# High-fidelity Sound Propagation in a Varying 3D Atmosphere

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## Abstract

A stable and high-order accurate upwind finite difference discretization of the 3D linearized Euler equations is presented. The discretization allows point sources, a varying atmosphere and curved topography. The advective terms are discretized using recently published upwind summation-by-parts (SBP) operators and the boundary conditions are imposed using a penalty technique. The resulting discretization leads to an explicit ODE system. The accuracy and stability properties are verified for a linear hyperbolic problems in 1D, and for the 3D linearized Euler equations. The usage of upwind SBP operators leads to robust and accurate numerical approximations in the presence of point sources and naturally avoids the onset of spurious oscillations.

## Keywords

Finite difference methods Aeroacoustics High-order accuracy Stability Boundary treatment Point sources## 1 Introduction

Numerical methods applicable to computational aeroacoustics is an active research field, requiring a deep knowledge of flow-physics, mathematical modeling, numerical analysis and computer science. A detailed review of the progress of computational aeroacoustics can be found in [10]. Commonly, different models are used for sound generation and sound propagation. Sound generation is a highly non-linear phenomenon while sound propagation over long distances is a mostly linear phenomenon. It is therefore natural to treat these two physical phenomena with different models and numerical techniques. In the present study the main focus is to capture sound propagation over long distances efficiently, under the assumption that the sound source can be accurately modeled as a point-source. Due to the linear nature of long-range sound propagation it can be modeled using the three dimensional (3D) linearized Euler equations. This model can give more accurate results than other commonly used models. Such models include for example, ray-tracing, and diffusion based models (see e.g. [37, 52]). The Euler equations includes effects from variations in the atmosphere and wind that are usually neglected in these types of models [19].

It is well known that compared to first- and second-order accurate methods, higher order methods capture wave dominated phenomena such as sound propagation more efficiently since they, for a given error tolerance, allow a considerable reduction in the degrees of freedom. In particular, high-order finite difference methods (HOFDM) are ideally suited for problems of this type. See the pioneering paper by Kreiss and Oliger [22]. The major difficulty with HOFDM is to obtain a stable boundary treatment, something that has received considerable past attention concerning hyperbolic and parabolic problems. (For examples, see [1, 7, 14, 20, 24, 46]). Another difficulty concerns non-smooth data such as point sources [42].

The main focus in this study is to develop a stable HOFDM for the linearized Euler equations in a 3D varying atmosphere with curved topography in the presence of point sources. A well-proven HOFDM for well-posed initial boundary value problems (IBVP), is to combine summation-by-parts (SBP) operators for approximating derivatives [23, 31, 45], and either the simultaneous-approximation-term (SAT) method [9], or the projection method [32, 40, 41, 42], to impose boundary conditions (BC). Recent examples of the SBP–SAT approach can be found in [5, 15, 21, 27, 28, 36, 38]. The following review papers [13, 49] are highly recommended for those interested in the SBP–SAT methodology. The SBP operators found in literature are essentially central finite difference stencils (see for example [4, 12, 23, 25, 26, 31, 45]). These SBP operators are closed at the boundaries with a careful choice of one-sided difference stencils, to mimic the underlying integration-by-parts formula in a discrete inner product. In this study, central-difference SBP operators defined on a regular equidistant grid will be referred to as *standard* SBP operators.

For linear IBVP with smooth data (here referring either to physical data or the underlying curvilinear grid) an SBP–SAT approximation constructed from standard SBP operators yields a stable and accurate approximation. For IBVP with non-smooth data or non-linear problems, the exclusive usage of standard SBP operators in combination with SAT or projection does not guarantee an accurate solution, even though the scheme is stable. To damp spurious oscillations, the addition of artificial dissipation is most often necessary. Adding robust and accurate artificial dissipation is far from trivial, and commonly involves tuning of parameters. It is imperative that the addition of artificial dissipation does not destroy the stability and accuracy properties or introduce stiffness, which in practice requires a careful boundary closure of the added artificial dissipation. A procedure for adding artificial dissipation to the standard SBP operators was introduced in [34]. In [27], SBP operators defined on a regular grid with non-central finite difference stencils in the interior are introduced, referred to as *upwind* SBP operators. An advantage of the upwind operators, compared to standard SBP operators, is that they naturally introduce artificial dissipation.

