Atmospheric Sound Propagation Over Large-Scale …scholarism.net/FullText/ijmcs20143102.pdfcomplex atmosphere nor irregular terrain properly. Unfortunately, due to a shortage of reliable

International Journal of Mathematics and Computer Sciences (IJMCS) ISSN: 2305-7661 Vol.31 July 2014 www.scholarism.net

837

Atmospheric Sound Propagation Over Large-Scale Irregular Terrain

Martin Almquist1 , Ilkka Karasalo

2 and Ken Mattsson

1

1Department of Information Technology, Uppsala University, Lägerhyddsvägen 2, 752 37 Uppsala, Sweden

2KTH Royal Institute of Technology, Stockholm, Sweden

Abstract

A benchmark problem on atmospheric sound propagation over irregular terrain has been solved using a stable

fourth-order accurate finite difference approximation of a high-fidelity acoustic model. A comparison with the

parabolic equation method and ray tracing methods is made. The results show that ray tracing methods can

potentially be unreliable in the presence of irregular terrain.

Keywords

Wave propagation Irregular terrain High-order finite difference methods

1 Introduction

High-fidelity simulations of sound propagation require an accurate treatment of the medium of propagation. In

the atmosphere the speed of sound typically varies in space, which causes refraction of sound rays. As sound

waves hit the surface of the earth they are partly reflected; the degree and direction of reflection depend on the

topography and the type of ground. Measurements have shown that in order to accurately predict sound pressure

levels (SPL), it is important to take the properties of both the atmosphere and the underlying terrain into account

(see for example [23, 35]). In realistic applications, the atmospheric variations can be complex and the terrain is

often irregular, which means both that the topography is non-trivial and that the type of ground can vary.

One high-fidelity model for sound propagation is based on the linearized Euler equations [36], which is most

often solved with the staggered grid finite difference time domain method originally presented in [44]. In [13] a

pseudo-spectral time domain method was used to solve the linearized Euler equations in urban courtyards. This

is an efficient approach when the underlying geometry fits into a block-Cartesian geometry.

Another high-fidelity model is the time-dependent acoustic wave equation. However, the wave equation has

been considered too computationally demanding in realistic large-scale 3-D settings. One way to reduce the

computational cost when the sound source consists of a single frequency is to Fourier transform the wave

equation, which yields the Helmholtz equation. From the Helmholtz equation, it is possible to further reduce the

computational cost via an assumption that reflections at boundaries are negligible. The resulting model is called

the parabolic equation (PE) model, often used in ocean acoustics (e.g. [5, 24, 37]) and aeroacoustics (e.g. [16–

18]). Because of the assumption about reflections, the PE method is only valid for very moderate topographies.

A significant drawback with the Helmholtz equation (and the PE method) is that time-dependent phenomena,

such as a varying atmosphere or sources (for instance wind turbines) can not easily be approximated.

However, despite the possible simplifications, the above models are all generally regarded as too

computationally expensive in realistic 3-D applications. Hence, most numerical methods for sound propagation

are based on much simpler models. A commonly used sound propagation model is NORD2000 [34], based on

ray tracing methods, which combine geometrical ray theory with the theory of diffraction (for examples of

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838

applications see [7, 22, 42, 43]). Ray tracing methods are computationally cheap, but they incorporate neither a

complex atmosphere nor irregular terrain properly.

Unfortunately, due to a shortage of reliable measurements over large-scale domains, it has been difficult to

validate the sound propagation models currently used in practical applications. An alternative to measurements

is validation through manufactured solutions (where one compares the numerical results with analytical

solutions), but this technique has not yet been widely accepted outside the field of applied mathematics.

In this paper we shall solve a realistic sound propagation problem with the time-dependent acoustic wave

equation as the underlying model. As mentioned above, this model is desirable from an accuracy perspective,

but it is computationally costly. It is therefore imperative to use an efficient numerical method, to minimize the

number of unknowns. It is well known that high-order (higher than second order) spatially accurate finite

difference schemes combined with high-order accurate time marching schemes are very well suited for wave

propagation problems on large domains (see the pioneering paper by Kreiss and Oliger [20]). To guarantee an

accurate approximation, it is necessary that the underlying numerical scheme can be proven stable, which is a

non-trivial task using high-order finite difference methods.

A robust and well-proven high-order finite difference methodology that ensures stability of time-dependent

partial differential equations (PDEs) is the summation-by-parts-simultaneous approximation term (SBP-SAT)

method. The SBP-SAT method combines semi-discrete operators that satisfy a summation-by-parts (SBP)

formula [21], with physical boundary conditions implemented using the simultaneous approximation term

(SAT) method [3].

An added benefit of the SBP-SAT method is that it naturally extends to multi-block geometries while retaining

the essential single-block properties: stability, accuracy, and conservation [4]. Thus, problems involving

complex domains or non-smooth geometries are easily amenable to the approach. References [12, 27, 28,32]

report applications of the SBP-SAT method to problems involving nontrivial geometries. Most of the published

results for the SBP-SAT method are for first order hyperbolic systems (and the Navier–Stokes equations). The

extension of the SBP-SAT technique to the second order wave equation is found in [28–31]. However, those

studies were restricted to 1-D concerning high-order finite difference methods (3-D results for the finite volume

technique were presented). The present study is a direct extension of the SBP-SAT method to the second order

wave equation on multidimensional curvilinear domains, including non-trivial boundary conditions.

In the first part of this paper, the SBP-SAT method is applied to the benchmark problem introduced in [35]. The

SBP-SAT method is here extended to the second order wave equation on a curvilinear 2-D domain with non-

trivial boundary conditions. A fourth-order accurate SBP-SAT approximation is implemented and verified by a

grid-convergence study against manufactured solutions. In the second part of this paper, the benchmark problem

in [35] is used to compare the PE and ray tracing methods with the newly implemented SBP-SAT method.

Conclusions as to the validity of the PE and ray tracing methods in the presence of a complex atmosphere and

irregular terrain are drawn.

The rest of the paper is organized as follows: In Sect. 2 we introduce some definitions and illustrate the SBP-

SAT method by applying it to a 1-D problem. In Sect. 3 we analyze a model problem in 2-D. We then introduce

the benchmark problem in Sect. 4, and describe how the SBP-SAT technique has been adapted to this problem.

The PE method and the ray interpolation methods are introduced in Sects. 5 and 6, respectively. The

implementation of the fourth-order accurate SBP-SAT method is verified in a series of convergence studies in

Sect. 7. In Sect. 8, we present the results from the benchmark problem, comparing the different numerical

methods. Conclusions are presented in Sect. 9. The finite difference (SBP) operators are listed in “Appendix”.

