To solve the linear acoustic equations for room acoustic purposes, the performance of the time-domain nodal discontinuous Galerkin (DG) method is evaluated. A nodal DG method is used for the evaluation of the spatial derivatives, and for the time-integration an explicit multi-stage Runge-Kutta method is adopted. The scheme supports a high order approximation on unstructured meshes. To model frequency-independent real-valued impedance boundary conditions, a formulation based on the plane wave reflection coefficient is proposed. Semi-discrete stability of the scheme is analyzed using the energy method. The performance of the DG method is evaluated for four three-dimensional configurations. The first two cases concern sound propagations in free field and over a flat impedance ground surface. Results show that the solution converges with increasing DG polynomial order and the accuracy of the impedance boundary condition is independent on the incidence angle. The third configuration is a cuboid room with rigid boundaries, for which an analytical solution serves as the reference solution. Finally, DG results for a real room scenario are compared with experimental results. For both room scenarios, results show good agreements.

1.
M. R.
Schroeder
, “
Novel uses of digital computers in room acoustics
,”
J. Acoust. Soc. Am.
33
(
11
),
1669
1669
(
1961
).
2.
M.
Vorländer
, “
Computer simulations in room acoustics: Concepts and uncertainties
,”
J. Acoust. Soc. Am.
133
(
3
),
1203
1213
(
2013
).
3.
L.
Savioja
and
U. P.
Svensson
, “
Overview of geometrical room acoustic modeling techniques
,”
J. Acoust. Soc. Am.
138
(
2
),
708
730
(
2015
).
4.
B.
Hamilton
, “
Finite difference and finite volume methods for wave-based modelling of room acoustics
,” Ph.D. dissertation,
The University of Edinburgh
, Edinburgh, Scotland,
2016
.
5.
V.
Valeau
,
J.
Picaut
, and
M.
Hodgson
, “
On the use of a diffusion equation for room-acoustic prediction
,”
J. Acoust. Soc. Am.
119
(
3
),
1504
1513
(
2006
).
6.
J. M.
Navarro
,
J.
Escolano
, and
J. J.
López
, “
Implementation and evaluation of a diffusion equation model based on finite difference schemes for sound field prediction in rooms
,”
Appl. Acoust.
73
(
6-7
),
659
665
(
2012
).
7.
D.
Botteldooren
, “
Finite-difference time-domain simulation of low-frequency room acoustic problems
,”
J. Acoust. Soc. Am.
98
(
6
),
3302
3308
(
1995
).
8.
J.
Sheaffer
,
M.
van Walstijn
, and
B.
Fazenda
, “
Physical and numerical constraints in source modeling for finite difference simulation of room acoustics
,”
J. Acoust. Soc. Am.
135
(
1
),
251
261
(
2014
).
9.
C.
Spa
,
A.
Rey
, and
E.
Hernandez
, “
A GPU implementation of an explicit compact FDTD algorithm with a digital impedance filter for room acoustics applications
,”
IEEE/ACM Trans. Audio Speech Lang. Process.
23
(
8
),
1368
1380
(
2015
).
10.
B.
Hamilton
and
S.
Bilbao
, “
FDTD methods for 3-D room acoustics simulation with high-order accuracy in space and time
,”
IEEE/ACM Trans. Audio Speech Lang. Process.
25
(
11
),
2112
2124
(
2017
).
11.
T.
Okuzono
,
T.
Yoshida
,
K.
Sakagami
, and
T.
Otsuru
, “
An explicit time-domain finite element method for room acoustics simulations: Comparison of the performance with implicit methods
,”
Appl. Acoust.
104
,
76
84
(
2016
).
12.
S.
Bilbao
, “
Modeling of complex geometries and boundary conditions in finite difference/finite volume time domain room acoustics simulation
,”
IEEE Trans. Audio Speech Lang. Process.
21
(
7
),
1524
1533
(
2013
).
13.
R.
Mehra
,
N.
Raghuvanshi
,
L.
Savioja
,
M. C.
Lin
, and
D.