In acoustics, efficient techniques to truncate unbounded domains is an important topic that deserves special attention. For wave equations, two commonly used techniques to truncate unbounded domains are: local high-order Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML). The latest developments for the ABC and PML methods are nicely summarized in [18]. As mentioned in [6], the generalization of these methods to more general hyperbolic systems such as the linearized Euler equations has not been completely successful. In [3] the PML method was generalized towards the linearized Euler equations with constant background flow but it is unclear if that method can be extended to the case with variable coefficients. An alternative to ABC and PML is found in [11], where a buffer region technique for computing compressible flows on unbounded domains is proposed. An extension of this technique for the linearized Euler equations with variable coefficients is something we would like to address in a coming study. A well-proven, first order accurate, non-reflecting BC for Euler and Navier–Stokes equations is to impose homogeneous characteristic BC (see for example [47, 50]). Since boundary treatment is not the main focus of the present study, we will employ this method.

The main result of this work is the formulation of a stable and high-order accurate upwind SBP–SAT discretization applicable to the linearized Euler equations in a 3D varying atmosphere and topography with a point source. We show that the usage of upwind SBP operators, as compared to standard SBP operators, leads to more accurate and robust approximations, in particular when modeling sound generation from point sources and in wave boundary interactions. The most significant difference between the schemes is observed on coarse grids, where spurious oscillations is the main error source in the standard discretization.

In Sect. 2 the SBP operators are introduced in one dimension (1D). To illustrate some of the difficulties related to the point source in a simple setting, the stability analysis for a 1D hyperbolic system is discussed in Sect. 3. The accuracy properties are verified by performing 1D numerical simulations, including point sources. The extension to the linearized Euler equations in a 3D varying atmosphere is introduced in Sect. 4 and the stability analysis is discussed in Sect. 5. Verification of accuracy and stability by numerical studies of the 3D Euler equations is performed in Sect. 6. Section 7 summarizes the work.

## 2 The SBP Operators

*N*equidistant grid points:

*H*is positive definite. The corresponding norm is \(\Vert v\Vert _{H}^2=v^T\,H\,v\). In the definitions and discretizations the following vectors (and matrix) will be used frequently:

### 2.1 Standard SBP Operators

The *standard* SBP operators can be found in earlier papers (see e.g. [4, 25, 29, 30, 31, 33, 35]). For completeness we restate the definition for these first derivative SBP operators:

### Definition 2.1

A difference operator \(D_1=H^{-1}\left( Q+\frac{B}{2}\right) \), approximating \(\partial /\partial \,x\), using a *p*th-order accurate interior stencil, is said to be a *p*th-order diagonal-norm first-derivative SBP operator if the diagonal matrix *H* defines a discrete inner product, and \(Q+Q^T=0\).

### 2.2 Upwind SBP Operators

In a recent study [27] diagonal-norm SBP operators with non-central interior stencils were introduced, referred to as *upwind* SBP operators,

### Definition 2.2

The difference operators \(D_{+}=H^{-1}\left( Q_{+}+\frac{B}{2}\right) \) and \(D_{-}=H^{-1}\left( Q_{-}+\frac{B}{2}\right) \) approximating \(\partial /\partial \,x\), using *p*th-order accurate interior stencils, are said to be *p*th-order diagonal-norm upwind SBP operators if the diagonal matrix *H* defines a discrete norm, \(Q_{+}+Q_{-}^T=0\), and \(\frac{Q_{\pm }+Q_{\pm }^T}{2}=\pm S\), where *S* is negative semi-definite.

