Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy &
Radiative Transfer
journal homepage: www.elsevier.com/locate/jqsrt
Review
Advances in the discrete ordinates and finite volume methods
for the solution of radiative heat transfer problems
in participating media
Pedro J. Coelho n
Mechanical Engineering Department, LAETA, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1,
1049-001 Lisboa, Portugal
a r t i c l e i n f o
abstract
Article history:
Received 6 March 2014
Received in revised form
16 April 2014
Accepted 21 April 2014
Available online 29 April 2014
Many methods are available for the solution of radiative heat transfer problems in
participating media. Among these, the discrete ordinates method (DOM) and the finite
volume method (FVM) are among the most widely used ones. They provide a good
compromise between accuracy and computational requirements, and they are relatively
easy to integrate in CFD codes. This paper surveys recent advances on these numerical
methods. Developments concerning the grid structure (e.g., new formulations for
axisymmetrical geometries, body-fitted structured and unstructured meshes, embedded
boundaries, multi-block grids, local grid refinement), the spatial discretization scheme,
and the angular discretization scheme are described. Progress related to the solution
accuracy, solution algorithm, alternative formulations, such as the modified DOM and
FVM, even-parity formulation, discrete-ordinates interpolation method and method of
lines, and parallelization strategies is addressed. The application to non-gray media,
variable refractive index media, and transient problems is also reviewed.
& 2014 Elsevier Ltd. All rights reserved.
Keywords:
Discrete ordinates method
Finite volume method
Spatial and angular discretization
Solution accuracy
Solution algorithm
Parallelization strategies
Contents
1.
2.
3.
4.
5.
6.
7.
n
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discretization procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Grid structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.
Cell-vertex methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.
Axisymmetrical geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.
Blocked-off region procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.
Embedded boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.
Body-fitted structured or unstructured grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.
Multi-block grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.
Grid adaptation and local grid refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Angular discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solution accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solution algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tel.: þ 351 218418194.
E-mail address: pedro.coelho@tecnico.ulisboa.pt
http://dx.doi.org/10.1016/j.jqsrt.2014.04.021
0022-4073/& 2014 Elsevier Ltd. All rights reserved.
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8.
9.
10.
11.
12.
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P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
Alternative formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.
Modified discrete ordinates and modified finite volume methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.
Even parity formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.
Discrete ordinates interpolation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.
Pseudo time stepping and method of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.
Other methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parallel implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.
Parallel implementation of the standard algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.
Parallel implementation of other solution algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.
Parallel implementation of alternative formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transient problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Application to non-gray media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Application to media with variable refractive index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction
Thermal radiation is an important heat transfer mode
that is present in many problems of practical relevance, such
as energy transfer in power plants, combustion chambers,
high-temperature heat exchangers, rockets, fires, etc. Thermal radiation is often combined with conduction and/or
convection, and other physical phenomena, such as turbulence and combustion may also be present. This implies that
the equations governing all the relevant phenomena need to
be solved simultaneously. This article, however, is only
concerned with thermal radiation in participating media,
which is governed by the radiative transfer equation (RTE).
The present work is concentrated on radiative heat
transfer problems. We shall not address developments in
the areas of atmospheric and solar radiation, or in other
areas where the Boltzmann equation needs to be solved,
e.g., neutron transport. Radiative transfer in porous media,
plasmas and light propagation, with or without polarization, is also excluded from the present review. Similarly,
applications to practical heat transfer problems, without
new developments in the solution methods, and to
coupled and inverse problems are also excluded from the
present survey, except for exemplification purposes.
Many methods have been developed for the solution of
radiative heat transfer problems in participating media.
Among these, the discrete ordinates method (DOM) [1,2]
and the finite volume method (FVM) [3,4] are among the
most widely used ones. They provide relatively good
accuracy for a wide range of problems, with moderate
computational requirements, and are relatively easy to
integrate in CFD codes. However, similarly to other methods for the solution of the RTE, that integration may lead
to a significant increase of computational time, particularly
for non-gray media, and to additional complexity when a
finer grid is used for the fluid flow than for radiation
calculations, e.g., to comply with the requirements of fine
boundary layer resolution. The progress achieved in
the DOM and FVM in the past few years is surveyed in
the present article. Work previous to year 2000 is not
addressed here, except when needed to complement the
description of the most recent one.
The DOM and FVM are addressed together in the
present paper, since they share many features, and the
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differences between them are small. The designation
‘discrete ordinates’ in the DOM strictly refers to the
angular discretization procedure, in which the RTE is
solved for a representative finite set of directions. A weight
is assigned to every direction, such that the sum of the
weights is equal to the area of the surface of a unit sphere,
and integrals over a solid angle are evaluated using a
quadrature method. In the DOM, the spatial discretization
is usually carried out using the finite volume/finite difference method, but other options are possible. Methods that
employ other spatial discretization procedures (e.g., finite
element methods, spectral methods, meshless methods)
while retaining discrete ordinates for angular discretization will not be addressed here. The designation ‘finite
volume’ in the FVM implies that both the spatial and
angular discretizations are performed using the finite
volume discretization procedure. The radiation intensity
over a solid angle is assumed to be constant, but its
direction is allowed to vary. Hence, the DOM and the
FVM differ on the angular discretization procedure, as
described in Section 2. Only a brief description is presented here. The reader is referred to references [1–4] or
text books [5,6] for further details, or to the references
cited below for specific developments.
It is probably fair to say that advances in the DOM and
FVM reported in the past decade or so have been more
significant than for other methods. This is a consequence of
the popularity of these methods, as mentioned above. Many
of these advances have been aimed at the mitigation of the
drawbacks of the methods, namely, by extending the
application to more complex grid structures, proposing
new spatial discretization schemes for the reduction of
false scattering, or other angular discretization methods
for the reduction of ray effects. Other developments are
concerned with the solution algorithm, improvement of the
accuracy, alternative formulations, parallel implementation,
application to non-gray or variable refractive index media,
and extension to transient problems. These advances are
surveyed in the remainder of this paper.
2. Discretization procedure
A brief overview of the discretization of the RTE using
the DOM and FVM is presented below. Although transient
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
123
problems, non-gray media and variable refractive index
media are addressed in the present review, we will only
consider the stationary RTE and gray media with unity
refractive index in the present section. The RTE for an
emitting–absorbing–scattering gray medium may be written as follows [5]:
Z
ss
s U ∇Iðr; sÞ ¼ βIðr; sÞ þκI b ðrÞ þ
Iðr; s0 ÞΦðs0 ; sÞ dΩ0
ð1Þ
4π 4π
following discretized equation for the DOM:
where IðsÞ is the radiation intensity in direction s, r is the
position vector, Ib is the blackbody radiation intensity, κ, β
and ss are the absorption, extinction and scattering coefficients of the medium, respectively, Φðs0 ; sÞ is the scattering
phase function and Ω is a solid angle. Integrating the RTE
in a control volume centered at grid node P and applying
the Gauss divergence theorem to the term on the left hand
side, yields
Z
Z
ss
s U nIðr; sÞ dA ¼ βVI P ðsÞ þ κVI b;P þ V
I P ðs0 ÞΦðs0 ; sÞ dΩ0
4π 4π
A
ð2Þ
where superscript m (1 rmrM) denotes the mth direction and wl is the quadrature weight for the lth direction.
In the FVM, Eq. (5) is integrated over a solid angle, often
referred to as control angle, ΔΩm, arising from the discretization of the entire spherical solid angle. It is assumed
that the value of the radiation intensity remains constant
within that control angle, like in the DOM. However, in
contrast to the DOM, the direction of the radiation intensity is allowed to vary within a solid angle. Hence, the
following discretized equations are obtained for the FVM:
where n denotes the outer unit vector normal to a cell
face, and V is the volume of the control volume under
consideration. Eq. (2) was obtained assuming that the
variables on the right side of Eq. (1) remain constant
within the control volume, following the standard finite
volume discretization procedure. The integral along the
boundary on the left side of Eq. (2) is now approximated
by a summation, yielding
Z
F
ss
∑ s U nf I f ðsÞAf ¼ βVI P ðsÞ þ κVI b;P þ V
I P ðs0 ÞΦðs0 ; sÞ dΩ0
4π 4π
f ¼1
F
½
þ
Inserting this equation into Eq. (3) yields
½
F
s Unf Af þβV I P ðsÞ ¼
∑
f ¼1
ðs U nf 4 0Þ
þ
ss
V
4π
Z
4π
I P ðs0 ÞΦðs0 ; sÞ dΩ0
F
∑
f ¼1
js U nf jI U;f ðsÞAf þ κVI b;P
ðs Unf o 0Þ
ð5Þ
The previous equations are valid for both the DOM and
the FVM. Now, the angular discretization, which differs in
the two methods, will be carried out.
In the DOM, Eq. (5) is replaced by a discrete set of M
coupled differential equations that describe the radiation
intensity field along M directions, and integrals over solid
angles are replaced by a quadrature of order M yielding the
F
¼
ðsm U nf 40Þ
∑
f ¼1
jsm U nf jI m
U;f Af þ κVI b;P
ðsm U nf o 0Þ
M
ss
V ∑ w I l Φðsl ; sm Þ
4π l ¼ 1 l P
2
ð6Þ
3
6 F
7m
m
4 ∑ Dcf Af þ βVΔΩm 5I P ¼
f ¼ 1
ðDm 4 0Þ
cf
þ
F
m
m
∑ jDm
cf jI U;f Af þ κVI b;P ΔΩ
f ¼ 1
ðDm o 0Þ
cf
M
ss
lm
V ∑ I l Φ ΔΩl ΔΩm
4π l ¼ 1 P
ð7Þ
where superscript m (1 rm rM) denotes the mth control
lm
m
angle and Dm
are defined as follows:
cf ; ΔΩ and Φ
Z
Dm
s Unf dΩm
ð8aÞ
cf ¼
ΔΩm
Z
ΔΩm ¼
ð3Þ
where subscript f denotes a cell face, whose area is Af, F is
the total number of cell faces of the control volume under
consideration, and If(s) is the mean radiation intensity at
cell face f along direction s. Different methods may be
employed to relate If(s) to the radiation intensity at the
grid nodes. In this section, for simplicity, we will use only
the step scheme, which approximates If(s) by the radiation
intensity at the grid node in the center of the upstream
control volume, IU,f, yielding
s Unf
s U nf
; 0 þI U;f max ;0
ð4Þ
I f ¼ I P max
js Unf j
js U nf j
m
sm U nf Af þ βV I P
∑
f ¼1
R
lm
Φ
¼
dΩm
ð8bÞ
ΔΩm
R
m
0
ΔΩl ΔΩm Φðs ; sÞ dΩ
m
l
ΔΩ ΔΩ
dΩl
ð8cÞ
The integrals in Eqs. (8a) and (8b) are evaluated analytically, while that in Eq. (8c) may require numerical integration or not, depending on the phase function.