2 The 1-D Problem

In this section we define the SBP-SAT method. To illustrate the power and simplicity of the method we shall

consider the following second-order hyperbolic problem:

autt=(bux)x,L0u=g0(t),L1u=g1(t),u=f1,ut=f2,0≤x≤1,x=0,x=1,0≤x≤1,t≥0,t≥0,t≥0,t=0,

















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839

(1)

where a(x)>0 and b(x)>0, and the boundary conditions are given by

L0uL1u=α0utt+α1ut+α2u+α3ux=g0(t),x=0=β0utt+β1ut+β2u+β3ux=g1(t),x=1.

(2)

In the present study we restrict ourselves to the case α3≠0,β3≠0, referred to as mixed boundary conditions. For

the treatment of Dirichlet conditions we refer to [29].

Before we start employing the SBP-SAT method, some definitions are needed. Let the inner product for real-

valued functions u,v∈L2[0,1] be defined by (u,v)=∫10uvc(x)dx,c(x)>0, and let the corresponding norm

be ∥u∥2c=(u,u). The domain (0≤x≤1) is discretized using the following N+1 equidistant grid points:

xi=ih,i=0,1,…,N,h=1N.

The approximate solution at grid point xi is denoted vi, and the discrete solution vector is v=[v0,v1,…,vN]T.

Similar to the continuous inner product, we define an inner product for discrete real-valued vector

functions u,v∈RN+1 by (u,v)Hc=uTHCv, where H is diagonal and positive definite andC is the projection

of c(x) onto the diagonal. The corresponding norm is ∥v∥2Hc=vTHCv.

Remark 1

The matrix product HC defines a norm if and only if HC is symmetric and positive definite. This can only be

guaranteed if H is a diagonal matrix (see [39] for a detailed study on this).

The following vectors will be frequently used:

e0=[1,0,…,0]T,eN=[0,…,0,1]T.

(3)

2.1 The SBP-SAT Method

SBP operators are essentially central finite difference stencils, closed at the boundaries with carefully chosen

one-sided difference stencils which mimic the underlying integration-by-parts formula in a discrete norm. In the

present paper we address the SBP operators by the accuracy of the central scheme and the type of norm which

they are based on. A 2pth order diagonal norm SBP operator is closed with pth order accurate one sided stencils

(see [30]). For first order hyperbolic problems, this implies that the convergence rate (i.e., global convergence)

drops to (p+1)th order when using a 2pth order diagonal norm SBP operator. For strongly parabolic problems

and second order hyperbolic problems the convergence rate instead drops to (p+2)th order (see [9, 40] for more

information on the accuracy of finite difference approximations).

To define the SBP-SAT method, we present Definitions 1–2 (first stated in [33] and [28]). We here say that a

scheme is explicit if no linear system of equations needs to be solved to compute the difference approximation.

Definition 1

A difference operator D1=H−1Q=H−1(Q¯−12e0eT0+12eNeTN) approximating ∂/∂x, using a 2pth-order

accurate stencil of minimal width, is said to be a 2pth-order accurate narrow-diagonal first-derivative SBP

operator if H is diagonal and positive definite and Q¯+Q¯T=0.

Definition 2

Let D(b)2=H−1(−M(b)−e0b0S0+eNbNSN) approximate ∂/∂x(b∂/∂x), where b(x)>0, using a 2pth-order accurate

stencil of minimal width. D(b)2 is said to be a 2pth-order accurate narrow-diagonal second-derivative SBP

operator, if H is diagonal and positive definite, M(b) is symmetric and positive semi-definite

and S0 and SN approximate the first-derivative operator at the boundaries.









840

The superscript (b) emphasizes that M(b) and D(b)2 depend on b(x). The explicit dependence can be found

in [26]. For completeness we have included the fourth order SBP operators (used in the present study) in

“Appendix”.

The following definition introduced in [26] is also central in this paper:

Definition 3

Let D1 and D(b)2 be 2pth-order accurate narrow-diagonal first- and second-derivative SBP operators.

If M(b)=DT1HBD1+R(b), and the remainder R(b) is positive semi-definite, D1 and D(b)2 are calledcompatible.

2.1.1 Continuous Analysis

Multiplying the first equation in (1) by ut and integrating by parts (referred to as “the energy method”) leads to

ddt(∥ut∥2a+∥ux∥2b)=2(buxut)x=1−2(buxut)x=0,

(4)

where (buxut)x=1 means (buxut) evaluated at x=1. We also identify

E=∥ut∥2a+∥ux∥2b,

(5)

as the total energy (kinetic and potential).

Multiplying the first equation in (1) by ut, integrating by parts and imposing the boundary conditions (2) leads

to

ddtE¯=BTx=0+BTx=1,

(6)

where

E¯=∥ut∥2a+∥ux∥2b+b(0)α3(α0u2t+α2u2)x=0−b(1)β3(β0u2t+β2u2)x=0,

(7)

and

BTx=0BTx=1=+2b(0)α1α3(ut−g02α1)2x=0−g20b(0)2α1α3=−2b(1)β1β3(ut−g12β1)2x=1+g21b(1)2β1β3.

(8)

Here we assume that α3≠0 and β3≠0. The following Lemma is central in the present study,

Lemma 1

Equation (1) with boundary conditions (2) has a bounded energy in terms of initial and boundary data

if α3≠0,β3≠0,α0α3≥0,α2α3≥0,β0β3≤0,β2β3≤0,α1α3<0 and β1β3>0 hold.

Proof

E¯ is non-negative and well defined if α3≠0,β3≠0,α0α3≥0,α2α3≥0,β0β3≤0,β2β3≤0 hold. By integrating (6) in

time, we obtain

E¯(t)+∫0t⎛⎝2b(1)β1β3(ut−g1(τ)2β1)2x=1−2b(0)α1α3(ut−g0(τ)2α1)2x=0⎞⎠dτ=E¯(0)+∫0t(g21(τ)b(1)2β1β3−g2

0(τ)b(0)2α1α3)dτ.



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841

If α1α3<0 and β1β3>0 hold we have a strong estimate of E¯(t) in terms of initial and boundary data.

Remark 2

The implication of Lemma 1 is that we can have at most linear time-growth of ∥u∥. Linear growth (in t) does not

violate well-posedness (see [10]). However, linear time-growth of ∥u∥ can only occur if we have f2≠0combined

with pure Neumann boundary conditions, i.e, α0=α1=β0=β1=0, so that we have a zero eigenvalue in the

spectrum [25].