One important property of the upwind SBP operators, compared to the standard operators, is the symmetric part of \(Q_\pm \) that when combined with flux splitting techniques, naturally introduce artificial dissipation which damps spurious oscillations. The usage of diagonal-upwind SBP operators in combination with the SAT technique of imposing BC allow for general stability proofs for linear hyperbolic and hyperbolic-parabolic systems of equations (see [27] for details).

### 2.3 Accuracy of SBP Operators

Let \(\tilde{D}\) denote either a standard SBP operator (i.e., \(D_1\), given by Definition 2.1) or an upwind SBP operator (i.e., \(D_{\pm }\), given by Definition 2.2). The following definition is relevant for the convergence properties of the SBP operators.

### Definition 2.3

Let \(\mathbf{x}^q\) be the projection of the polynomial \(\frac{x^q}{q\,!}\) onto the discrete grid-points, i.e., the vector **x**. We say that \(\tilde{D}\) is *p*th-order if \(\tilde{D}\mathbf {x}^{q} = q\mathbf {x}^{q-1}\) for \(q=0\ldots p\), in the interior and for \(q=0 \ldots p/2\) or \(q=0 \ldots (p-1)/2\) for odd *p*, at the boundaries.

It is important to keep in mind that the formal boundary accuracy alone does not dictate the expected convergence rate of the numerical approximation. The expected convergence rate can be shown to be higher. In [48] it is shown that a point-wise stable approximation of an IBVP involving derivatives up to order *q* yields a convergence rate of order \(q+r\), where *r* is the order of accuracy at the boundaries. Let *p* denote the interior accuracy of a diagonal-norm SBP operator (both standard and upwind). The boundary accuracy for diagonal-norm SBP operators is restricted to (*p* / 2)th-order accuracy when p is even (see [31]), and \((p-1)/2\) when *p* is odd. The expected convergence rate for first order hyperbolic problems is \((r+1)\)th, and \((r+2)\)th for parabolic problems and second order hyperbolic problems. As an example both the 5th order upwind SBP operator and the 4th order standard SBP operator has \(r=2\) which gives the expected convergence rate 3 for first order hyperbolic IBVP. In the present study point-wise stability is not proven, but the numerical convergence studies presented in Sect. 6 indicate that the expected convergence rates from the assumption of point-wise stability are obtained.

## 3 A Scalar 1D Case

Since the 3D linearized Euler equations is a hyperbolic system of equations with varying coefficients, the discretization scheme is relativity complex (see Sects. 4 and 5). The motivation for the proposed numerical method is more clearly seen by first analyzing a simplified 1D scalar hyperbolic problem with constant coefficients. This simpler problem shares some of the important numerical difficulties with the full 3D linearized Euler equations, such as the numerical approximation of point sources. After comparing the upwind and standard discretization for the simplified 1D problem we will do a similar comparison for the 3D linearized Euler equations in Sect. 6.

*g*(

*t*) is a smooth function with \(g(0)=0\). Here

*f*(

*x*) is the initial data and \(g_l(t)\) the boundary data.

### 3.1 Point Source Discretization

*d*is the discretization of \(\delta \). However, these moment conditions alone do not guarantee a high order approximation for hyperbolic problems discretized with standard SBP operators, since there is no damping mechanism for spurious oscillations related to the Nyquist mode, (sometimes referred to as the \(\pi \)-mode). Spurious oscillations are typically triggered by non-smooth features, for example point sources. In [42] a set of additional smoothness conditions were introduced, to prevent spurious oscillations when combining point sources and standard SBP operators. To enforce the smoothness conditions of order \(P>0\) we introduce the following additional constraint

Apart from added complexity, an issue is that the smoothness conditions are constructed for the periodic problem, which introduces some complications if the sources are placed near physical boundaries. In this special case a modified boundary approximation of the smoothness conditions obtained by investigating the singular values of the difference operator is required (See [42] for details). In the present study we introduce an approach which naturally works for sources placed at the boundary.