3. Grid structure
Both the DOM and FVM were originally applied to
Cartesian or axisymmetrical geometries, even though
the FVM was originally formulated for general control
volumes. Both methods were subsequently extended to
more complex grid structures, and since the spatial discretization is generally accomplished using finite volumes,
most of the developments described below can be applied
to both methods. Most of the progress described below
concerning the grid structure originated in CFD, where the
finite volume method is widely employed.
3.1. Cell-vertex methods
In general, the radiation intensity is computed at the grid
nodes located at the center of the control volumes of the grid
under consideration. This option is referred to as a cellcentered grid. Alternatively, the grid nodes may be placed at
the vertices of the disjoint subdomains, referred to as elements, that define the mesh, and the radiation intensity is
calculated at the vertices of the elements, yielding the
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P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
written using angular coordinates defined by either Cartesian or cylindrical base vectors. In the latter case, an angular
redistribution term appears in the radiative transfer
equation (RTE), which accounts for the variation of the
direction of propagation of the radiation intensity in a
cylindrically axisymmetrical coordinate system, even
though the physical direction of propagation does not
change. The angular redistribution term involves the azimuthal derivative of the radiation intensity, and has traditionally been discretized in the DOM using a recursive
relation derived in [1], which is enforced by the principle
of conservation of energy for isotropic radiation, to determine the coefficients of that term. Ben Salah et al. [14]
so-called vertex-centered grid. The control volume that surrounds a grid node in vertex-centered grids is defined either
by joining together the centers of the elements that share that
grid node or by connecting the centers of the elements that
share the grid node to the midpoints of the sides of those
elements. In the latter case, the method is sometimes referred
to as a control volume finite element method. Cell-vertex
methods were used, for example, in [7–13].
3.2. Axisymmetrical geometries
In the case of axisymmetrical geometries, the vector
along the direction of propagation of radiation may be
Active
region
Inactive
region
Block 1
Block 2
Block 3
Fig. 1. Schematic of grid structures. (a) Blocked-off. (b) Embedded boundaries. (c) Body-fitted structured. (d) Body-fitted unstructured. (e) Multi-block.
(f) Local grid refinement.
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
derived explicit expressions for the coefficients of the
discretized angular redistribution term using the FVM,
which satisfy the recursive relation derived in [1].
A comparison of this recursive relation with two other
methods proposed in the literature for the simulation of
axisymmetrical geometries is reported in [15]. The other
two methods are presented in [16,17], both of them in the
framework of the FVM. Chui et al. [16] used angular
coordinates based on a Cartesian base vector, so that the
angular redistribution term does not appear. They
employed a mapping procedure to obtain the radiation
intensity in two spatial and two angular coordinates by
calculating the radiation intensity in three spatial coordinates and one angular coordinate. Murthy and Mathur [17]
also relied on Cartesian base vectors to express the rays'
direction, but developed a conservative discretization
scheme of the angular redistribution term that allows
the solution of axisymmetrical problems for all ordinate
directions using a two-dimensional mesh, in contrast with
Chui et al. [16]. Similar accuracy was obtained by Kim and
Baek [15] for the different methods.
A modified discretization method is proposed in [18] for
the RTE in two-dimensional axisymmetrical meshes using
the FVM. The directions of propagation of radiation intensity
are determined relative to Cartesian-base vectors, and are
fixed even when the spatial location changes, as in [16,17]. A
two-dimensional mesh is used, along with two angular
coordinates, as in [17]. However, in contrast with Ref. [17],
the proposed method eliminates control-angle overlap
caused by misalignment of solid angles with the faces of
control volumes in the angular direction. Errors due to the
curvature of cell faces are also eliminated.
Kim [19] reported an alternative formulation for the
FVM in axisymmetrical cylindrical enclosures, using
cylindrical base vectors for both spatial and angular
coordinates. A mapping was used to maintain a spatial
and angular two-dimensional solution procedure, while
the angular redistribution term was determined without
any artifice from angular and geometrical considerations
by means of angular edge directional weights, thus generalizing the method presented in [14].
A method to solve the RTE in axisymmetrical geometries by means of a general three-dimensional FVM solver
that extends the two-dimensional geometry by one cell in
the third direction (i.e., the grid has only one cell in the
tangential direction) has recently been proposed [20]. At
the two boundaries that contain the symmetry axis, a
symmetry boundary condition is applied.
3.3. Blocked-off region procedure
Cartesian or cylindrical grids are easier to employ than
body-fitted structured or unstructured meshes, and they
were often preferred in the past. In the blocked-off method
[21,22], curved or straight inclined boundaries are
approximated in a stepwise fashion, while maintaining
Cartesian or cylindrical grids (see Fig. 1a). A loss of
accuracy occurs due to this approximation. Obstructions
within the domain or baffles, defined as obstructions with
a negligible thickness, may be treated using a similar
procedure. In the blocked-off procedure, the domain
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contains active regions, where the solution is sought, and
inactive regions, lying within obstacles, where the solution
is also obtained, even though it is not meaningful. Wasteful computations and memory storage are needed to treat
the inactive regions, but the mathematical procedure is
relatively simple. Examples of application of this method
may be found in [23–26].
3.4. Embedded boundaries
Embedded boundaries are aimed at an improvement of
the blocked-off procedure. A Cartesian (or cylindrical)
coordinate system is still employed, but an exact treatment of straight inclined boundaries is used, as shown in
Fig. 1(b), in contrast to the blocked-off procedure. Curved
boundaries are approximated as piecewise straight lines,
which may be skewed relatively to the Cartesian directions. Hence, irregular polygonal control volumes may
appear along the boundaries, and the RTE is integrated
over these irregular control volumes, like in the case of
arbitrary control volumes. Byun et al. [27] compared the
results obtained using embedded boundaries, as formerly
described in [28], a blocked-off procedure for Cartesian
coordinates and a body-fitted mesh. They concluded that
Cartesian meshes with embedded boundaries and bodyfitted meshes yield similar results, while the blocked-off
procedure produces some errors, especially for the radiative heat fluxes on skewed boundaries.
3.5. Body-fitted structured or unstructured grids
The DOM was originally developed for regular geometries using Cartesian or cylindrical coordinates, and only
during the nineties was extended to body-fitted coordinates (see Fig. 1c and d). The application of the DOM to
two-dimensional, planar or axisymmetrical, or threedimensional unstructured grids is reported in [29].
A comparison of three different formulations of the DOM
for two-dimensional complex geometries is presented in
[30], namely, the discrete ordinates interpolation method
[31], which will be addressed below, a finite volume
spatial discretization for orthogonal curvilinear coordinates [32], and a finite volume spatial discretization
for unstructured grids mapped using triangular control
volumes [33].
In contrast with the DOM, the FVM was originally
formulated for complex geometries, even though the first
applications were restricted to regular geometries, as
mentioned above. More recently, an unstructured radiative
heat transfer module was developed [34] and coupled
with the national combustion code developed at NASA
[35], allowing for general symmetrical, periodic or wall
boundary conditions. A pixelation approach was used to
improve the accuracy of treating reflecting walls and
symmetry or periodic boundaries. Unstructured grids were
also used in [36] to simulate radiative transfer in twodimensional geometries with obstacles. Kim et al. [37]
extended the FVM to two-dimensional polygonal unstructured meshes, which are generated from unstructured
triangular meshes by connecting adjacent centroids of
the triangular control volumes. Similar polygonal
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P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
unstructured meshes were used by Kim et al. [38] for
axisymmetrical enclosures. Hybrid unstructured meshes
comprising prismatic, pyramidal and tetrahedral control
volumes were recently employed [39,40]. Other applications using unstructured grids are those relying on a
control volume finite element formulation, which has been
mentioned above [8,9,11,12].
3.6. Multi-block grids
Spatial-multiblock procedures may be used as an alternative to local grid refinement by restricting grid refinement to regions where fine grids are needed for numerical
accuracy reasons (see Fig. 1e). They are also useful in
geometrically complex domains, where a single structured
grid may be difficult or even impossible to generate.
Neighboring blocks may either overlap or not. The former
option is simpler to implement, but the latter is more
flexible and saves some memory and computing time.
Gridlines at the interface between blocks may be either
continuous or not. Continuous gridlines at the interface are
straightforward to implement and ensure complete conservation of radiative energy. Talukdar et al. [41] used
continuous gridlines at the interface, and overlapping
blocks. Discontinuous gridlines at the interface require
an interpolation method to exchange data between neighboring blocks. The procedure proposed by Chai and Moder
[42] and also used by Byun et al. [27] allows for nonoverlapping discontinuous gridlines, and ensures conservation of heat transfer rate, net radiant power, and other
full-range and half-range moments across every block
interface.
The multi-block strategy may also be applied to the
angular discretization, as shown in [43], where a coarse
angular discretization was used in optically thick regions,
and a fine angular discretization in optically thin ones. A
procedure that ensures integral conservation of heat
transferred between neighboring blocks was developed.
3.7. Grid adaptation and local grid refinement
Grid adaptation is a technique used to concentrate grid
nodes in regions where higher resolution is needed during
the solution procedure, either by redistributing grid nodes
or by grid refinement (see Fig. 1f). It is often combined
with local grid refinement, which restricts a fine grid to
regions of the computational domain where the spatial
discretization error is guessed or estimated to be high,
while using a coarse grid elsewhere. In this case, the grid
structure is characterized by a nested hierarchy of refined
subgrids. A coarse mesh covers the entire computational
domain. Then, a finer refinement level is placed at the
desired locations by dividing the control volumes of the
original grid into smaller ones. Typically, a control volume
is divided in two, four or eight equally sized control
volumes for one-, two- and three-dimensional problems,
respectively. These refined regions do not need to be
contiguous. This procedure may be repeated, yielding
regions of higher refinement level.