2.1.2 Semi-Discrete Analysis

The discrete approximation of (1) using the SBP-SAT method is

Avtt=D(b)2v+τ0H−1e0(L0v−g0)+τ1H−1eN(L1v−g1),

(9)

where e0 and eN are defined in (3). (We assume the same initial conditions v=f1,vt=f2 as in the continuous

case). The matrix A has the values of a(x) injected on the diagonal. The semi-discrete boundary operators that

mimic (2) are given by

L0vL1v=α0(vtt)0+α1(vt)0+α2v0+α3S0v=β0(vtt)N+β1(vt)N+β2vN+β3SNv.

(10)

Applying the energy method by multiplying (9) by vTtH and adding the transpose leads to

ddtEH=−2b0(vt)0S0v+2bN(vt)NSNv+2τ0(α0(vt)0(vtt)0+α1(v2t)0)+2τ0(α2(vt)0v0+α3(vt)0S0v−(vt)0g0)+2τ1(β0(v

t)N(vtt)N+β1(v2t)N)+2τ1(β2(vt)NvN+β3(vt)NSNv−(vt)Ng1),

(11)

where

EH=∥vt∥2Ha+vTM(b)v.

Lemma 2

Equation (9) with boundary operators (10) exactly mimics the continuous energy estimate (6)

if τ0=b0α3,τ1=−bNβ3, and is thus stable if the conditions in Lemma 1 hold.

Proof

insert τ0=b0α3,τ1=−bNβ3 in (11) to obtain

ddtE¯H=BT0+BTN,

(12)

where

E¯H=∥vt∥2Ha+vTM(b)v+b0α3(α0(v2t)0+α2(v2)0)−bNβ3(β0(v2t)N+β2(v2)N),

and

BT0BTN=+2b0α1α3((vt)0−g02α1)2−g20b02α1α3=−2bNβ1β3((vt)N−g12β1)2+g21bN2β1β3.

Equation (12) is a semi-discrete analogue to (6), and stability follows if the conditions in Lemma 1 hold.

3 Analysis in 2D














842

In this section we analyze the scalar 2-D wave equation with mixed boundary conditions. To allow for complex

domains, we transform the equation given on a curvilinear domain to an equation on the unit square. We then

derive an energy estimate for the continuous case. After discretizing the model in space with the SBP-SAT

method, we prove stability by exactly mimicking the continuous energy estimate.

3.1 Definitions

To make the notation more compact we introduce the Kronecker product:

C⊗D=⎡⎣⎢⎢c0,0D⋮cp−1,0D⋯⋯c0,q−1D⋮cp−1,q−1D⎤⎦⎥⎥,

(13)

where C is a p×q matrix and D is an m×n matrix. We also let IN be the (N+1)×(N+1) identity matrix.

If the problem is given on a curvilinear domain Ω (referred to as the physical domain) we transform it to the unit

square, Ω′. The unit square is discretized using the (Nξ+1)(Nη+1) grid points:

(ξi,ηj)=(iNξ,jNη),i=0,1,…,Nξ,j=0,1,…,Nη.

The boundaries of Ω′ are denoted by W (west), N (north), E (east) and S (south), respectively, as shown in Fig. 1.

The approximate solution at a grid point (ξi,ηj) is denoted by vij, and the discrete solution vector

is v=[v00,…,v0Nη,v10,…,vNξNη]T. The matrix RW is defined so that RWv is a vector with the same length

as v and the same elements on the positions corresponding to the west boundary, but zeros everywhere else. The

matrices RN,RE and RS are defined similarly for the north, east and south boundaries, respectively.

Fig. 1

The mapping between cartesian (left) and curvilinear (right) coordinates

By D1ξ we denote the 2-D version of the narrow-stencil first-derivative SBP operator D1, approximating ∂∂ξ.

Similarly, D(b)2ξ approximates ∂∂ξ(b∂∂ξ). In the same manner, we let Hξ denote the 2-D version of the

diagonal matrix H, applied in the ξ-direction. D1η,D(b)2η and Hη are defined similarly for the η-direction.

To simplify the notation (without any restriction) we here assume Nξ=Nη=N. The 2-D operators can be neatly

expressed in terms of the 1-D operators using the Kronecker product:

D1ξHξEWRWEERE=D1⊗IN,D1η=IN⊗D1=H⊗IN,Hη=IN⊗H=e0⊗IN,ES=IN⊗e0=EWETW,RS=ESETS=eN

⊗IN,EN=IN⊗eN,=EEETE,RN=ENETN,

(14)

where the vectors e0 and eN are defined in (3). Assuming that the coefficient b is constant, we can also write

D(b)2ξ=D(b)2⊗IN,D(b)2η=IN⊗D(b)2.

(15)

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843

In the case of a variable coefficient b, however, Eq. (15) does not hold. To cover also that case, we introduce the

notation

b(ξ)i(η)=b(ξi,η),b(η)i(ξ)=b(ξ,ηi),i=0,1,…,N.

(16)

We also define

d(b)ij=⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢(D(b(η)0)2)i,j(D(b(η)1)2)i,j⋱(D(b(η)N)2)i,j⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥.

(17)

We then have

D(b)2ξ=⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢d(b)11d(b)21⋮d(b)(N+1)1d(b)12d(b)22⋮d(b)(N+1)2⋯⋯⋱⋯d(b)1(N+1)d(b)2(N+1)⋮d(b)(N+1)(N

+1)⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥

(18)

and

D(b)2η=⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢D(b(ξ)0)2D(b(ξ)1)2⋱D(b(ξ)N)2⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥.

(19)

3.2 The Continuous Problem

We consider the following problem:

utt=bΔuγ1u+γ2∇u⋅n+γ3ut=0u=f1,ut=f2,(x,y)∈Ω,(x,y)∈∂Ω,(x,y)∈Ω,t≥0t≥0t=0,

(20)

where b(x,y)>0. We have chosen homogeneous boundary conditions to avoid unnecessary notation in the

analysis. Similarly to the 1-D analysis in Sect. 2.1, the analysis holds for inhomogeneous conditions as well. We

also limit our present study to the case γ2≠0, which includes the important case of Neumann

conditions (γ1=0,γ2=1,γ3=0).

We can add dissipation to (20) by adding a term b∇⋅(σ∇ut),σ(x,y)≥0 to the right hand side of the PDE. The added

dissipation term will be used to create absorbing layers at artificial boundaries in Sect. 4.2. Including the

dissipation term, the problem reads

utt=bΔu+b∇⋅(σ∇ut)γ1u+γ2∇u⋅n+γ3ut=0u=f1,ut=f2,(x,y)∈Ω,(x,y)∈∂Ω,(x,y)∈Ω,t≥0t≥0t=0.