Manocha
, “
An efficient GPU-based time domain solver for the acoustic wave equation
,”
Appl. Acoust.
73
(
2
),
83
94
(
2012
).
14.
C.
Spa
,
A.
Garriga
, and
J.
Escolano
, “
Impedance boundary conditions for pseudo-spectral time-domain methods in room acoustics
,”
Appl. Acoust.
71
(
5
),
402
410
(
2010
).
15.
M.
Hornikx
,
W.
De Roeck
, and
W.
Desmet
, “
A multi-domain Fourier pseudospectral time-domain method for the linearized Euler equations
,”
J. Comput. Phys.
231
(
14
),
4759
4774
(
2012
).
16.
M.
Hornikx
,
C.
Hak
, and
R.
Wenmaekers
, “
Acoustic modelling of sports halls, two case studies
,”
J. Build. Perform. Simul.
8
(
1
),
26
38
(
2015
).
17.
B.
Cockburn
and
C.-W.
Shu
, “
TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework
,”
Math. Comput.
52
(
186
),
411
(
1989
).
18.
Y.
Reymen
,
W.
De Roeck
,
G.
Rubio
,
M.
Baelmans
, and
W.
Desmet
, “
A 3D discontinuous Galerkin method for aeroacoustic propagation
,” in
Twelfth International Congress on Sound and Vibration
, Lisbon, Portugal (
2005
).
19.
J.
Nytra
,
L.
Čermák
, and
M.
Jícha
, “
Applications of the discontinuous Galerkin method to propagating acoustic wave problems
,”
Adv. Mech. Eng.
9
(
6
),
168781401770363
(
2017
).
20.
A.
Modave
,
A.
St-Cyr
, and
T.
Warburton
, “
GPU performance analysis of a nodal discontinuous Galerkin method for acoustic and elastic models
,”
Comput. Geosci.
91
,
64
76
(
2016
).
21.
Y.
Reymen
,
M.
Baelmans
, and
W.
Desmet
, “
Efficient implementation of Tam and Auriault's time-domain impedance boundary condition
,”
AIAA J.
46
(
9
),
2368
2376
(
2008
).
22.
K.-Y.
Fung
and
H.
Ju
, “
Broadband time-domain impedance models
,”
AIAA J.
39
(
8
),
1449
1454
(
2001
).
23.
V. E.
Ostashev
,
D. K.
Wilson
,
L.
Liu
,
D. F.
Aldridge
,
N. P.
Symons
, and
D.
Marlin
, “
Equations for finite-difference, time-domain simulation of sound propagation in moving inhomogeneous media and numerical implementation
,”
J. Acoust. Soc. Am.
117
(
2
),
503
517
(
2005
).
24.
H. L.
Atkins
and
C.-W.
Shu
, “
Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations
,”
AIAA J.
36
(
5
),
775
782
(
1998
).
25.
J. S.
Hesthaven
and
T.
Warburton
,
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis and Applications
(
Springer-Verlag
,
New York
,
2007
), Chaps. 2,3,4,6,10.
26.
J. S.
Hesthaven
and
C.-H.
Teng
, “
Stable spectral methods on tetrahedral elements
,”
SIAM J. Sci. Comput.
21
(
6
),
2352
2380
(
2000
).
27.
R. J.
LeVeque
,
Finite Volume Methods for Hyperbolic Problems
(
Cambridge University Press
,
Cambridge
,
2002
), Chap. 2.
28.
P.
Lasaint
and
P.-A.
Raviart
, “
On a finite element method for solving the neutron transport equation
,” in
Mathematical Aspects of Finite Elements in Partial Differential Equations
(
Elsevier
,
Amsterdam
,
1974
), pp.
89
123
.
29.
F. Q.
Hu
and
H.
Atkins
, “
Eigensolution analysis of the discontinuous Galerkin method with nonuniform grids: I. One space dimension
,”
J. Comput. Phys.
182
(
2
),
516
545
(
2002
).
30.
M.