This approach was used by Jessee et al. [44] and Howell
et al. [45], who carried out the local grid refinement
adaptively, during the course of the solution procedure,
based on an estimation of the solution error. A multi-level
algorithm was used to obtain the solution of the RTE.
Recently, a multi-block based adaptive mesh refinement
method was employed in [46]. The blocks of the grid are
organized in a hierarchical quad-tree data structure,
restricting the number of control volumes by dynamically
adapting the mesh to satisfy the refinement criteria. An
adaptive procedure for unstructured hybrid grids was
reported in [40].
4. Spatial discretization
Similarly to the FVM, the DOM often employs a finite
volume spatial discretization, although other methods
may also be employed. The spatial discretization of the
RTE requires the calculation of the radiation intensity at
the cell faces of the control volumes, and the discretization
schemes employed for this purpose are addressed in the
present subsection. All schemes surveyed here are applicable to both the DOM and the FVM.
Earlier works used the step, the diamond or the
exponential scheme, or variants of these, to evaluate the
radiation intensity at cell faces of the control volumes.
The step scheme, which is the counterpart of the upwind
scheme in CFD, introduces excessive numerical smearing,
also referred to as false diffusion or false scattering. The
diamond scheme, which is similar to the central differences scheme in CFD, is unbounded, and may yield
physically unrealistic solutions. Unrealistic solutions may
be prevented by setting to zero negative radiation intensities that may appear during the solution procedure.
However, this practice may yield non-physical spatial
oscillations. A variable weight scheme that combines the
step and the diamond schemes has been proposed, but
does not satisfactorily overcome the drawbacks of those
two schemes. A positive scheme, which guarantees positive radiation intensities, but not necessarily bounded
ones, has also been used. The exponential scheme and
variants are potentially more accurate in one-dimensional
computations, but not always in multidimensional ones,
Table 1
Average absolute error ( 102) of the radiation intensity field along
directions α ¼451 and α¼ 301 [51].
Discretization
scheme
α¼ 451 α¼ 301 Discretization
scheme
α ¼451 α ¼301
STEP
MINMOD
GAMMA
CLAM
NOTABLE
MUSCL
SMART
CUBISTA
WACEB
VONOS
SUPERBEE
Van Albada
OSHER
11.27
4.89
4.19
3.29
3.24
2.74
2.19
3.02
2.53
1.62
1.24
4.02
3.86
3.35
3.04
3.55
2.53
1.76
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
10.02
5.30
4.80
4.13
4.16
3.70
3.45
3.96
3.64
2.82
2.75
4.61
4.43
UMIST
KOREN
UNO2
SONIC A
SONIC B
N
S-MINMOD
S-CLAM
S-SUPERBEE
S-Van Albada
S-OSHER
S-UMIST
S-KOREN
4.17
3.85
4.17
3.32
2.82
5.36
3.55
3.03
2.39
3.27
3.10
3.36
2.92
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
101
S10
1 st
X
U
v
r
orde
X
U
v
Average emission error (%)
where unbounded solutions may occur. A discussion of
these schemes has been published in [47].
The discretization of the first order derivatives of the
radiation intensity is similar to the discretization of the
convective terms of transport equations in CFD. Accordingly, schemes developed in CFD may be used in radiative
transfer. This was first demonstrated by Liu et al. [48] and
Jessee and Fiveland [49], who applied to the discretization
of RTE several high-order resolution schemes formulated
according to the normalized variable diagram (NVD)
proposed by Leonard [50]. This formulation establishes
a set of criteria that discretization schemes need to satisfy
to guarantee boundedness and at least second-order
accuracy.
A comprehensive comparison of discretization schemes
is reported in [51], which includes the MINMOD, GAMMA,
CLAM, NOTABLE, MUSCL, SMART, CUBISTA, WACEB and
VONOS (see [51] for the references to these schemes). An
example of the results obtained is presented below. A twodimensional square enclosure of unit side length with
black walls is considered. The medium is transparent and
radiation propagates along direction s ¼cos αiþsin αj. The
radiation intensity field is defined as I¼H(y yL) where
yL ¼0.5þ (x 0.5) tan α, and H is the Heaviside function.
The radiation intensity field at the boundary is prescribed
and the RTE is solved using the DOM along s direction. The
average absolute error of the predicted radiation intensity
is given in Table 1 for a uniform grid with 25 25 control
volumes and for several discretization schemes. The error
is much larger for the step scheme than for all other
schemes, particularly when α¼451. The CLAM scheme,
which was recommended in [49], is not among the most
accurate ones, but is rather stable and relatively economical, while other good NVD schemes, namely, CUBISTA,
MUSCL, WACEB and SMART are more accurate, but more
time consuming.
The high-resolution schemes based on the NVD treat
the radiation across a control volume face as locally onedimensional. Bounded skew high order resolution
schemes were developed for CFD [52] and applied to the
RTE by Coelho [53]. The skewed schemes, despite being
more accurate, are more computationally demanding,
particularly for fine grids, and have not been further
employed.
Total variation diminishing (TVD) schemes (see, e.g.,
[54]), originally developed to solve hyperbolic equations,
prescribe alternative criteria that discretization schemes
should satisfy to ensure accuracy, monotonicity and
entropy preservation. These schemes were formerly developed for compressible flows, aiming at a good resolution of
very steep gradients characteristic of shock waves, and
later extended to incompressible flows. A few schemes
(e.g., MINMOD, CLAM, MUSCL) satisfy both NVD and TVD
criteria. Several TVD schemes (SUPERBEE, Van Albada,
OSHER, UMIST, KOREN) have been applied to radiative
heat transfer benchmark problems in [46,51,56,57]. The
SUPERBEE yielded the most accurate results among the
TVD schemes for the tests carried out in [51] (see Table 1),
but it is computationally demanding.
Fig. 2 shows results obtained by Godoy and Desjardin
[57] for a cubical enclosure containing a gray medium in
127
v
U
X
100
v
U
X
2n
v
U
X
do
e
rd
v
U
X
10-1
X
U
v
10-2
0.01
0.05
r
step
minmod
MC
Ospre
superbee
1.5
UMIST
van Albada
van Leer
ChOsh
Koren
0.1
0.15 0.2
cell width (m)
Fig. 2. Average relative error of the emissive power as a function of
cell size for regular 3D Cartesian meshes and several discretization
schemes [57].
radiative equilibrium. The walls are black, three of them
have an emissive power of unity and the other ones are
cold. The absorption coefficient of the medium is 1 m 1.
The calculations were performed using the DOM and the
S10 quadrature. The average relative error of the emissive
power of the medium is shown in Fig. 2 as a function of
grid size for the step and several TVD schemes. All these
schemes exhibit an order of convergence between 1.6 and
1.7, and are much more accurate than the step scheme,
whose order of accuracy is close to 1.0. Most TVD limiters
have approximately the same performance. The SUPERBEE
limiter is slightly more accurate for moderate pure absorbing–emitting and isotropic media, but not in the case of
anisotropic media.
The accuracy of TVD schemes decreases in the vicinity
of smooth extrema. The essentially non-oscillatory (ENO)
schemes [58] aim at overcoming this disadvantage of TVD
schemes, and yield a uniformly high-order accurate discretization scheme. While the NVD and TVD schemes
relate the dependent variable at a cell face to its values
at two upstream and one downstream grid nodes, the ENO
schemes involve three upstream nodes and two downstream nodes. ENO schemes were tested in [51,56], and no
improvement over the TVD schemes was found for the test
cases reported there.
Schemes based on the NVD and TVD criteria involve
stencils with grid nodes aligned along lines normal to the
cell faces. In contrast, genuinely multidimensional schemes
involve stencils with grid nodes in a quadrangular arrangement around the central grid node. The S-Van Albada
scheme was originally employed in [55] to solve the RTE,
showing several advantages over the TVD schemes,
namely, minimal cross-stream dissipation and dispersion
errors. Ismail and Salinas [59] found that genuinely multidimensional schemes along with the Van Albada limiter
yield results more accurate than those obtained using the
CLAM scheme. Genuinely multidimensional schemes were
also employed by Coelho [51], who concluded that they
perform particularly well for problems with discontinuities. In the example shown in Table 1, the S schemes
consistently perform better than the corresponding TVD
schemes. Genuinely multidimensional schemes are rather
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P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
stable, but computationally demanding. As a consequence,
the first-order N scheme is not competitive in comparison
with NVD or TVD schemes, in contrast to the second-order
S schemes, especially the S-SUPERBEE scheme. The genuinely multidimensional schemes have only been applied to
two-dimensional problems, in contrast to previous
schemes, which have been used to solve one-, two- and
three-dimensional problems.
Apart from the genuinely multidimensional schemes,
the discretization schemes mentioned above are readily
applicable to curvilinear structured grids, either orthogonal or non-orthogonal, even though most works on radiative heat transfer in curvilinear grids have relied on the
basic spatial discretization schemes or on the exponential
schemes mentioned above.
Most applications of the DOM or FVM to unstructured
grids have relied on the step scheme [29,36,37] and on
slightly different versions of the exponential scheme
[10,60]. A skew upwinding scheme was reported in
[13,61]. The diamond scheme is not directly applicable to
unstructured grids, but a modified version referred to as
diamond mean flux was proposed in [62]. Joseph et al. [63]
reported a comparison of the step, diamond mean flux and
exponential scheme of Sakami and Charrete [64]. Capdevilla et al. [65] tested four different high order spatial
schemes for the discretization of the RTE using the FVM,
but these schemes are not bounded.
The application of NVD and TVD schemes to unstructured meshes is reported in [66,67]. The implementation
of these schemes in unstructured grids requires approximations, since in general there is no upwind neighbor of
the two grid nodes straddling a cell face. A method to
overcome this difficulty was proposed in [68] for NVD
schemes, an equivalent implementation was reported in
[69] for TVD schemes, and an improved method was
described in [70], all of them for CFD problems. These
methods along with a new one were compared in [66,67]
for the solution of the RTE. It was found that although the
NVD and TVD schemes perform much better than the step
and diamond mean flux schemes, they are not as accurate
as in Cartesian coordinates, and their order of convergence
is lower than in that case. The implementation proposed in
[68] has also been used in the recent work of Lygidakis and
Iokonos [40] on unstructured grids.