(21)

We now transform the problem to the unit square. Assume that there is a smooth one-to-one mapping

{x=x(ξ,η)y=y(ξ,η),

from Ω′ to Ω. The Jacobian J of the transformation is

J=xξyη−xηyξ.

The scale factors η1 and η2 of the transformation are defined as

η1=x2ξ+y2ξ−−−−−−√,η2=x2η+y2η−−−−−−√.






844

(22)

Since the mapping is one-to-one, the Jacobian is everywhere non-zero. By the chain rule, we have

{uξ=uxxξ+uyyξuη=uxxη+uyyη,

which is equivalent to

⎧⎩⎨⎪⎪⎪⎪ux=1J(uξyη−uηyξ)=1J((uyη)ξ−(uyξ)η)uy=1J(uηxξ−uξxη)=1J((uxξ)η−(uxη)ξ).

(23)

Replacing u with ux and uy in (23) yields

uxxuyy=1J(1J(uξyη−uηyξ)yη)ξ−1J(1J(uξyη−uηyξ)yξ)η=1J(1J(uξxη−uηxξ)xη)ξ−1J(1J(uξxη−uηxξ)xξ)η.

(24)

By adding uxx and uyy and rearranging terms, the first equation in (21) can be written as

J~utt=Δ~u+Δ~σut,(ξ,η)∈Ω′,

(25)

where we have defined

Δ~uΔ~σuα1=(α1uξ)ξ+(βuξ)η+(βuη)ξ+(α2uη)η,=(σα1uξ)ξ+(σβuξ)η+(σβuη)ξ+(σα2uη)η,=1J(x2η+y2η),β=−1J(xηx

ξ+yηyξ),α2=1J(x2ξ+y2ξ),

and

J~=Jb.

Using Eq. (23) to transform ∇u⋅n in the second equation in (21) yields the transformed boundary conditions:

⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪γ1η2u−γ2(α1uξ+βuη)+γ3η2ut=0,γ1η2u+γ2(α1uξ+βuη)+γ3η2ut=0,γ1η1u−γ2(α2uη+βuξ)+γ3

η1ut=0,γ1η1u+γ2(α2uη+βuξ)+γ3η1ut=0,(ξ,η)∈W(ξ,η)∈E(ξ,η)∈S(ξ,η)∈N.

(26)

The complete transformed problem is given by (25), (26) and the initial conditions stated in (21). Applying the

energy method (here assuming that also the time derivative of the boundary condition (26) holds) leads to

ddtE=−∫Wγ3+σγ1γ2η2u2tdη−∫Eγ3+σγ1γ2η2u2tdη−∫Nγ3+σγ1γ2η1u2tdξ−∫Sγ3+σγ1γ2η1u2tdξ−∫Ω′[utξutη]T[σα1σβ

σβσα2][utξutη]dΩ′,

(27)

where

E=12⎛⎝⎜∫Ω′J~u2tdΩ′+∫Ω′[uξuη]T[α1ββα2][uξuη]dΩ′+BT⎞⎠⎟,

(28)

and

BT=∫Wγ1γ2η2u2dη+∫Eγ1γ2η2u2dη+∫Nγ1γ2η1u2dξ+∫Sγ1γ2η1u2dξ+∫Wσγ3γ2η2u2tdη+∫Eσγ3γ2η2u2tdη+∫Nσγ3γ2η1

u2tdξ+∫Sσγ3γ2η1u2tdξ.










845

(29)

The matrix [α1ββα2] is positive definite since α1>0 and α1α2−β2=(xξyη−xηyξ)2=J2>0. Thus, the problem has

an energy estimate if the relations

γ1γ2≥0,γ3γ2≥0

(30)

hold. The last term in (27) implies damping of the energy for σ>0.

3.3 The Semi-Discrete Problem

In the semi-discrete setting we use the following notation for the matrices corresponding to the continuous

variable coefficients, for readability purposes: If λ denotes a variable coefficient in the continuous setting, we

here denote the matrix with the values of λ(ξ,η) at the grid points injected on the diagonal by λ. There is no risk

of confusion since it will always be clear from context whether we are in a continuous or semi-discrete setting.

The semi-discrete version of (26) is given by

⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪LWv=RW{γ1η2v−γ2(α1Sξv+βD1ηv)+γ3η2vt}=0LEv=RE{γ1η2v+γ2(α1Sξv+βD1ηv)+γ3η2v

t}=0LSv=RS{γ1η1v−γ2(α2Sηv+βD1ξv)+γ3η1vt}=0LNv=RN{γ1η1v+γ2(α2Sηv+βD1ξv)+γ3η1vt}=0.

(31)

In the numerical scheme we also impose the time-derivative of the boundary conditions when σ>0. For instance,

if we have the boundary condition Lv=f, we impose both Lv=f and Lvt=ft using the SAT technique.

The semi-discrete approximation of (25) and (26) using the SBP-SAT method is

J~vtt=D(α1)2ξv+D1ξβD1ηv+D1ηβD1ξv+D(α2)2ηv+D(σα1)2ξvt+D1ξσβD1ηvt+D1ησβD1ξvt+D(σα2)2ηvt+τ1H−

1ξLWv+τ1H−1ξLEv+τ2H−1ηLSv+τ2H−1ηLNv+τ3σH−1ξLWvt+τ3σH−1ξLEvt+τ4σH−1ηLSvt+τ4σH−1ηLNvt.

(32)

The first main result of the present study is stated in the following theorem:

Theorem 1

The scheme (32) is stable if τ1=τ2=τ3=τ4=−1γ2 and (30) holds.

Proof

Applying the energy method by multiplying (32) by vTtHξHη and adding the transpose leads to

ddtEH=vTt(1+τ1γ2)Hηα1(−RW+RE)Sξv+vTt(1+τ2γ2)Hξα2(−RS+RN)Sηv+vTt(1+τ3γ2)Hησα1(−RW+RE)Sξvt+vT

t(1+τ4γ2)Hξσα2(−RS+RN)Sηvt+vTt(1+τ1γ2)Hη(−RW+RE)βD1ηv+vTt(1+τ2γ2)Hξ(−RS+RN)βD1ξv+vTt(1+τ3γ2)

Hη(−RW+RE)σβD1ηvt+vTt(1+τ4γ2)Hξ(−RS+RN)σβD1ξvt+vTt(τ1γ3+τ3σγ1)Hηη2(RW+RE)vt+vTt(τ2γ3+τ4σγ1)H

ξη1(RS+RN)vt+vTtHηM(σα1)ξvt−vTtHξM(σα2)ηvt+2(D1ξvt)TσβHξHη(D1ηvt),

where

EH=12vTtHξHηJ~vt+12(vTHηM(α1)ξv+vTHξM(α2)ηv+2(D1ξv)TβHξHη(D1ηv))+12(−vTτ1γ1Hηη2(RW+RE)v−v

Tτ2γ1Hξη1(RS+RN)v)+12(−vTtτ3σγ3Hηη2(RW+RE)vt−vTtτ4σγ3Hξη1(RS+RN)vt).