Our approach is to replace the standard SBP operators (5) with upwind SBP operators (6), recently introduced in [27]. These operators introduce artificial dissipation which removes spurious oscillations without the additional smoothness conditions. In this work, the discretizations of the delta distribution combined with the standard and upwind operators will be denoted \(d_S\) and \(d_U\) respectively. In the standard case the discrete delta distribution \(d_S\) obeys both the smoothness and moment conditions up to the interior accuracy order of the SBP operators in the scheme. In the upwind case the smoothness conditions are unnecessary. Hence, \(d_U\) only obeys the moment conditions.

### 3.2 Two Different Schemes

### 3.3 Stability Analysis

*g*(

*t*) to zero. Therefore, the only difference in the stability analysis of the schemes (5) and (6) arise from the differences between the operators. Multiplying the homogeneous version of (5) by \(v^TH\) and adding the transpose yields,

### 3.4 Numerical Experiments in 1D

*f*(

*x*)) and boundary data (\(g_l(t)\)) are set to zero. The semi-discretizations are integrated in time using the classical 4th order Runge–Kutta method, with a time step \(dt=0.1h\).

*k*is calculated as

*dim*is the dimension of the problem (in 1D \(dim=1\) and in 3D \(dim=3\)). The convergence results from the standard and upwind SBP schemes are presented in Tables 1 and 2 respectively. Both SBP–SAT approximations converge as expected (since the pulse does not interact with the boundary the interior accuracy of the operators \(k=p\) is expected). This verifies that the smoothness conditions are not necessary when employing upwind SBP operators.

\(l_2\)-errors and convergence rates, when solving (5) with the 2nd, 4th, and 6th order standard SBP operators, **with** smoothness conditions

| 2nd order | 4th order | 6th order | |||
---|---|---|---|---|---|---|

\(log_{10}(e)\) | | \(log_{10}(e)\) | | \(log_{10}(e)\) | | |

101 | \(-\,0.64\) | − | \(-\,1.67\) | − | \(-\,2.45\) | − |

201 | \(-\,1.24\) | 1.97 | \(-\,2.85\) | 3.90 | \(-\,4.08\) | 5.45 |

401 | \(-\,1.84\) | 2.01 | \(-\,4.05\) | 3.98 | \(-\,5.84\) | 5.83 |

801 | \(-\,2.45\) | 2.00 | \(-\,5.25\) | 4.01 | \(-\,7.69\) | 6.14 |

\(l_2\)-errors and convergence rates, when solving (6) with the 3rd, 5th and 7th order upwind SBP operators, **without** smoothness conditions

| 3rd order | 5th order | 7th order | |||
---|---|---|---|---|---|---|

\(log_{10}(e)\) | | \(log_{10}(e)\) | | \(log_{10}(e)\) | | |

101 | \(-\,1.12\) | − | \(-\,2.09 \) | − | \(-\,2.64\) | − |

201 | \(-\,2.03\) | 2.68 | \(-\,3.52\) | 4.77 | \(-\,4.79\) | 7.12 |

401 | \(-\,2.92\) | 2.98 | \(-\,5.03\) | 5.00 | \(-\,6.95\) | 7.18 |

801 | \(-\,3.82\) | 2.96 | \(-\,6.52\) | 4.96 | \(-\,9.03\) | 6.91 |

## 4 Euler Equations

*u*,

*v*,

*w*the velocity components in the x, y and z direction and

*p*is the pressure. The coefficient matrices are given by,

### 4.1 Background State

When linearizing the Euler equations (13) a known steady state solution to the equations is assumed. Different forcing functions \(\hat{\mathbf {F}}\) are chosen depending on what kind of waves the model targets. For testing purposes, we have followed the sound propagation model in [19], and set the forcing term to zero. This choice does not change the stability analysis, so the model can easily be changed to target other types of waves.

*Bo*) is parallel to the

*xy*-plane and the altitude increases with

*z*the following analytic solution

*u*(

*z*) and

*v*(

*z*) are real valued functions. With this solution, various cases of wind speeds and speed of sound profiles varying with altitude can be modeled. However, in the case of a curved bottom boundary, we need to add the constraint \(u(z)=0\) and \(v(z)=0 \) when \(z\in Bo\) to obey the wall boundary condition. In this case a realistic analytic background state with wind is difficult to obtain. For the general case, we would need a background state consistent with the model. This can be achieved by an accurate steady state solution to the non-linear Euler equations. However, this is not within the scope of this project.