5. Angular discretization
The angular discretization of the RTE requires the
selection of a finite number of directions of propagation
of radiation intensity and the associated quadrature
weights in the DOM, and the selection of discrete solid
angles, also referred to as control angles, in the FVM. In
general, any angular discretization method employed in
the FVM may also be applied in the DOM, but the reverse
is not true. In fact, although the weight of a quadrature in
the DOM may be thought of as a solid angle, its boundaries
are not always defined geometrically, preventing in
such a case its direct application in the FVM. The angular
discretization is largely arbitrary, but there are some
recommended rules. A detailed list of guidelines is presented in [71].
The simplest angular discretization method consists of
the division of the angular domain into a finite number of
discrete, nonoverlapping, solid angles defined by the
intersection of lines of constant latitude and lines of
constant longitude. This choice is typical of the FVM, but
it may also be employed in the DOM. However, most
applications of the DOM use the SN [1,72] or the TN [73]
quadratures. More recent alternative angular discretization methods are surveyed below.
Kim and Huh [74] proposed a quadrature, referred to as
FTn FVM, in the framework of the FVM, somewhat similar
to the spherical rings arithmetic progression quadrature
(SRAPN) introduced in [75]. In the latter, a hemisphere is
divided into N spherical rings, starting from the top of the
sphere, where the spherical ring degenerates on a crown.
The spherical rings are divided into a different number of
identical solid angles, which increases in arithmetic progression from the top of the hemisphere to the bottom.
The centers of the solid angles obtained in this way define
the discrete directions, while the area of each solid angle,
i.e., the quadrature weight, is the same for all discrete
directions. There are only two minor differences between
FTn FVM and SRAPN. The polar angle is uniformly divided
in the FTn FVM, while the division is non-uniform in
SRAPN, and the azimuthal angle for the first octant is not
subdivided for the spherical ring at the top, while it is
uniformly divided into 2 angles in the SRAPN. The application of these quadratures to the DOM is straightforward.
A quadrature for the DOM that also employs, in every
octant, directions associated with solid angles defined by
the intersection of lines of constant latitude and lines of
constant longitude was used in [76]. However, these solid
angles are not subject to any rules like those used in
[74,75]. The only restriction is that the directions must be
invariant to any rotation of 901. The weights used for the
evaluation of the heat flux and for the calculation of the inscattering term are evaluated following the procedure
used in the FVM, i.e., assuming that the radiation intensity
does not change within a solid angle. This means that the
quadrature weights are obtained from analytical calculation of the integrals that appear in the definition of heat
flux and in-scattering term after setting the radiation
intensity to unity. Accordingly, the resultant method may
be regarded as a hybrid of the DOM and FVM.
Rukolaine and Yuferev [77] proposed two different
piecewise quasilinear angular (PQLA) quadratures that
are somewhat similar to the TN quadrature. The accuracy
of PQLA quadratures is not as good as that of SN and TN
quadratures with a similar number of discrete directions,
but they allow the DOM solution of radiation problems
with specular reflective boundaries in complex geometries, since the angular dependence of the radiation
intensity may be expressed analytically. Li et al. [78]
proposed two spherical symmetrical equal dividing (SSDN)
quadratures, both based on geometrical considerations,
with equal weights for all directions. The number of
discrete directions is limited to 96, and the accuracy is
reported to be similar to that of the SN quadratures.
A comparison of a wide range of quadratures, namely,
SN, TN, PQLA, double cyclic triangle (DCT) [79] and two
types of Lebedev quadratures [80,81] is presented in [71].
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
6. Solution accuracy
Many recent works on the DOM and FVM have aimed
at the improvement of the solution accuracy, like those
mentioned above concerned with the spatial and the
angular discretization. In this subsection, other articles
reporting modifications to achieve more accurate solutions, either for general or for particular problems, are
addressed.
False scattering (see, e.g., [83]) may be successfully
reduced using appropriate discretization schemes, as
described above. However, other proposals have been
made to reduce false scattering. Li et al. [84] proposed
the double rays method to reduce false scattering in the
framework of the discrete ordinates interpolation method,
which is an alternative formulation of the DOM, as
described in Section 8.3. Although the underlying principle
of this method is quite simple, the interpolation procedure
is cumbersome, and the computer code becomes complicated, even in two-dimensional problems, the only ones
considered in this paper. A hybrid scheme, which is much
simpler to implement, but only able to reduce false
scattering in arbitrarily specified directions, while retaining false scattering in the other discrete directions, is
reported in [85]. This scheme may be useful to deal with
collimated irradiation. Estimation of false scattering in the
DOM and FVM is presented in [86,87].
Ray effects are also discussed in [83]. Ray effects can be
reduced by increasing the number of discrete directions, at
the expense of additional computational time. An investigation of the reduction of ray effects with the increase of
the number of discrete directions is presented in [88].
Estimation of ray effects in the DOM and FVM is presented
in [86]. The modified discrete ordinates or finite volume
methods, addressed in Section 8.1, successfully mitigate
ray effects arising from discontinuities or sharp gradients
of the temperature of the boundaries.
An improved DOM for mitigation of ray effects based on
the concept of discrete directions with infinitely small
weights is described in [89]. The method employs a ray
tracing procedure with a large number of discrete directions to calculate the heat fluxes incident on the boundary
after the DOM solution has been determined using a
conventional solution procedure. The method was applied
to rectangular enclosures with black walls and an isotropic
scattering medium. A drastic reduction of ray effects
originated by non-smooth radiative emission at the
boundary or in the medium was observed, with a minor
increase in the computational time. The effectiveness of
the method depends on how fast the ray tracing calculations can be performed. The accuracy and the computational requirements in more demanding problems,
including gray boundaries, anisotropic scattering, nonhomogeneous and non-isothermal media, and complex
geometries have not been investigated.
Ray effects and false scattering have opposite effects
and tend to compensate each other, as discussed in
[83,90]. False scattering tends to smoothen the radiation
intensity field, while ray effects tend to enhance discontinuities or gradients of the radiation intensity field. This
implies that simultaneous spatial and angular refinement,
or a more accurate spatial differencing scheme and angular refinement, should be used to improve solution accuracy. If only one of these two refinements is made, the
solution accuracy may decrease [90].
In general, the DOM does not strictly conserve energy
in the case of anisotropic scattering, i.e., there is no
guarantee that the integral of the scattering phase function
over a unity spherical surface yields 4π, even though it is
possible to use a correction factor of the in-scattering term
to enforce conservation [91], as demonstrated in [92].
However, the correction factor introduces changes in the
overall shape and asymmetry factor of the phase function
[93], and this may yield large errors in the case of highly
anisotropic phase functions. Comparative calculations of
the DOM and FVM for strongly anisotropic media show
some advantages of the FVM for such media [93]. A new
phase function normalization was proposed by Hunter and
Guo [94–96] that ensures conservation of both the scattered energy and the overall asymmetry factor after
discretization. An example of the effectiveness of this
normalization procedure is shown in Fig. 3, which refers
to a cubic enclosure containing a cold medium with one
diffusely emitting hot wall at zn ¼z/L¼ 0, L being the side
length. The optical thickness of the medium is τ¼10.0 and
the scattering albedo is ω¼1.0. The Henyey–Greenstein
0.25
Old Norm
New Norm
Monte Carlo
0.2
Q (x*,y * = 0.5, z* =1)
It was found that the best accuracy among the studied
quadratures, with up to 100 discrete directions, was the
Lebedev quadrature of the Chebyshev type LC11, which has
96 directions, and integrates exactly all moments up to
order 11, except the first order moment.
It is well known that the widely used SN quadratures
yield physically unrealistic negative weights when the
number of directions becomes large. Hunter and Guo
[82] compared four quadratures that do not suffer from
this limitation, namely, the TN, SRAPN, and two types of
Gauss–Legendre quadratures: the Legendre equal-weight
and the Legendre–Chebyshev quadratures. They concluded
that they all have accuracy equal to or better than SN for up
to 288 directions and for the investigated test cases.
129
g = 0.93
0.15
0.1
g = 0.80
S12
HG Phase Fun.
0.05
0
τ = 10.0, ω = 1.0
g = 0.20
0
0.1
Positive Scheme
0.3
0.2
0.4
0.5
x*
Fig. 3. Incident heat flux on the top wall of a cubic enclosure [96]. The
asymmetry factor g of the Henyey–Greenstein phase function is normalized according to [91] (old norm) or to [94] (new norm). The DOM
predictions from [96] are compared with Monte Carlo results from [93].
130
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
phase function was considered with different values of the
asymmetry factor. The radiative heat flux at the center of
the wall at zn ¼1 predicted by the DOM with the S12
quadrature is plotted in Fig. 3, and compared with Monte
Carlo benchmark results. It can be seen that the old
normalization [91] and the new one [94–96] yield similar
results for an asymmetry factor g¼ 0.20, but the differences between the two normalization procedures become
increasingly larger as the asymmetry factor approaches
unity. In such a case, the new normalization factor yields
much better results than the old one.
A drawback of the normalization technique proposed
by Hunter and Guo [94–96] is the need to predetermine a
normalization matrix. This may be avoided using a simpler
normalization technique recently proposed [97], which
simultaneously conserves scattered energy and the asymmetry factor of the phase function, and retains most of the
phase-function shape. The method was applied to the
Henyey–Greenstein phase function, yielding results similar
to those obtained using the previous normalization technique, but with a lower computational burden.
A similar problem may occur in the FVM. Even though the
FVM accurately conserves scattered energy, the phase function asymmetry is not exactly conserved, leading to a significant change in the scattering effect for highly anisotropic
scattering media. The phase function normalization reported
in [94–96] was extended to the FVM, and the advantage of the
normalization for such media was demonstrated [98].
The application of the DOM and the FVM to problems
with external collimated radiation is not straightforward,
particularly if the direction of the collimated beam is
different from the discrete directions of the quadrature set.
In the case of the FVM, a small solid angle that contains the
direction of propagation may be used [99], and good results
have been reported even for highly anisotropic media, without phase function normalization [100]. In the DOM, the
direction of the collimated radiation beam may be expressed
as an average combination of neighboring directions [101]. A
more elegant approach is to add the direction of the
collimated beam to the DOM quadrature set, and assign an
infinitely small weight to that direction [102].
7. Solution algorithm
The discrete set of algebraic equations in the DOM and
FVM is generally solved using a Gauss–Seidel method that
solves all the equations for a discrete direction following
an optimal sweeping order. This is often referred to as
space-marching (SM) or mesh sweeping algorithm. In the
case of regular geometries meshed using Cartesian or
cylindrical coordinates, and for a given discrete direction,
the optimal marching procedure starts from a control
volume located at a corner of the computational domain.