By choosing τ1=τ2=τ3=τ4=−1γ2 we obtain an energy estimate completely analogous to (27). If (30) holds, we

have a non-growing energy. □

4 Model Problem











846

We begin this section by stating the problem, which can be solved by any appropriate method, in Sect. 4.1. In

Sect. 4.2 we derive a time-domain model of the problem. Section 4.3 is devoted to the details of applying the

SBP-SAT method to this model.

4.1 Problem Description

The benchmark problem introduced in [35] consists of the following components:

A 3-D topography, which is assumed to be axisymmetric. Figure 2 shows the vertical cross section of

the topography, which is available for download at [41].

A point source emitting spherical waves with a frequency of 50 Hz, located at range r=0 m,

height z=10 m.

Constant sound speed c=340 m/s.

The boundary condition at the ground, which in frequency domain is described by Eq. 15 in [35]. In

time domain, it is described by (37)–(39).

The objective is to compute the propagation loss P1(r) 1 meter above the ground, for range r< 2,000 m. Let A be

the amplitude of the point source and let u1(r) be the amplitude of the sound pressure field 1 m above ground at

range r. The definition of propagation loss yields

P1(r)=−20log(u1(r)A).

(33)

The propagation losses obtained with the PE approach in [35] and with the SBP-SAT method in the present

paper are available for download at [41].

Fig. 2

The topography

4.2 The SBP-SAT Model

When deriving the SBP-SAT model, we assume that propagation of sound waves is governed by the acoustic

wave equation

utt=bΔu,

(34)

where u is the acoustic pressure and b is the square of the wave velocity. As in Sect. 3.2, we introduce

dissipation,














847

utt=bΔu+b∇⋅(σ∇ut).

(35)

Expressing Eq. (35) in cylindrical coordinates (r,ϕ,z) and assuming symmetry in the azimuthal direction (the ϕ-

direction) results in the axisymmetric two-dimensional restriction of (35),

1butt=1r(rur)r+uzz+1r(σrur)r+(σuz)z.

(36)

We construct the physical domain by introducing artificial boundaries in the manner shown in Fig. 3. In the

simulations, the west boundary was placed at r=1 m and the north boundary at heights ranging

from z=300 to z=750 m. The dot at the z-axis marks the location of the source, just outside the west boundary of

the domain.

Fig. 3

Qualitative description of the physical domain. The dot at the z-axis represents the point source

The boundary condition at the south boundary is a locally-reacting impedance condition given by (see [35])

(pω0c−χ2)u+∇u⋅n+qcut=0,

(37)

where c is the wave speed, n is the unit outward normal, ω0 is the angular frequency of the source, χ is the

curvature and p and q are real numbers that satisfy the relation

p+qi=iZ^,

(38)

where Z^ is the normalized sound impedance, Z^=18.3+17.5i. The curvature χ is defined as

χ(r)=−Hrr(r)(1+H2r(r))3/2,

(39)






848

where H(r) denotes the height of the ground at horizontal position r.

The north and east boundaries are artificial boundaries that are introduced to truncate the unbounded domain.

We need to avoid reflections at these boundaries. Absorption of waves at artificial boundaries is an important

numerical issue. One possible approach is to apply an absorbing boundary condition (ABC), for example the

first order Engquist Majda ABC [8],

⎧⎩⎨⎪⎪⎪⎪(ut+b√∇u⋅n)|z=zmax=0,(ut+b√∇u⋅n+b√2ru)|r=rmax=0.

(40)

However, a first order Engquist Majda ABC is perfectly absorbing only at normal incidence. At 45∘ incidence

on a straight boundary, 17 % of the incoming wave is reflected, and close to glancing the reflection coefficient

tends to unity. Another, more efficient approach is to introduce an absorbing layer (AL) close to the artificial

boundary. This can be done by smoothly increasing the dissipation coefficient σ from zero to some fixed value.

The dissipation must increase quickly enough that the waves are damped efficiently inside the layer, but it must

also increase smoothly enough that we avoid reflections at the interface between the AL and the interior domain.

The wider one can afford to make the layer, the easier it is to find a function σ that fulfills both criteria. This AL

is a special case of the method presented by Appelö and Colonius [1]. Appelö and Colonius also slowed down

waves inside the layer by stretching the grid at the boundaries and included higher-order dissipation operators

for better performance. Since the focus in the present study is not on optimal absorbing layers, we here settle for

a simple version of their AL. We will verify that the truncation of the domain does not affect the solution by

placing the artificial boundaries at different locations in the simulations.

At the west boundary, the boundary condition is determined by the source. Consider a point source with

amplitude A and frequency f. At a distance r~ from the source, the acoustic pressure is given by

u(s)(r~,t)=Ar~sin(2πf(t−r~c)).

(41)

Now let the source be located at (r,z)=(0,z0). The distance r~ from the source to a point (r,z) is given by

r~=r2+(z−z0)2−−−−−−−−−−√.

(42)

Combining (41) and (42) yields

u(s)(r,z,t)=Ar2+(z−z0)2−−−−−−−−−−√sin⎛⎝⎜2πf⎛⎝⎜t−r2+(z−z0)2−−−−−−−−−−√c⎞⎠⎟⎞⎠⎟.

(43)

We will impose this boundary data on the west boundary. This can be done using a Dirichlet boundary condition

by extending the SAT technique developed in [29]. However, in the present study we instead impose the data

using a mixed boundary condition (which allows for a stronger energy estimate in terms of boundary data),

ut+b√∇u⋅n+b√2ru=u(s)t+b√∇u(s)⋅n+b√2ru(s),r=rmin.

(44)

We now introduce logical coordinates (ξ,η) and perform a transformation from the unit

square (0≤ξ≤1,0≤η≤1) onto the physical domain (r(ξ,η),z(ξ,η))∈Ω. Equation (36) transforms into

J~utt=Δ~u+Δ~σut,(ξ,η)∈Ω′,








849

(45)

where we have defined

J~Δ~uΔ~σu=rJb,=(α1uξ)ξ+(βuξ)η+(βuη)ξ+(α2uη)η,=(σα1uξ)ξ+(σβuξ)η+(σβuη)ξ+(σα2uη)η,

and

α1=rJ(r2η+z2η),β=−rJ(rξrη+zξzη),α2=rJ(r2ξ+z2ξ).