### 4.2 Curvilinear Transform

### 4.3 Well-Posed Boundary Conditions

*Ea*(east),

*So*(south),

*No*(north),

*Bo*(bottom) and \(T o\) (top) as shown in Fig. 4 where the computational 3D domain \(\Omega '\) is \(W e<\xi <Ea\), \(So<\eta <No\) and \(Bo<\zeta <T o\).

*x*,

*y*and

*z*as

### 4.4 The Continuous Energy Analysis

In this section an energy analysis is shown for the Euler equations (16) with the BC in (17) and (18). To simplify the notation in the coming energy analysis, we will assume constant coefficient matrices. We stress that the energy analysis can be performed also for the variable coefficient case (see for example [39]).

The following inner product is used:

### Definition 4.1

Let \(\mathbf u,v \in L^2[\Omega ^{1,1,1}_{0,0,0}]\), where \(\mathbf u =[\mathbf u ^{(1)},\mathbf u ^{(2)},\ldots ,\mathbf u ^{(m)}]^T\) and \(\mathbf v =[\mathbf v ^{(1)},\mathbf v ^{(2)},\ldots , \mathbf v ^{(m)}]^T\)are vector valued functions with *m* components. Further, let the inner product be defined as \((\mathbf u,v )= \int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{} \mathbf u ^T\mathbf v \, dx \, dy \, dz\), with corresponding norm \(||\mathbf u ||^2 =(\mathbf u,u )\).

## 5 Semi-Discrete Analysis in 3D

In this section a standard and an upwind energy stable 3D SBP–SAT discretzation of (16) is introduced. To simplify notation in the energy analysis, we will only consider the case with constant background flow. We stress that the energy analysis can be performed also for the variable coefficient case, by a skew-symmetric splitting of the equations (see for example [39]). The skew-symmetric splitting is straightforward and unrelated to the numerical difficulty of imposing well-posed BC and will therefore not be pursued here. Hence, to simplify the notation in the coming energy analysis, \(\mathbf {A}\), \(\mathbf {B}\) and \(\mathbf {C}\) are assumed constant coefficient matrices (here consistent with \(\mathbf {E}=0\)). Before we present the discretizations some necessary definitions and notation are introduced.

### 5.1 Definitions in 3D

*C*is a \(p \times q\) matrix and

*D*an \(m \times n\) matrix. The Kronecker product fulfills the relations \((A\otimes B)(C \otimes D)=AC\otimes BD\) and \((A\otimes B)^T=A^T \otimes B^T\). Let \(I_n\) be the \(n\times n\) identity matrix for any

*n*. To extend the 1D difference operators to 3D the following notation is introduced. For any \(N_\xi \times N_\xi \) matrix

*P*let

*P*let

*P*let

*V*be extended at the boundary by

### 5.2 Stability Analysis

### Lemma 1

### Proof

## 6 Numerical Experiments in 3D

### 6.1 Convergence

*xz*-plane of the pressure component in the reference simulation at 4 different times. The resulting convergence rates and errors are displayed in Tables 3 and 4.

\(l_2\)-errors and convergence rates with the standard scheme (24) on a curvilinear domain computed against a reference solution in 3D, with 2nd, 4th, and 6th order SBP operators

| 2nd order | 4th order | 6th order | |||
---|---|---|---|---|---|---|

\(log_{10}(e)\) | | \(log_{10}(e)\) | | \(log_{10}(e)\) | | |

17 | \(-\,1.56\) | − | \(-\,2.06\) | − | \(-\,2.02\) | − |

33 | \(-\,2.07\) | 1.76 | \(-\,3.07\) | 3.49 | \(-\,3.11\) | 3.77 |

65 | \(-\,2.67\) | 2.03 | \(-\,4.07\) | 3.40 | \(-\,4.37\) | 4.28 |

129 | \(-\,3.28\) | 2.03 | \(-\,5.01\) | 3.18 | \(-\,5.72\) | 4.54 |

\(l_2\)-errors and convergence rates with the upwind scheme (25) on a curvilinear domain computed against a reference solution in 3D, with 3rd, and 5th order SBP operators