That corner is selected according to the sign of the
direction cosines of the direction under consideration, in
such a way that the upstream cell faces lie on the
boundary of the domain. The solution of the discrete
equations for all the remaining control volumes proceeds
in the direction of orientation of the direction cosines, in
such a way that the upstream cell face intensities of the
visited control volume are available either from the
boundary conditions or from the calculations performed
for the previously visited control volumes. Then, a similar
procedure is undertaken for all the other directions. After
all directions have been scanned, the radiation intensities
leaving the boundary surfaces are updated using the
boundary conditions. The iteration process continues until
the convergence criterion has been satisfied. We will refer
to this solution procedure as the standard solution algorithm [1,2]. The standard solution algorithm may also be
applied to complex geometries using structured or
unstructured grids, but the sweeping order needs to be
determined for every direction, and stored. The sweeping
order may be found as described, e.g., in [10,35,63].
The SM algorithm becomes slow when the coupling
between the discrete directions is strong, as in the case of
strongly scattering media and/or highly reflecting diffuse
boundaries. In fact, the in-scattering term of the radiative
transfer equation (RTE) and the radiation intensities leaving the boundaries are usually calculated using the values
available from the previous iteration, i.e., an explicit
treatment is used, causing an increase of the number of
iterations required to achieve convergence. The convergence rate also decreases with the increase of the optical
thickness of the medium when the temperature field is
unknown, e.g., in radiative equilibrium problems. In such a
case, the RTE and the conservation equation for energy are
solved simultaneously. The convergence rate is also
affected when high-order resolution schemes are used
and implemented using a deferred correction procedure.
Acceleration methods to overcome or mitigate the
decrease of the convergence rate for optically thick media
have been proposed in [103]. More recently, Koo et al.
[104] proposed a fully implicit method aimed at the
improvement of convergence when the temperature field
is unknown and the optical thickness of the medium is
large. An improved version of an implicit scheme formerly
reported in [105], which was found to fail for pure
scattering in the case of optically intermediate media
and fine grids, is described in [106].
An alternative to the SM algorithm and to the acceleration schemes is the use of nonstationary iterative
methods to solve the system of governing discrete equations. These include, among others, the generalized minimum residual (GMRES) [107], the generalized conjugate
gradient (GCG) [108], the generalized conjugate gradient
least-squares (GCG-LS) [109] and the conjugate gradient
squared (CGS) [110] methods. In general, these methods
are optimally suited for large sparse systems of equations,
and converge faster than stationary iterative methods, like
the Gauss–Seidel method. Their convergence rate may be
further improved using preconditioning.
The conditioned CGS method was used in [8,9,11,12]. Ben
Salah et al. [9] found no significant variation of iterations and
CPU time with the increasing of the scattering coefficient
when the conditioned CGS solver was used. However,
Axelsson [109] claims that Lanczos-type methods, like this
one, are not based on any minimization property and may
exhibit rather erratic convergence behavior.
Krishnaprakas et al. [111] compared three different generalized conjugate gradient methods of the Krylov subspace
family for the solution of 2D radiation problems using the
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
FVM, namely, the GMRES, GCG and GCG-LS methods. All
these methods were employed using successive underrelaxation preconditioning, and restarted after p iterations,
where p denotes the dimension of the Krylov subspace. The
equations for all directions and all control volumes were
solved simultaneously, which is only feasible due to the small
number of equations (grid size and directions) employed. It
was found that preconditioning greatly improves the convergence rate, but the under-relaxation parameter plays a
major role, and there is no rule to obtain the optimum value.
They attempted to use the CGS method, but the convergence
was poor and erratic.
Godoy and Desjardin [57] used the GMRES method for
3D radiation problems using the DOM. The equations for
all directions and all control volumes were solved simultaneously using a domain decomposition strategy and
parallel computing. Otherwise, it would not have been
possible to store all the data in a single processor, as
pointed out in [34]. Once an initial guess of the radiation
intensities is obtained, the iterative procedure consists of a
step of Newton's method, followed by p iterations of
GMRES, which are repeated until convergence is achieved.
A Jacobian-free Newton–Krylov GMRES method is used in
[112]. This method avoids the need to form and store a large
Jacobian matrix. Products of the Jacobian matrix by a Krylovgenerated vector are approximated by using either semiexact or numerical formulations, allowing for large memory
savings. It was concluded that a numerical approximation of
the Jacobian-vector products is preferred over the semi-exact
approximation only if the length of the Krylov space is small.
A similar method has been used in [46], where a pseudo-time
marching algorithm was employed, based on Newton's
method to relax the semi-discrete form of the governing
equations to steady-state. The implementation uses a
Jacobian-free inexact Newton method coupled with GMRES.
Again, the Jacobian is not explicitly formed, and only matrixvector products are calculated at each iteration. A combined
block incomplete lower–upper local preconditioner and an
additive Schwarz global preconditioner were used to improve
the convergence of the iterative linear solver.
131
Li et al. [113] proposed a Schur-decomposition method
for the direct solution of the discretized algebraic equations obtained using a Chebyshev collocation spectral
method for the spatial discretization, while retaining the
discrete ordinates method for the angular discretization.
Multigrid methods have been employed to improve the
convergence rate of radiative transfer problems
[34,55,114]. Murthy and Mathur [34] used an algebraic
multigrid procedure that constructs coarse level equations
by grouping a number of fine-level discrete equations. A
convergence acceleration procedure, referred to as the
coupled ordinates method, and based on an algebraic
multigrid method, was reported in [114] and applied to
solve the coupled RTE and energy equations. Balsara [55]
used the full approximation storage (FAS) multigrid in
conjunction with GMRES. The RTE was solved for 2D
problems with isotropic scattering, but the coupling with
the energy equation was not addressed. Good convergence
rate was found for both transparent and participating
media, including strongly absorbing and/or scattering
media. The convergence rate was even improved by
increasing the scattering coefficient.
The solution algorithms mentioned above may be faster
than the SM algorithm or not, depending on the problem
under consideration. In general, it is expected that
they may be more advantageous in comparison with the
SM algorithm when the coupling between the radiation
intensities in different directions becomes stronger.
A comparison of different solution algorithms is presented
in [46] for two 2D problems in a unit square enclosure
with black walls. In the first test case, all walls are cold,
while the medium is hot, emits and absorbs, but does not
scatter. In the second test case, the bottom wall is hot,
while the others are cold. A purely scattering medium,
with the highly anisotropic phase function B2 defined in
[91], is considered. Table 2 shows the influence of the
optical thickness of the medium, τ, grid size and spatial
discretization scheme on the CPU time for three different
algorithms: the SM algorithm, the Newton–Krylov GMRES
method reported in [46] (NK-GMRES) and the FAS
Table 2
Comparison of CPU times (s) for different solution algorithms and test cases as a function of grid size and optical thickness of the medium.Adapted from
[46].
Test case
Algorithm
τ ¼ 0.01
τ¼ 10.0
32 32
64 64
128 128
32 32
64 64
128 128
1
SM – step
SM – CLAM
SM – GM
NK-GMRES – step
NK-GMRES – TVD
Multigrid – step
Multigrid – TVD
0.0
0.6
0.3
1.6
1.9
12.9
48.7
0.1
3.5
1.7
11.2
12.8
69.7
272.8
0.5
31.3
14.5
78.1
90.7
495.5
1612.2
0.0
0.4
0.3
1.0
2.5
3.3
25.4
0.1
2.6
1.5
5.8
8.4
14.5
58.3
0.5
19.5
11.4
38.2
52.1
88.6
313.5
2
SM – step
SM – CLAM
SM – GM
NK-GMRES – step
NK-GMRES – TVD
Multigrid – step
Multigrid – TVD
0.0
0.7
1.0
2.6
4.8
14.7
55.8
0.3
6.0
6.2
17.4
30.8
79.0
301.1
1.2
52.5
51.2
105.4
252.5
516.5
1672.0
1.8
6.6
5.1
2.1
3.2
19.8
32.6
8.5
47.2
27.9
12.0
24.0
71.7
100.1
40.0
439.2
176.4
82.6
155.0
334.8
576.1
132
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
multigrid solver. The step, CLAM and a genuinely multidimensional (GM) schemes were used along with the SM
algorithm, while the step and a TVD scheme were used for
the other two algorithms. The DOM with the S6 quadrature
was employed to perform the calculations. The convergence criterion of the SM algorithm was a maximum
absolute change in radiation intensity between iterations
lower than 10 5. In the FAS multigrid and in the Newton–
Krylov method, a reduction of residuals by 10 orders of
magnitude was demanded. In the first test problem, the
SM algorithm is the fastest one, for all grid sizes and for
both optically thin and thick media, while the multigrid
algorithm is the most time consuming. In test case 2,
where a purely scattering medium is considered, the SM
solution algorithm is still the most computationally economical for the optically thin medium, but the Newton–
Krylov GMRES method is the fastest for the optically thick
medium. In all cases, and for a given algorithm, the step
scheme requires less computational time than the high
order schemes to achieve the same convergence criteria.
However, the results of the step scheme are far less
accurate, and in general require more computational time
than the high order schemes to attain a given accuracy.
8. Alternative formulations
8.1. Modified discrete ordinates and modified finite volume
methods
The modified discrete ordinates method, MDOM [115],
based on the modified differential approximation [116], is
aimed at the mitigation of ray effects originating from
discontinuities or abrupt changes of the wall temperature.
The radiation intensity is split into two components: the
component coming from the walls is calculated analytically, and the medium intensity component is determined
using the standard DOM. The model was originally developed for two-dimensional enclosures with black walls and
homogeneous, isotropically scattering, gray media, and
extended in [117] to three-dimensional enclosures. The
method is also applicable to nonhomogeneous and/or
anisotropic scattering media, and to gray walls.