To summarize, the model that we solve with the SBP-SAT method is the Eq. (45) with the boundary conditions

(37) and (44) at the south and west boundaries, and either the ABC (40) or the AL at the north and east

boundaries. In the ABC approach, σ is identically zero. In the AL approach, σ is non-zero close to the north and

east boundaries. The Eq. (45) has the same form as (25), and all the boundary conditions (37), (44) and (40) are

of the mixed type analyzed in Sect. 3. Thus, the analysis performed in Sect. 3, proving well-posedness for the

continuous problem and stability for the discrete scheme, holds for this model too.

4.3 Implementation Details

We have implemented a fourth-order SBP-SAT method of the model problem. The spatial discretization is thus

fourth-order accurate in the interior scheme and second-order accurate in the boundary closures. For

completeness we have included the operators (first presented in [26]) in “Appendix”. The classical fourth-order

accurate Runge–Kutta method was used for discretization in time.

In order to apply the solver to the model problem, a computational grid must be constructed in the physical

domain. Generating a good grid on a complex domain is not a trivial task. If the grid is not smooth enough, the

convergence rate will decrease. In this case, the terrain profile has only two continuous derivatives, and hence

we suspect that the grid will not support high-order accuracy. We have used Pointwise, a commercial software

for creating grids. Figure 4 shows an example of a coarse grid generated in Pointwise. Table 1 lists the number

of gridpoints required when using 6, 9 and 12 points per acoustic wavelength, with the north boundary at a

height of 500 m.

Fig. 4

Coarse example grid

Table 1

Number of grid points corresponding to different resolutions. North boundary at z=500 m














http://link.springer.com/article/10.1007/s10915-014-9830-4/fulltext.html#Tab1


850

Resolution (points per wavelength) Nr Nz Nr×Nz

6 1,801 451 0.81×106

9 2,701 676 1.83×106

12 3,601 901 3.24×106

In the simulations, we time-advanced until the solution became periodic in time and then computed the

amplitude of the sound waves by measuring |v|max, the maximum absolute value of the solution during one

period, one meter above ground. The propagation loss P (measured in dB) was computed as

P=−20log(|v|maxA),

(46)

where A is the amplitude of the point source.

5 The Parabolic Equation Method

The PE method used in the model problem of Sect. 8 is described briefly below. More details can be found

in [19].

As an initial step, a smooth approximation h(r) of the ground height as function of range is computed from the

data, using an interpolating or a variance-reducing B-spline expansion [6, Ch. XI], the choice depending on the

smoothness of the data. The geometry is then mapped from the physical (r,z) domain to a rectangle in

the (ξ,η) plane by an orthogonal curvilinear transform

r=r(ξ,η),z=z(ξ,η),rξrη+zξzη=0,

(47)

such that η is constant along the boundaries of the computational domain. Assuming cylindrical symmetry, the

Helmholtz equation for the complex pressure u(ξ,η) is

f(f−1uξ)ξ+f(g−1uη)η+k2a2u=0,

(48)

where, using the unit dB/wavelength for the attenuation α,

fa=ρa/rb,g=ρb/ra,=(r2ξ+z2ξ)1/2,b=(r2η+z2η)1/2,k=ωc(1+iαlog(10)40π).

The PE approximations are derived by writing (48) in the form

T2u=(1−L)u−k−20Ru,

(49)

where k0 is a reference wavenumber, T and L are differential operators, and R a function:

Tu=−ik−10f1/2(f−1/2u)ξ,

(50)






851

Lu=−k−20[f(g−1(uη))η+k2a2u]+u,

(51)

R=(3f2ξ−2ffξξ)/(4f2).

(52)

Discarding the term Ru, which is small since R∼ξ−2 as ξ grows, Eq. (49) is simplified to

T2u=(1−L)u.

(53)

The PE schemes compute one-way solutions to (53) by solving

Tu=1−L−−−−√

(54)

with the pseudo differential operator 1−L−−−−√ replaced by a rational function of L,

Tu=Pm(L)Qn(L),

(55)

analogously with [2]. Pm and Qn are polynomials of degrees m and n in the Padé approximation

1−x−−−−√=Pm(x)/Qn(x)+O(xm+n+1),x→0.

(56)

Thus the JEPE PE-approximations are

u=Qn(L)v,

(57)

TQn(L)v=Pm(L)v,m=max(n,1),n=0,1,2,…

(58)

Increasing the Padé order n reduces the phase error as function of elevation angle, but also increases the

computational work. In practice, Padé orders n=0,1,2 are the most frequently used and correspond to the narrow

angle (15∘), the wide angle (35∘) and the very wide angle (55∘) approximations, cf. [15, Sec. 6.2.4].

Equation (58) with initial conditions at ξ=ξ0 and boundary conditions at η=0 (the upper boundary)

and η=−H (the ground) is solved using the method of lines. Thus, u,T,L and the boundary conditions are

discretized vertically using a centered second-order finite difference scheme [38, Sec. 9]. The vertically

discretized form of Eq. (58) is a system of ODEs (omitting the indices m and n)

ddξDQ(L)w=ik0D[P(L)−Q(L)]ww(ξ0)=w0

(59)

for the scaled and wavenumber-shifted complex pressure

w(ξ)=e−ik0ξ1/2v(ξ).

http://link.springer.com/search?dc.title=Discarding&facet-content-type=ReferenceWorkEntry&sortOrder=relevance









852

(60)

D and L are diagonal and tri-diagonal matrix-valued functions of ξ, respectively, with D real and the imaginary

part of L diagonal and non-positive. The initial profile w0 is computed from the given source data (height and

vertical directivity), by low-pass filtering w.r.t. vertical wave number to the validity interval of the PE scheme.

Equation 59 is then solved by a two-step fourth order A-stable second derivative method by Jeltsch (method J4

in [14]).

6 The Ray Interpolation Method

In the ray interpolation method the ground height is described by a smooth function of range identical to that in

the PE method described in Sect. 5. A ray trajectory (r(s),z(s)) where s is arc length, is a solution to the ODE

system [15, Sec. 3.2.1]

dr/dsdz/dsdϕ/ds=cos(ϕ)=sin(ϕ)={sin(ϕ)∂c/∂r−cos(ϕ)∂c/∂z}/c.