| 3rd order | 5th order | ||
---|---|---|---|---|

\(log_{10}(e)\) | | \(log_{10}(e)\) | | |

17 | \(-\,1.96\) | − | \(-\,2.37\) | − |

33 | \(-\,2.62\) | 2.29 | \(-\,3.46\) | 3.76 |

65 | \(-\,3.45\) | 2.82 | \(-\,4.52\) | 3.59 |

129 | \(-\,4.33\) | 2.96 | \(-\,5.57\) | 3.54 |

\(l_2\)-errors and convergence rates with the standard scheme (24) in a cuboid domain with wind, computed against a reference solution in 3D, for 2nd, 4th, and 6th order SBP operators

| 2nd order | 4th order | 6th order | |||
---|---|---|---|---|---|---|

\(log_{10}(e)\) | | \(log_{10}(e)\) | | \(log_{10}(e)\) | | |

17 | \(-\,1.66\) | − | \(-\,1.79\) | − | \(-\,1.83\) | − |

33 | \(-\,1.81\) | 0.53 | \(-\,2.31\) | 1.80 | \(-\,2.47\) | 2.21 |

65 | \(-\,2.23\) | 1.41 | \(-\,3.32\) | 3.41 | \(-\,3.53\) | 3.59 |

129 | \(-\,2.81\) | 1.94 | \(-\,4.41\) | 3.70 | \(-\,4.73\) | 4.04 |

\(l_2\)-errors and convergence rates with the upwind scheme (25) in a cuboid domain with wind computed against a reference solution in 3D, for 3rd, and 5th order SBP operators

| 3rd order | 5th order | ||
---|---|---|---|---|

\(log_{10}(e)\) | | \(log_{10}(e)\) | | |

17 | \(-\,1.93\) | − | \(-\,2.04\) | − |

33 | \(-\,2.21\) | 1.00 | \(-\,2.66\) | 1.14 |

65 | \(-\,2.80\) | 1.99 | \(-\,3.78\) | 3.82 |

129 | \(-\,3.61\) | 2.71 | \(-\,4.95\) | 3.91 |

*xz*-plane, at 4 different times in the reference simulation. The convergence results are presented in Tables 5 and 6.

For both settings the scheme with the standard operators shows the expected convergence (\(k^{(2)}=2, \ k^{(4)}=3\) and \(k^{(6)}=4\)), while the upwind scheme shows a convergence rate higher than the expected rates (\(k^{(3)}=2\) and \(k^{(5)}=3\)). We can not explain this higher convergence, however it has been observed in earlier studies with the upwind operators [27].

### 6.2 Resolution on Coarse Grids

## 7 Conclusions and Future Work

The main motivation in the present study has been to derive a provably stable high-order accurate SBP–SAT approximation of the linearized Euler equations in a 3D varying atmosphere and curved topography, including point sources. This is achieved by utilizing novel upwind SBP operators with built in artificial damping.

The upwind SBP–SAT discretization of the 3D Euler equations leads to highly robust and accurate approximations, verified through numerical computations in 1D and 3D. Numerical experiments show that in the presence of point sources, the usage of upwind SBP operators efficiently avoid the onset of spurious oscillations. At the boundaries, the upwind SBP–SAT approximation yield a more efficient artificial damping of reflected waves when imposing CBC, as compared to the exclusive usage of central-difference (standard) SBP operators. In a coming study we hope to extend the current model with more accurate non-reflecting boundary conditions. We also aim for an extension of the upwind SBP–SAT methodology towards more general and non-linear problems in 3D, such as the compressible Navier–Stokes equations.

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