An example of application of the MDOM taken from
[90] is presented here. It consists of a two-dimensional
square enclosure of unity side and black walls. The top
wall is hot and the others are cold. The absorption and
extinction coefficients of the medium are κ¼ 0 and
ss ¼1 m 1, and the scattering is isotropic. The incident
heat flux, q, on the bottom boundary, normalized by the
emissive power of the top boundary, is displayed in Fig. 4
for different meshes, SN quadratures and spatial discretization schemes. The quasi-exact solution [118] is taken as
reference. The DOM predictions obtained using the step
scheme and a coarse grid (15 15 control volumes) are
smooth, but do not match the reference solution, and
become slightly worst for finer angular discretizations, due
to the interaction between false scattering and ray effects
[90]. If the CLAM scheme is used and/or a finer mesh is
employed (125 125 control volumes), the solution exhibits oscillations arising from ray effects for both S8 and S16.
The predicted solution only approaches closely the quasi-
analytical one if the finest grid is used together with a very
fine angular discretization (see Fig. 4c). In contrast, the
MDOM yields an accurate solution even for the coarse grid
and S8 quadrature.
The MDOM was applied by Sakami and Charette [119]
to two-dimensional enclosures of irregular geometry containing an emitting–absorbing–scattering medium, considering both isotropic and anisotropic scattering. The
calculation of the in-scattering term for the wall component of the radiation intensity, as well as the calculation of
the heat flux incident on the walls, is more difficult in this
case, due to the irregular geometry of the enclosure. These
calculations were performed as in the zone method,
breaking the boundary of the enclosure into sub-surfaces
of uniform leaving intensity, and evaluating numerically
the integrals by means of Gaussian quadrature. An application to complex two-dimensional enclosures with obstacles is reported in [120].
Two-dimensional irregular geometries were studied by
Baek et al. [121] using similar ideas, but applying the
modified finite volume method (MFVM). They employed
the Monte Carlo method instead of the zonal method to
calculate the in-scattering term and the incident heat flux,
and the standard FVM instead of the standard DOM to
solve the governing equation for the medium radiation
intensity component. Amiri et al. [26] also used the MDOM
to solve radiation problems in complex enclosures, but
relied on the blocked-off concept to approximate inclined
or curved boundaries.
The MDOM or the MFVM, as used in the works mentioned
above, are unable to mitigate ray effects originating from
sharp gradients of the emissive power of the medium. A new
modified version that successfully mitigates ray effects originated from discontinuities or abrupt changes of both the wall
and medium emissive powers was proposed in [90]. The
method was developed for two-dimensional rectangular
enclosures with black walls, containing a homogeneous,
isotropically scattering, gray medium. The extension to nonhomogeneous media, anisotropic scattering and gray boundaries is described in [122].
8.2. Even parity formulation
The even parity formulation of the RTE was originally
introduced to solve neutron transport problems, and later
brought to the heat transfer community. It is based on the
transformation of the RTE (first-order integro-differential
equation) into a second-order integro-differential equation.
In this way, a hyperbolic type equation (RTE) is transformed
into an elliptic (or parabolic, depending on the spatial
discretization scheme) equation, i.e., an initial value problem is replaced by a boundary value problem. The former
may yield physically unrealistic negative intensities when a
diamond scheme is used or ray effects when a step scheme
is employed. The even-parity equations do not yield unrealistic intensities, since the governing differential equations
involve second-order derivatives, yielding a positive definite and self-adjoint system of equations.
The spatial discretization of the even parity equations
has been carried out using the finite volume method
[123,124] or the finite element method [125]. The even-
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
133
0.30
0.30
0.25
0.25
q 0.20
0.20
exact
15x15,step
15x15,clam
125x125,step
125x125,clam
0.15
exact
15x15,step
15x15,clam
125x125,step
125x125,clam
0.15
0.10
0.5
0.6
0.7
0.8
0.9
1
0.5
0.30
0.30
0.25
0.25
0.20
0.20
0.6
0.7
0.8
0.9
1
q
exact
15x15,step,S8
15x15,step,S16
125x125,step,S8
exact
15x15,step
125x125,step,S8
0.15
0.15
0.5
0.6
0.7
0.8
0.9
x
1
0.5
0.6
0.7
0.8
0.9
1
x
Fig. 4. Incident heat flux on the bottom boundary of a square enclosure with a top hot wall containing a purely isotropic scattering medium [90]. (a) DOM,
S8 quadrature. (b) DOM, S16 quadrature. (c) DOM, 100 polar angles (Nθ) and 100 azimuthal angles (Nϕ) per octant. (d) MDOM.
parity formulation usually requires more CPU time and more
iterations to converge, especially for optically thin media,
and the accuracy is often lower than for the standard
formulation.
8.3. Discrete ordinates interpolation method
The discrete ordinates interpolation method (DOIM) was
originally developed for the even parity formulation of the
RTE [126], and later extended to the standard formulation
of the RTE [31]. In the DOIM, the angular discretization is
performed as in the DOM, but the spatial discretization
method is different, and does not use the concept of control
volume. The radiation intensity is determined at the grid
nodes, located at the intersection of the gridlines, using the
integral form of the RTE. The radiation intensity at the
upstream location, which is the nearest point over a gridline
along the direction of propagation of radiation, is determined from interpolation of the radiation intensity at the
neighboring upstream grid nodes.
A comparison between the DOIM applied to the standard RTE and to the even-parity formulation is reported in
[127], and a comparison between the DOIM and two other
versions of the DOM for two-dimensional curved geometries may be found in [30]. An extension to twodimensional unstructured grids is presented in [128],
while extensions to three-dimensional grids are reported
in [129,130]. A major drawback of the DOIM is that it is not
conservative, i.e., it does not guarantee conservation of
energy over a control volume. Moreover, when the DOIM
is used together with a finite volume method for coupled
fluid flow and heat transfer problems, it is necessary to
provide the radiation intensity at the centers of control
volumes or cell faces, whichever are missing. An interpolation scheme to provide the missing radiation intensities is proposed in [131] and tested in one-dimensional
problems. The results obtained are accurate and free from
physically unrealistic intensities.
8.4. Pseudo time stepping and method of lines
The discrete ordinates with time stepping [132], DOTS,
and the method of lines (MOL) solution of the DOM
[133,134] transform the original boundary value problem
governed by the RTE to an initial-value problem by adding
a time derivative term to the RTE. This technique is widely
134
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
the three-point upwind scheme (DSS014) yields an oscillatory solution, which was attributed to the strong temperature gradients in a purely scattering medium. All but
the S4 quadrature gave accurate results.
Q z* (L 0 /2,L 0 /2,z)
used in CFD. A similar technique was employed in [40], but
using the transient RTE. The time is a new independent
variable that does not affect the results, since the solution
of the original boundary value problem is obtained as the
time gets sufficiently large, such that steady-state conditions are reached.
In the DOTS method [132], the time derivative of the
radiative energy flux and an artificial viscosity term are
added to the RTE. The resultant equation is integrated over
a control volume using the central differences scheme. In
Ref. [132] the integration in time was carried out using the
Euler forward-difference scheme, and a local time step
weight function was used for each control volume and
direction of propagation. In [40] the time integration was
carried out using a four-stage Runge–Kutta method.
In the MOL solution of the DOM, only the time
derivative of the radiation intensity is added to the RTE.
The spatial derivatives are calculated using a Taylor series
[133] or two- and three-point upwind differencing
schemes [134]. The resulting set of ordinary differential
equations (ODEs) is integrated until steady-state using a
publicly available solver for ODEs. This solver chooses the
time steps in a way that maintains the accuracy and
stability of the evolving solution. Selçuk and Kırbaş [133]
claim that the results are significantly better than those
computed using the finite volume method when the
medium is optically thin.
An example of application of the MOL solution of the
DOM taken from [134] is illustrated in Fig. 5. The enclosure
is cubic with side length Lo, black and cold walls, except
the bottom wall at z¼0, which is hot. It contains a pure
isotropically scattering medium with an optical thickness
of unity. The DOM calculations were carried out using a
mesh with 253 control volumes, a two-point upwind
scheme and the S8 quadrature. Monte Carlo results
of Kim and Huh [74] were taken as a reference. The
predicted dimensionless heat flux along the vertical centreline of the enclosure is shown in Fig. 5 for different
quadratures and discretization schemes. The two-point
upwind scheme (DSS012) produces good results, while
8.5. Other methods
A second-order formulation of the RTE with radiation
intensity as the dependent variable, in contrast with the
even-parity formulation, has been proposed by Zhao and
Liu [135], who used the finite element method for spatial
discretization and discrete ordinates for angular discretization. This formulation of the RTE was employed by
Hassandazeh and Raithby [136], who concluded that,
despite the accuracy and smoothness of the results
obtained, the computational cost is high, mainly due to
the elliptic nature of the governing differential equation,
and the large bandwidth and lack of diagonal dominance
of the system of discretized equations.
Several other methods for the solution of the RTE have
been developed that employ different spatial discretization schemes, while retaining the discrete ordinates angular discretization procedure. These methods, formerly
developed for computational mechanics or CFD problems
are generally referenced by the name of the spatial
discretization method employed (e.g., finite elements,
spectral elements, meshless methods), and they will not
be addressed here.
9. Parallel implementation
The paralellization of the DOM and FVM was addressed
in [137–139] using either domain decomposition or angular decomposition. The latter is more amenable to parallelization, yielding higher speedup, but the number of
processors is restricted to the number of discrete directions, and the data for the whole spatial domain must be
stored in every processor. In addition, the domain decomposition is usually employed in CFD, and therefore this
decomposition method is often preferred, even though its
1.0
1.0
0.8
0.8
X
X
X
X
X
0.6
X
0.6
X
X
X
X
0.4
0.2
X
0.4
MC
DSS012
DSS014
X
X
0.2
X
0.0
0.0
0.2
0.4
0.6
z/L0
0.8
1.0
0.0
0.0
X
MC
S4
S6
S8
S10 (LSH)
0.2
0.4
X
X
0.6
X
X
X
X
X X
X X
X
0.8
1.0
z/L0
Fig. 5. Dimensionless heat flux along the vertical centerline of a cubic enclosure containing a purely scattering medium predicted using the MOL solution
of the DOM [134]. (a) Influence of the discretization scheme. (b) Influence of the order of quadrature.
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
parallel efficiency is usually lower. Hence, recent developments have investigated improved sweeping strategies,
and the parallelization of alternative solution algorithms
or alternative formulations.
9.1. Parallel implementation of the standard algorithm
When the domain decomposition parallelization
method is used, it is not possible to maintain the convergence rate of a sequential calculation in the case of more
than four processors in 2D problems or eight processors in
3D problems if the parallelization process described in
[137–139] is used. However, a few studies have been
reported that attempt to distribute as efficiently as possible the workload among the processors in order to
increase the speedup of the calculations. Among these,
Yildiz and Bedir [140] and Chandy et al. [141] used a
wavefront calculation procedure for 2D or 3D rectangular
domains.