(61)

c=c(r,z) is the sound speed and ϕ=ϕ(s) the elevation angle of the ray. In the high-frequency limit, the ray

trajectories are streamlines of the acoustic intensity field i.e., propagation paths of acoustic energy. The

wavefield at a point (r,z) is then a sum of contributions from all rays passing through (r,z)—all eigenraysfrom

the source to (r,z)). Each eigenray contributes to the sum with the field inside an infinitesimal tube surrounding

the ray. With a mono-frequency monopole source with amplitude P0 at 1 m range, the value of the ray-tube field

along an eigenray (r(s),z(s)) is

P(f,s)=P0α(s)ei2πfτ(s)eiπnc(s)/2Πnb(s)j=1γj

(62)

where

ffrequency

(63)

α(s)=cosϕ0rA(s)ray tube area factor

(64)

ϕ0=ϕ(0)launch angle

(65)

A(s)=−sin(ϕ)∂r/∂ϕ0+cos(ϕ)∂z/∂ϕ0

(66)

τ(s)travel time along ray

(67)

nc(s)number of caustic points along ray

(68)

nb(s)number of ground reflections along ray

(69)






853

γjreflection coefficient for j′th ground reflection

(70)

The wavefield P(f,r) at points (r,h(r))0<r<R on the ground from a source at (0,zs) is computed in two steps.

First, a pre-defined number K of ray paths (rj(s),zj(s)) are computed by solving the ray ODEs with a fourth order

Runge–Kutta method [11, p. 178] with variable stepsize and local error control. The rays start from the

source (rj(0),zj(0))=(0,zs) with uniformly distributed vertical launch angles Φj,j=1,…,K, and the ODE system

(61) is augmented with one equation each for the travel time τ(s) along the ray and the partial

derivatives ∂r(s)/∂ϕ0,∂z(s)/∂ϕ0,∂ϕ(s)/∂ϕ0 with respect to launch angle ϕ0. Rays are reflected at the ground and

terminated at the maximal range r=R or at the upper boundary z=Z of the computational region. The number and

the locations of caustic points and ground reflections along each ray are determined.

Then, for each point (rj,h(rj)) on a receiver grid, the field P(f,r) is obtained as a sum of contributions of the

form (62) approximating the eigenrays by cubic interpolation to appropriate ray subsets.

7 Convergence Results

In this section we verify the implementation of the fourth-order SBP-SAT method and investigate the quality of

the grid in a series of convergence studies. We will calculate the convergence rate q as

q=log10(∥vref−v(N2)∥h∥vref−v(N1)∥h)/log10(N1N2)1/d,

(71)

where d is the dimension (d=2 here), vref is a reference solution, v(N) is the corresponding numerical solution

with N grid points and ∥vref−v(N)∥h is the discrete l2 norm of the error.

7.1 Convergence Study Without Absorbing Layer

To separate the effects of the grid from the numerical method, we here present a convergence study on a smooth

curvilinear grid. We use the analytical solution

u(a)(r,z,t)=Ar2+(z−z0)2−−−−−−−−−−√sin⎛⎝⎜2πf⎛⎝⎜t−r2+(z−z0)2−−−−−−−−−−√c⎞⎠⎟⎞⎠⎟

(72)

which is the pressure field created by a point source with frequency f and amplitude A, located at (r,z)=(0,z0).

We here set A=1,f=12.5 Hz and z0=10 m. In the computations we use the same setup as for the benchmark

problem, except that we here set the dissipation coefficient σ to zero everywhere (in order to have an analytical

solution) and impose the analytical solution u(a) as initial and boundary data. The setup with the smooth domain

and the initial condition is shown in Fig. 5. The convergence results are presented in Table 2. We note that we

obtain the expected fourth order convergence rate.







854

Fig. 5

The analytical solution at time t=0 on a smooth grid

Table 2

log(l2−errors) and convergence rates on a smooth grid, without absorbing layer

Nr×Nz logel2 q

81×21 -0.26 0.00

161×41 -1.30 3.46

321×81 -2.62 4.45

641×161 -3.73 3.64

1,281 × 321 -4.88 3.80

Next, we investigate the quality of the grid generated (using the commercial grid-generator Pointwise) for the

benchmark problem by running a convergence study with exactly the same setup on that grid. The results are

presented in Table 3. We note that we obtain approximately third order convergence on this grid and draw the

conclusion that the grid, as expected, is not smooth enough to support high-order accuracy (higher than third

order). When solving the benchmark problem, we will thus have to make do with third order convergence.

Table 3

log(l2−errors) and convergence rates on the grid generated for the benchmark problem, without absorbing layer



855

Nr×Nz logel2 q

226×56 0.33 0.00

451×111 -0.53 2.87

901×221 -1.37 2.77

1,801 × 441 -2.19 2.72

7.2 Convergence Study with Absorbing Layer

To verify also the implementation of the AL, we here perform a convergence study with a non-zero dissipation

coefficient σ on the benchmark grid. The solution obtained with 3,601 × 881 grid points was used as a reference

solution. The results are presented in Table 4. Similar to Table 3, we obtain slightly less than third order

convergence on this grid.

Table 4

log(l2−errors) and convergence rates on the grid generated for the benchmark problem, with absorbing layer

Nr×Nz logel2 q

226×56 0.46 0.00

451×111 -0.32 2.59

901×221 -1.09 2.56

1,801 × 441 -1.84 2.48

8 Computations

We have solved the benchmark problem described in Sect. 4 for two different sound speed profiles:

profile 1: Constant profile, c=340 m/s

profile 2: Linear profile, c=c0+kz with c0=340 m/s and k=0.1 s−1.

To guarantee a correct solution it is important to verify: (1) that the SPL at ground level is grid-converged, and

(2) that reflections at artificial boundaries are negligible. We begin this section by investigating the effects of the

artificial boundary treatment, and then perform a grid-convergence study. The grid-converged results obtained

with the SBP-SAT method are then compared with the results obtained with the PE and ray tracing methods.

8.1 Domain Truncation

To investigate the effect of the artificial boundaries, we computed the propagation loss for different locations of

the north boundary, for case 1. In Fig. 6 we compare the first order Engquist Majda ABC with the AL approach

to truncate the domain at the north boundary. The effects of the reflections using the first order Engquist Majda

ABC decrease as we move the north boundary higher, but even with the north boundary at a height of 2,000 m

the reflections interfere with the interior waves and cause rapid oscillations in the SPL at ground level. The






856

spurious reflections when using the AL are much smaller. After 500 m, the results do not change visibly. In the

remaining computations we place the north boundary at z=500 m and employ the AL approach to truncate the

domain.

Fig. 6

Propagation loss measured 1 m above ground for different locations of the north boundary, using a first order

Engquist Majda ABC and b an AL

Remark 3

We also extended the domain and moved the east boundary further to the right, but the location of the east

boundary turned out to have no impact on the SPL at ground level.


857

8.2 Grid-Convergence

To verify that the discretization errors are negligible, we study how the computed propagation loss varies with

grid refinement. The results are shown in Fig. 7. The curve obtained using 4.5 grid points per wavelength

deviates significantly from the others, while the curves corresponding to 6 and 12 grid points per wavelength are

almost identical, i.e., indicating grid convergence. In the remaining simulations, grids with 12 points per

wavelength were used.

