 2 c 2 0 dx 1 + e -iωt R 2 s(x -x 1 ) +∞ t t 1 e iωt 1 2π t 2 1 -|x 1 | 2 c 2 0 3 2 dt 1 dx 1 .
We can now show that both of these quantities vanish at infinity. Let us first focus on the first term in R std

R 2 s(x -x 1 ) 2π t 2 -|x 1 | 2 c 2 0 dx 1 .
As we are interested in the limit as t → +∞ and as the space integral is over a compact domain, we can assume that 3t

R 2 s(x -x 1 ) 2π t 2 -|x 1 | 2 c 2 0 dx 1 1 π|t| R 2 |s(x -x 1 )|dx 1 ----→ t→+∞ 0,
as the last integral is over a compact domain. By taking the limit in the previous inequality, we have

lim t→+∞ R 2 s(x -x 1 ) 2π t 2 -|x 1 | 2 c 2 0 dx 1 = 0.
We can now work on the second term.

As we are interested in the limit as t → +∞ and as the space integral is over a compact domain, we can assume once again that 3t = 0, where p std,ω := W std (•, ω) * s is the outgoing solution of the standard Helmholtz equation.

Step 3: Limiting amplitude principle for the convected Helmholtz equation. We have

p(x, t) = R 2 R G conv (x 1 , t 1 )g(x -x 1 , t -t 1 )dt 1 dx 1 = R 2 R G std ( x 1 , t 1 )g A -1 ( x -x 1 ), 1 α ( t -t 1 ) + 1 αc 0 M 0 • ( x -x 1 d t 1 d x 1 = R 2 +∞ | x 1 | c 0 g A -1 ( x -x 1 ), 1 α ( t -t 1 ) -1 αc 0 M 0 • ( x -x 1 ) 2π t 2 1 -| x 1 | 2 c 2 0 d t 1 d x 1
using that g(x, t) = s(x)e -iωt H(t), where s is a function with compact support, = R 2 s(A -1 ( xx 1 ))e

-iω( 1 α t-1 αc 0

M 0 • x) e -i ω αc 0 M 0 • x 1 t- M 0 c 0 •( x-x 1 ) | x 1 | c 0 e i ω α t 2π t 2 1 -| x 1 | 2 c 2 0 d t 1 d x 1 = R 2 s(A -1 ( x -x 1 ))e -iω( 1 α t-1 αc 0 M 0 • x) e -i ω αc 0 M 0 • x 1 i 4 H (1) 0 ω αc 0 | x 1 | + r conv d x 1 = R 2 1 α s(x -x 1 )e -iωt e -i ω α 2 c 0 M 0 •x 1 i 4 H (1) 0 ω αc 0 |Ax 1 | dx 1 + R conv ,
where r conv and R conv are defined as r std and R std in the previous step. By using similar arguments to the ones in the previous step, we can show that lim t→+∞ R conv (•, t) = 0.

We now define the time-harmonic Green function by Step 4: Equivalence under Prandtl-Glauert-Lorentz transformation.

W conv (x, ω) = 1 α exp - iω α 2 c 0 M 0 • x i 4 H
We notice that

W conv (x, ω) = 1 α exp - iω α 2 c 0 M 0 • x i 4 H (1) 0 ω αc 0 |Ax| = 1 α exp - iω α 2 c 0 M 0 • x W std x, ω α ,
We recall that s is defined by We have

p std, ω ( x) = α exp iω M 0 • x α 2 c 0 p conv,ω (x), indeed p std, ω ( x) = R 2 s( x -x 1 )W std ( x 1 , ω)d x 1 , = R 2 α exp iω M 0 • (x -x 1 ) α 2 c 0 s(x -x 1 )α exp iω M 0 • x 1 α 2 c 0 W conv (x 1 , ω) dx 1 α , = α exp iω M 0 • x α 2 c 0 R 2 s(x -x 1 )W conv (x 1 , ω)dx 1 .
So the outgoing solutions are mapped to one another through the Prandtl-Glauert-Lorentz transformation.

Figure 1 :

 1 Figure 1: Flow around an aircraft, the flow is uniform outside K

Figure 2 :Lemma 3 .

 23 Figure 2: Σ and Σ for M 0 = [0.6 0.2] T and R = 2

Figure 3 :

 3 Figure 3: Comparison between two ABCs for M = 0.6 and He = 3

Figure 4 :

 4 Figure 4: Local error E loc in the domain for M = 0.6 and He = 3

Figure 5 :

 5 Figure 5: Local error E Σ on Σ for M = 0.6 and He = 3. The red circle is the same on the two pictures.

-Figure 6 :

 6 Figure 6: Comparison between two ABCs for M = 0.8 with R = 3

Figure 7 :Figure 8 :

 78 Figure 7: Results and error for ABC1 with M = 0.8 and He = 10

Figure 9 :

 9 Figure 9: Numerical examples for various values of θ 0 with M = 0.6 and ABC1

Figure 10 :

 10 Figure 10: Interferences between multiple point-sources for M = 0.6 with ABC1

  For a point x = (x, y), we define the translated polar coordinates around x C by r := |x -x C |, and θ := arctan y -y C x -x C .