Ainsworth
, “
Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods
,”
J. Comput. Phys.
198
(
1
),
106
130
(
2004
).
31.
F. Q.
Hu
,
M.
Hussaini
, and
P.
Rasetarinera
, “
An analysis of the discontinuous Galerkin method for wave propagation problems
,”
J. Comput. Phys.
151
(
2
),
921
946
(
1999
).
32.
F.
Hu
and
H.
Atkins
, “
Two-dimensional wave analysis of the discontinuous Galerkin method with non-uniform grids and boundary conditions
,” in
8th AIAA/CEAS Aeroacoustics Conference & Exhibit
(
2002
), p.
2514
.
33.
T.
Toulorge
and
W.
Desmet
, “
Optimal Runge-Kutta schemes for discontinuous Galerkin space discretizations applied to wave propagation problems
,”
J. Comput. Phys.
231
(
4
),
2067
2091
(
2012
).
34.
H.
Atkins
, “
Continued development of the discontinuous Galerkin method for computational aeroacoustic applications
,” in
3rd AIAA/CEAS Aeroacoustics Conference
(
1997
), p.
1581
.
35.
B.-T.
Chu
and
L. S.
Kovásznay
, “
Non-linear interactions in a viscous heat-conducting compressible gas
,”
J. Fluid Mech.
3
(
5
),
494
514
(
1958
).
36.
K. W.
Thompson
, “
Time dependent boundary conditions for hyperbolic systems
,”
J. Comput. Phys.
68
(
1
),
1
24
(
1987
).
37.
B.
Engquist
and
A.
Majda
, “
Absorbing boundary conditions for numerical simulation of waves
,”
Proc. Natl. Acad. Sci.
74
(
5
),
1765
1766
(
1977
).
38.
B.
Gustafsson
,
High Order Difference Methods for Time Dependent PDE
(
Springer-Verlag
,
Berlin
,
2007
), Chap. 2.
39.
S.
Bilbao
,
B.
Hamilton
,
J.
Botts
, and
L.
Savioja
, “
Finite volume time domain room acoustics simulation under general impedance boundary conditions
,”
IEEE/ACM Trans. Audio Speech Lang. Process.
24
(
1
),
161
173
(
2016
).
40.
D.
Dragna
,
K.
Attenborough
, and
P.
Blanc-Benon
, “
On the inadvisability of using single parameter impedance models for representing the acoustical properties of ground surfaces
,”
J. Acoust. Soc. Am.
138
(
4
),
2399
2413
(
2015
).
41.
S. C.
Reddy
and
L. N.
Trefethen
, “
Stability of the method of lines
,”
Numer. Math.
62
(
1
),
235
267
(
1992
).
42.
H. O.
Kreiss
and
L.
Wu
, “
On the stability definition of difference approximations for the initial boundary value problem
,”
Appl. Numer. Math.
12
(
1-3
),
213
227
(
1993
).
43.
C.
Bogey
and
C.
Bailly
, “
A family of low dispersive and low dissipative explicit schemes for flow and noise computations
,”
J. Comput. Phys.
194
(
1
),
194
214
(
2004
).
44.
C.
Geuzaine
and
J.-F.
Remacle
, “
GMSH: A 3-D finite element mesh generator with built-in pre- and post-processing facilities
,”
Int. J. Numer. Meth. Eng.
79
(
11
),
1309
1331
(
2009
).
45.
X.
Di
and
K. E.
Gilbert
, “
An exact Laplace transform formulation for a point source above a ground surface
,”
J. Acoust. Soc. Am.
93
(
2
),
714
720
(
1993
).
46.
H.
Kuttruff
,
Acoustics: An Introduction
(
CRC Press
,
Boca Raton, FL
,
2006
), Chap. 9.
47.
J.
Saarelma
,
J.
Botts
,
B.
Hamilton
, and
L.
Savioja
, “
Audibility of dispersion error in room acoustic finite-difference time-domain simulation as a function of simulation distance
,”
J. Acoust. Soc. Am.
139
(
4
),
1822
1832
(
2016
).
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