Fig. 7

Convergence study. The graphs show the propagation loss 1 m above the ground for different levels of grid

refinement

8.3 Comparison of Models

Figure 8 shows the propagation loss, obtained with the SBP-SAT method, in the entire domain up to a height of

100 m. The effects of the refraction that occurs with profile 2 is most apparent far away from the source.




858

Fig. 8

Propagation loss (dB) for different sound speed profiles, a profile 1 and b profile 2

In Fig. 9 we compare the result obtained with the SBP-SAT method with the result published in [35] and the

result obtained with the PE method described in Sect. 5, for profile 1. We observe that the results are in fairly

good agreement. The maximum difference between the SBP-SAT and the PE methods in Sect. 5 and in [35] are

8 and 4 dB, respectively.







859

Fig. 9

Propagation loss 1 m above ground for SBP-SAT and PE, with profile 1

In Fig. 10 we compare the ray tracing methods with the SBP-SAT method, for profile 1. The difference between

the computed SPL using the SBP-SAT method and the most accurate ray tracing method is always greater than

15 dB beyond 1,100 m. We also note that the results obtained with the different ray tracing methods differ

significantly from one another, and they all under-predict the SPL.

Fig. 10

Propagation loss 1 m above ground for the SBP-SAT method and the ray tracing methods, with profile 1

In Fig. 11 we compare the SBP-SAT method with the PE and ray tracing methods, with profile 2. The PE

method again shows reasonable agreement with the SBP-SAT method. The ray tracing methods are here in




860

better agreement with the SBP-SAT method than they were in Fig. 10 (profile 1), but again they under-predict

the SPL, except ”Ray interpolation” that now over-predicts (except at the very end of the domain).

Fig. 11

Propagation loss 1 m above ground for the SBP-SAT method, the PE and ray tracing methods, with profile 2

9 Conclusion

The theory surrounding the SBP-SAT technique has been extended with a result that proves the stability of the

SBP-SAT method for the second order wave equation on a curvilinear 2-D domain with mixed boundary

conditions. A fourth-order accurate SBP-SAT method has been applied to the benchmark problem on

atmospheric sound propagation introduced in [35]. Since the SBP-SAT method is here applied to the full wave

equation model, it can be used as a reference against which simpler (and computationally cheaper) methods can

be validated.

The present study has shown that, when applying the SBP-SAT method to sound propagation problems, the

following should be considered:

The introduction of artificial boundaries must not affect the solution. One way to achieve this is with

carefully constructed absorbing layers.

A grid generated for a realistic topography might not support high-order accuracy.

The SPL must be grid-converged.

The results presented in Figs. 10 and 11 show that ray tracing methods are not reliable for prediction of SPL in

the case of irregular terrain. The PE methods show reasonable agreement with the SBP-SAT method, both with

constant speed of sound and with a linear sound speed profile, which is expected since the topography in this

problem is rather gentle. One would expect the PE methods, and the ray tracing methods in particular, to be

more unreliable in the case of more pronounced topography. This is something we hope to address in a coming

study.

Appendix: Finite Difference Operators

For completeness we present the fourth order SBP operators. Here h denotes the grid-spacing. The interior

stencils (in D1 and M(b)) are the standard central 4th order accurate finite difference stencils. At the boundaries

we use one-sided stencils that are formally second order accurate. The discrete norm H is defined:






861

The first derivative SBP operator is given by,

The third-order accurate boundary derivative operator S0 is given by,

S0=1h[116−332−1300…]

The interior stencil of −hM(b) at row i is given by (i=7…N−6):

mi,i−2mi,i−1mi,imi,i+1mi,i+2=16bi−1−18bi−2−18bi=16bi−2+16bi+1+12bi−1+12bi=−124bi−2−56bi−1−56bi+1

−124bi+2−34bi=16bi−1+16bi+2+12bi+12bi+1=16bi+1−18bi−18bi+2.

The left boundary closure of −hM(b) (given by a 6×6 matrix) is given by

m1,1m1,2m1,3m1,4m1,5m1,6m2,2m2,3m2,4m2,5m2,6m3,3m3,4m3,5m3,6m4,4m4,5m4,6m5,5m5,6m6,6=1217b1

+59192b2+27010400129345067064608b3+694623760312070402387648b4=−5968b1−6025413881211265549

76b3−5374166637042184992b4=217b1−59192b2+21331800516049630912b4+20839385998024815456b3=36

8b1−124472400121126554976b3+75280666721126554976b4=4957908710149031312b3−4957908710149031

312b4=−1784b4+1784b3=34813264b1+92582828316238757669235228057664b3+2360243299962031278205


862

871342944b4=−59408b1−2929461579460729725717938208b3−294467388102329725717938208b4=−591088

b1+2602973192328912556411742685888b3−608341868138411278205871342944b4=−132818869266337594

290333616b3+132818869266337594290333616b4=−86732904112b3+86732904112b4=151b1+59192b2+1377

705022330059726218083221499456b4+56446113384296b5+3782888823025465122092707643413496776874

56b3=1136b1−125059743572b5−48363400904421872275525802884687299744b3−172204932779818917715

3814624b4=−1053241207733542840005263888b4+16139767610328843057963657098519931984b3+564461

4461432b5=−9601191280713392b4−33916692148b5+3323505419126452850508784b3=31088b1+507284006

600757858213475219048083107777984b3+18691032230716b5+124b6+195006219843699738346176140288

32b4=−495927181498464461320965546238960637264b3−16b6−1599871490964937594290333616b4−37517

7743572b5=−3683952230716b5+752806667539854092016b3+10636498712336b4+18b6=8386761355510099

813128413970713633903242b3+22247172617734372763180339520776b4+56b6+124b7+280535371786b5=−

35039615213452232b4−16b7−1309181092513226425254392b3−11187492230716b5−12b6=32906368004458

7b4+55801816692148b5+56b7+124b8+66020484313226425254392b3+34b6

The corresponding right boundary closure is obtained by replacing bi→bN+1−i for i=1,…,8 followed by a

permutation of both rows and columns. Let mi,j be the entry at row i and column j in M(b). The matrix M(b) is

symmetric, which means that it is completely defined by the entries on and above the main diagonal,

i.e., mj,i=mi,j,i=1,…,N,j=i,…,N.

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Documents

Atmospheric Sound Propagation Over Large-Scale …scholarism.net/FullText/ijmcs20143102.pdfcomplex atmosphere nor irregular terrain properly. Unfortunately, due to a shortage of reliable