Yildiz and Bedir [140] assign a level number to every
control volume of the mesh and for every direction. The
level number quantifies how far a control volume is from
the upstream boundary of the domain for the direction
under consideration. In every iteration of the solution
algorithm, the calculations in a processor only start when
the radiation intensity at all upstream boundaries of the
control volumes of the lower level are available for at least
one direction. A processor remains idle when there is no
such direction. For a particular direction, the calculations
in a processor start from a control volume in the corner of
the subdomain assigned to that processor and propagate
in a wavefront, sweeping control volumes of increasing
level. A shifting procedure, analogous to pipelining in
vectorial machines, is proposed to maximize the ratio of
the time the processors are busy to the total run time. The
calculation of the level number of the control volumes and
utilization of the processors may be determined a priori
for different subdomain partitions and discrete directions.
Chandy et al. [141] developed a marching method
referred to as staged technique, which relies on a priority
queuing system in which the calculations are organized
and prioritized dynamically based on data availability.
Their study is developed for non-scattering media and
upwind finite-differencing schemes. The method is very
similar to that used in [140], but it is unclear whether the
sweeping order is identical in both cases or not. Calculations performed for a 3D problem with black boundaries
using a mesh with 128 128 128 control volumes, the S6
quadrature, and 32 processors reveal that the algorithm of
Gonçalves and Coelho [137] and Burns and Christon [139]
yields a speedup of about 2, while this staged technique
achieves a speedup of about 10. An increase of the parallel
efficiency with the problem size was found.
In the previous works, the computational domain is
divided into a number of rectangular continuous subdomains equal to the number of processors, and each subdomain is assigned to a different processor. Bailey and
Falgout [142] compare a few algorithms in which the
computational domain is divided into a number of rectangular continuous subdomains that may exceed the number
of processors, and several discontinuous subdomains may
135
be assigned to the same processor, with the same number
of subdomains per processor. The sweeping algorithms
compared in [142] are the Koch–Baker–Alcouffe algorithm
[143], a data-driven algorithm [144] and the Compton and
Clouse algorithm [145]. A theoretical analysis of the scaling
of sweep algorithms for the solution of the Boltzmann
equation has been reported in the framework of neutron
transport problems [142], but the analysis is readily
applicable to the RTE and the DOM and FVM. Although
the parallel efficiency of these sweeping algorithms may
be higher than those of the algorithms mentioned above,
which assign every subdomain to a different processor,
they are not suitable for coupled fluid flow-radiative
transfer problems, because in these problems the subdomains are generally continuous.
Poitou et al. [146] compared different parallel decompositions, namely, angular, domain, simultaneous angular
and domain, simultaneous angular and wavelength, and
simultaneous angular, wavelength and domain decompositions. They performed a coupled large eddy simulation of
a turbulent reactive flow with radiative heat transfer. They
used the DOM with the S4 quadrature, a global model with
5 spectral quadrature points for the evaluation of the
radiative properties and an unstructured mesh with 2.6
million control volumes and 25 partitioned subdomains.
The calculations were performed in a cluster that combines distributed memory in the different nodes of the
cluster, which communicate using the message passing
interface (MPI) library, and shared memory for the cores of
a node. Open multi-processing (OpenMP) was used to
implement the angular decomposition, while MPI was
employed for the wavelength and domain decompositions.
The domain decomposition strategy is similar to that used
in the past [137–139], but the radiation intensity field is
stored at the interfaces between neighboring subdomains
(virtual boundaries) and updated at the end of each
iteration. In a coupled fluid flow/radiative transfer simulation, the number of iterations required to achieve the
convergence substantially decreases from the first call of
the radiation solver to subsequent calls when the radiation
intensity field at the interfaces is stored. This allows a good
parallel efficiency to be achieved up to 25 cores with the
spatial domain decomposition strategy. In the case
of a higher number of cores, the efficiency decreases,
since the calculation time becomes too short compared
to the communication time. However, it was found that
a combination of angular, domain and wavelength decompositions is the best strategy, yielding very good efficiency
up to 1200 processors for the studied problem and for the
cluster used in the calculations.
Plimpton et al. [147] presented algorithms to improve
the parallel efficiency of the domain decomposition
method for the solution of the Boltzmann transport
equation in unstructured grids using the DOM. The algorithms are readily applicable to structured grids and to the
solution of the RTE. A basic parallel sweeping algorithm is
described that allows different processors to perform
simultaneous sweeps for different directions and spectral
bands. Two improvements of this basic algorithm are
reported, namely, a simple geometrical heuristics for
prioritizing the control volume/direction tasks each
136
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
the proposed algorithms for participating media is yet to
be demonstrated.
processor works on, and a partitioning algorithm that can
reduce the time the processors are idle waiting for another
processor's computations. Parallel efficiencies over 50% for
the basic parallel sweeping algorithm and up to 80% with
the two improvements were obtained for a mesh with
about 3 million control volumes using 2048 processors. A
similar parallel sweeping algorithm for unstructured grids
is presented in [148]. However, the ordering of the control
volumes in [147] is based on geometrical considerations,
while in [148] it is based on graph theory.
Recently, Colomer et al. [149] reported several variants
of the algorithms developed in [147,148], which exploit
different possibilities of alternating the communication
and calculation stages, as well as the effect of delaying the
communications among processors by means of buffering.
They found that one of the proposed algorithms, which
consists of completing all the solvable tasks for all ordinates before any communications take place, consistently
yields the best parallel efficiency. The calculations were
carried out using up to 2560 processors. Different geometries, meshes and quadratures were used. An example of
the results obtained using the best algorithm is illustrated
in Fig. 6 for a sphere containing a transparent medium.
Fig. 6(a) shows the results obtained using an unstructured
mesh with about 1.5 105 tetrahedra. It shows that superlinear speedup for S4 (up to 768 processors), S8 and S12 was
achieved. The higher the order of quadrature, the higher
the speedup is, due to the reduced idle time. However, the
speedup tends to stagnate when the number of processors
becomes too high and the workload per processor too
small. Fig. 6(b) shows the normalized time for the same
problem when the mesh size is varied while the ratio of
the number of unknowns to the number of processors is
fixed. The solution time increases only by a factor of about
2.2 for S12 and about 2.8 for S4 when the mesh size and the
number of processors increase by a factor of 160. These
results prove the excellent performance of the parallel
algorithm, in comparison with previous ones that
often exhibit a modest increase of speedup with the
increase of the number of processors for the domain
decomposition parallelization. However, only transparent
media have been considered, so that the performance of
9.2. Parallel implementation of other solution algorithms
The parallelization of nonstationary iterative methods
for the solution of the system of discrete algebraic equations is addressed in [150–152]. Parallel calculations using
the GMRES were reported in [57], but no details about the
parallel efficiency were given.
Liu et al. [150] applied the domain decomposition
parallelization method to unstructured meshes for 2D
and 3D problems, which were solved using the FVM. The
solver is a preconditioned conjugate gradient method. The
parallel efficiency decreased significantly with the increase
of the number of processors. In the case of a 3D problem
solved in a computer with 18 processors, using a mesh
with about 28 103 control volumes and the S4 quadrature, the parallel efficiency ranged from about 40% to
less than 20%, depending on the radiative properties of the
medium.
Krishnamoorthy et al. [151] solved the same problem as
Burns and Christon [139] using the DOM. They employed
two different solvers, namely, GMRES and BiCGSTAB [153]
with point Jacobi or block Jacobi preconditioning. The
block Jacobi was found to be more efficient than the point
Jacobi preconditioning. The parallel performance of BiCGSTAB was slightly better than that of GMRES for a small
number of processors, but little differences between the
two solvers were found for a large number of processors.
Calculations performed for a fixed number of 373 control
volumes per processor and a quadrature with 80 directions
yielded parallel efficiencies that ranged from 71% for 8
processors to 4% for 125 processors, taking as a reference
(efficiency of 100%) calculations performed using 2 processors. In the case of a fixed number of 1213 control
volumes per processor, the parallel efficiencies increased
to 91% for 8 processors and 67% for 125 processors, taking
again the solution for 2 processors as the reference. Nongray media were considered in [152] using the same
parallelization procedure.
4096
speedup
3072
2.5
normalized time
3584
3
S12
S8
S4
Linear
2560
2048
1536
S4
S8
S12
2
1.5
1024
512
128
128
1
512 768 1024
1536
number of CPUs
2048
2560
16 256 512
1024
1536
2048
2560
number of CPUs
Fig. 6. Influence of the number of processors on the speedup for a fixed mesh (a) and on the normalized computing time for a fixed number of unknowns
per processor (b) [149].
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
8
5000
1
1
400
8
4
Strong scalability
on NASA Pleiades
64 x 64 x 16 mesh
S10 quadrature
4
8
2
1
200
1
2
4
8
8
4
1
2
Ideal
1 Thread
2 Thread
4 Thread
8 Thread
4
8
1
2
8
2
4
1
1
1 1248218421842
1
Time per Newton iteration (s)
Speed up compared to 1 CPU core
600
137
4500
1
1
4000
3500
1
3000
1
2500 11
2000
1500
1
Weak scalability
on NASA Pleiades
Constant load:
64 x 64 x 16 mesh
S10 quadrature
2
2
2
2
2
2
22
44
44
44
500 8888 8
4
4
4
1
2
4
8
1 Thread
2 Thread
4 Thread
8 Thread
1000
8
8
8
8
0
500
1000
1500
2000
Total number of CPU cores
1
500
1000
1500
2000
Total number of CPU cores
Fig. 7. Influence of the number of CPU cores on the speedup for a fixed mesh (a) and on the computing time per Newton iteration for a fixed per-node-load
(b) [112].