Figure 11 :

 11 Figure 11: Sketch of the settings: in blue: streamlines of v 0 , in red: artificial boundary Σ, in green: artificial boundary in PGL coordinates, in magenta: point-source, in black: obstacle

  Figure 12: Point-source in a potential flow around a circular obstacle for M ∞ = 0.4

  where H is the Heaviside function and r = |x|. For the proof, we refer to [DJ06, Th. 1] or [Ver10, Sec. A.4.2].

  , we also define r2 = x2 + ŷ2 , and θ = arctan ŷ x . (A.2b) It is straightforward to check that r = r α , and θ = θ.

  used the following change of variables t = r cosh θ and the integral form of the Hankel function, see [OLBC10, Eq. 10.9.10].

  also obtain the limiting amplitude principle for the convected Helmholtz equationlim t→∞ p(x, t) -p conv,ω (x)e -iωt L 2 (R 2 ) = 0,where p conv,ω = W conv (•, ω) * s is the outgoing solution of the convected Helmholtz equation.

  s( x, ω) := α exp iω M 0 • x α 2 c 0 s(x, ω),and we definep std, ω := W std (•, ω) * s = R 2 s(• -x 1 )W std ( x 1 , ω)d x 1 .

Table 1 :

 1 Relative error E O in the domain for M = 0.4

Table 2 :

 2 Relative error E O in the domain for M = 0.6

	1.2
	0.2
	-0.8
	-1.8
	-1.8	-0.8	0.2	1.2

Table 3 :

 3 Relative error E O

  Notice that the last integral is finite as the support of s is compact. By taking the limit in the previous inequalities, we have lim (x, t)| = 0, ∀x ∈ R 2 .

		t→+∞ R 2	|s(x -x 1 )|	t	+∞	t 1 e iωt 1 2π t 2 c 2 0 1 -|x 1 | 2	2 3	dt 1 dx 1 = 0.
	We therefore have
						lim
	Now, we define the time-harmonic Green function by
						W std (x, ω) =	i 4	H 0 (1)	ω c 0	|x| .
	We end up with the limiting amplitude principle
					lim
							2 1 4	>	3t 2 4	>	|x 1 | 2 0 c 2	,
	which leads to				t 1 1 -|x 1 | 2 2π t 2 0 c 2	3 2	<		t 1 π 4 t 3 1	=	4 πt 2 1	.
	We therefore have
		t	+∞		t 1 e iωt 1 2π t 2 1 -|x 1 | 2 0 c 2	3 2	dt 1 <	4 π	t	+∞	1 t 2 1	dt 1 =	4 πt	----→ t→+∞	0.
	Finally we have
	R 2	|s(x -x 1 )|	t	+∞	t 1 e iωt 1 2π t 2 c 2 0 1 -|x 1 | 2	3 2		dt 1 dx 1	R 2	|s(x -x 1 )|	4 πt	dx 1
														4 πt R 2	|s(x -x 1 )|dx 1 ----→ t→+∞	0.

t→+∞ |R std t→∞ p std (x, t) -p std,ω (x)e -iωt L 2 (R 2 )

See https://www.plafrim.fr.

We have multiplied the error by two, as the authors consider a symmetric domain and only compute the error on a half domain.

Acknowledgement

The authors would like to thank Florian Faucher for his help with the numerical implementation of HDG formulations with ABCs in hawen. The authors are also very thankful to the anonymous reviewers whose comments helped improve and clarify this manuscript.

Experiments presented in this paper were carried out using the PlaFRIM experimental testbed, supported by Inria, CNRS (LABRI and IMB), Université de Bordeaux, Bordeaux INP and Conseil Régional d'Aquitaine (see https://www.plafrim.fr).

Funding

Nathan Rouxelin was supported by a grant from e2s-UPPA (see https://e2s-uppa. eu).

Appendix A. Proof of Lemma 2

In this appendix, we prove the result of Lemma 2. As the numerical examples of this paper are in 2D, we have chosen to prove the result in the same dimension. Extension to 3D is straightforward and would involve the standard 3D Green functions instead of the Hankel functions.

We apply the limiting amplitude principle which consists in studying the long-term behaviour of the time-domain solution with the following source term

where s is a function in L 2 (O) with compact support.