The Jacobian-free Newton–Krylov methods were also
parallelized [46,112]. A combined memory-shared and
memory distributed computer system was used by Godoy
and Liu [112]. A speedup of about 600 was achieved using
2048 CPU cores in a 256-node 8-core/node computer
system for a non-homogeneous purely scattering 3D
problem discretized using 64 64 16 control volumes
and the S10 quadrature, as shown in Fig. 7(a). Fig. 7(b)
shows the time required per Newton iteration for a
constant per-node-load of a 64 64 16 mesh and the
S10 quadrature. The number of subdomains is equal to the
ratio of the total number of CPU cores to the number of
threads. It can be seen that the increase of the number of
threads per node yields a smaller and more linear behavior
in the increase of the computational time per Newton
iteration as the number of subdomains increases. Charest
et al. [46] reported a parallel efficiency greater than 85% on
up to 256 processors for a two-dimensional square enclosure containing an emitting–absorbing medium. The calculations were performed using a uniform mesh with
512 512 control volumes and the S6 quadrature.
9.3. Parallel implementation of alternative formulations
A parallel implementation of the DOTS formulation is
reported in [154]. The spatial and the angular discretization were carried out using the finite volume method, and
the pseudo-time discretization was performed using the
explicit Euler method. A shared-memory vector machine
with 16 processors was used. Parallel efficiencies up to 95%
were obtained for 16 processors using the spatial domain
decomposition, which largely exceeds the efficiency
obtained using the standard formulation. This is attributed
to two reasons. One is the explicit nature of the algorithm.
The calculation of the radiation intensity at a control
volume in a time step only requires data from the previous
time step, which are fully available. This means that the
convergence rate, i.e., the number of iterations required to
achieve a converged solution is the same regardless of the
number of processors. The other reason is the use of a
shared-memory machine that almost eliminates the need
for additional storage and communications overhead.
10. Transient problems
The steady-state RTE accurately describes radiative
transfer in many unsteady problems, since the characteristic time scale of radiative transport is often too small
compared with other time scales of the problem under
consideration. However, there are several cases where that
disparity of time scales does not exist, and the transient
RTE must be solved. This typically occurs in problems
involving short pulses of light with duration similar to or
smaller than the time needed for the photons to propagate
through the medium. These problems may occur in a
wide variety of areas, such as biomedical diagnosis and
treatment (e.g., optical tomography, laser–tissue interaction, laser ablation), remote sensing and laser materials
processing.
The first application of the DOM to the solution of the
transient RTE is reported in [155]. The accuracy of the P1
and P3 models, diffuse approximation, two-flux method
and DOM in the prediction of transient radiative transfer
in a one-dimensional slab was compared, and it was found
that the DOM predictions were more accurate than the
others. Subsequently, Mitra and Churnside [156] used the
DOM to analyze one-dimensional transient radiative transfer in oceanographic optical remote sensing.
Guo and Kumar [157] were the first to apply the DOM
to two-dimensional rectangular enclosures containing
absorbing, emitting, and anisotropically scattering media
subject to diffuse and/or collimated laser irradiation, and
to extend the analysis to three-dimensional problems
[158,159]. The Duhammel's superposition theorem was
used in [158] to determine the transient response of pulse
radiation. The results compared favorably with those
obtained from the solution of the transient RTE with the
time-dependent boundary condition. Guo and Kim [159],
who were concerned with radiative transfer in biological
tissues, pointed out that the air and a biological tissue have
138
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
different refractive indices, and treated the boundary at
the interface between the air and the tissue as specular
reflecting using Snell's law and the Fresnel equation. A
diffuse boundary was also considered for comparison.
They found that the effect of the boundary condition is
significant, and that reflectance and transmittance for the
specularly reflecting boundary are greater than for a
diffusely reflecting boundary.
First order spatial discretization schemes, such as the
step scheme, perform poorly. As an example, it has been
found that when radiation from a short pulse laser
propagates through a medium, the transmitted heat fluxes
emerge earlier than the minimum time required by the
radiation to leave the medium. An accurate spatial discretization scheme is needed to resolve the steep radiation
wave front originating from a temporal square pulse with
little numerical diffusion and oscillation errors.
Sakami et al. [160] used a piecewise parabolic interpolation method to achieve this goal, and report excellent
agreement with the results obtained using Monte Carlo or
an integral formulation for one-dimensional media. An
extension of this work to a two-dimensional medium
subjected to a collimated beam is reported in [161]. A
Strang-type splitting method was employed. In each time
step, the radiation intensity was calculated by first sweeping the mesh along the x-direction, fixing the y coordinate
in each sweep and neglecting the y-derivatives; and then
sweeping the mesh along the y-direction while neglecting
the x-derivatives. Sakami et al. [162] used the same
method to investigate short-pulse laser propagation
through tissues for the detection of tumors and inhomogeneities in tissues. Das et al. [163,164] and Trivedi et al.
[165] compared experimentally measured scattered optical signals originating from short pulse laser irradiation in
a tissue medium containing inhomogeneities with accurate numerical solutions of the transient RTE obtained
using the method formerly reported in [161].
Transient radiative transfer in purely scattering 3D
media is addressed in [56] using the MOL solution of the
DOM. Several discretization schemes, including first and
fourth order finite difference schemes, TVD and ENO
schemes were employed. It was concluded that the Van
Leer TVD scheme performed the best regarding the accuracy and computational efficiency. Further comparisons of
the performance of different spatial discretization schemes
for transient problems are reported in [166,167].
Boulanger and Charette [168] adapted the formulation
of Sakami and co-workers [160–162] to multi-dimensional
non-homogeneous media of arbitrary optical distribution.
They considered a Gaussian shape laser pulse and compared the results with those of the more common square
pulse. The main features of the temporal signature are
similar for both pulse shapes. The method was used in
[169,170] to solve inverse problems, namely, to determine
the optical properties inside a medium from a given set of
measurements at the boundaries, and in [171] to recover
the position of heterogeneities in one- and twodimensional turbid media, based on long-term back-scattered photons, in order to exploit the feasibility of using
direct local reflectance imaging of tissues using short pulse
lasers.
Chai and co-workers [172–174] were the first to apply
the FVM to the solution of the transient RTE for one-, twoand three-dimensional problems. A comparison of the
DOM, FVM and discrete transfer method (DTM) in the
calculation of the irradiation of a short pulse laser is
reported in [175]. In all the cases studied in [175], the
results from the three methods were found to match very
well with each other, but the DOM was found to be
computationally the most efficient. An example of these
results for a planar absorbing and scattering medium is
shown in Fig. 8. A square short pulse collimated radiation
incident on the top boundary propagates through
the medium. The figure shows the temporal variation of
the transmittance and reflectance, normalized by the
magnitude of the incident radiation, for an extinction
coefficient β¼5 m 1, a scattering albedo ω¼1, and isotropic scattering (a¼ 0), as a function of the angle θ
between the collimated incident radiation and the
normal to the boundary. The maximum values of the
0.08
0.5
DOM
DTM
FVM
0.07
θ = 0º
β = 5.0, ω = 1.0
0.05
Reflectance
Transmittance
0.06
a = 0.0
0.04
DOM
DTM
FVM
0.4
45º
0.03
β = 5.0, ω = 1.0
0.3
a = 0.0
0.2
θ = 0º
60º
0.02
45º
0.1
0.01
0
0
60º
0
10
20
30
Time
40
50
60
0
10
20
30
40
50
60
Time
Fig. 8. Influence of the solution method and angle of incidence on the normalized transmittance and reflectance from a one-dimensional absorbing and
scattering medium subject to a collimated incident short pulse laser [175].
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
transmittance and reflectance decrease with an increase in
the angle of incidence, due to the reduction of the
radiative energy that penetrates into the medium through
the top boundary. The time at which the transmittance
signal appears increases for larger incidence angles.
Muthukumaran and Mishra [176–181] studied the
interaction of a short-pulse laser train of step or Gaussian
temporal profiles in one- and two-dimensional media. The
presence of inhomogeneities in the medium was investigated by Muthukumaran and Mishra [178–181]. In all
cases, the FVM was applied using the fully implicit method
for time discretization and the diamond scheme for spatial
discretization. A comparison between the DOM and FVM
for one-dimensional planar media subjected to a shortpulse laser with either a step or a Gaussian temporal
profile is presented in Mishra et al. [182]. The temporal
evolutions of transmittance, reflectance and incident
radiation in the medium were compared for two cases,
namely, a single short-pulse laser and a multi-pulse laser.
A close agreement between the results obtained using the
two methods was found for all the studied cases.
Akamatsu and Guo [183] applied the DOM along with
Duhamel's superposition theorem, formerly used in the
case of irradiation from a single pulse [158], to investigate
ultrafast radiative transfer in a 3D non-emitting, highly
scattering medium subjected to pulse train irradiation.
This work has been extended to the study of collimated
irradiation of ultrafast square pulse trains [184]. A comparison between the Duhamel's superposition theorem and
the direct simulation of transient radiative transfer in a 3D
absorbing and scattering medium, subjected to a diffuse
square pulse train, showed that Duhamel's superposition
theorem is more efficient than the direct simulation, and
yields more accurate results [185].
The characteristics of the time-varying transmittance
and reflectance signals from a short pulse laser in onedimensional participating media were investigated in
[186]. A new non-dimensional number was proposed to
characterize those signals. Recent works by Bhowmik et al.
[187] and Marin et al. [188] investigate the temporal
139
variations of transmittance and reflectance in biological
tissues, namely, in skin and liver.
The solution of the transient RTE in cylindrical coordinates using the DOM may be found in [189]. An application
to bio-heat transfer in skin tissues irradiated by a short
pulse laser is reported in [190]. A comparison between the
DOM and FVM for cylindrical coordinates shows that the
two methods yield similar results for the considered
problems, but the FVM requires more memory and computational requirements [191].
Measurements in biological media based on a collimated radiation beam, whose intensity is modulated in
amplitude at a given frequency, have some advantages
compared to time domain measurements. The transient
RTE may be solved in the space–frequency domain, as
demonstrated by Ren et al. [192] and Elaloufi et al. [193].
The space–time and the space–frequency formulations
of the transient RTE were compared using the DOM
[194–196]. The space–frequency formulation of the transient RTE provides accurate solutions, without physically
unrealistic transmitted radiation at early time periods
[194,197]. This precision cannot be achieved with a
space–time formulation, even if high-order resolution
schemes or flux limiters are used [197]. However, the
space–frequency formulation is time consuming, due to
the large number of angular frequencies needed to correctly represent the incident pulse. Rousse [197] reported
an increase of the computational time by a factor of about
five when the frequency-based approach is used instead of
the time-domain formulation.
Fig. 9 shows an example of results reported in [194] for
a plane parallel layer constituted by an absorbing and
linearly anisotropic scattering medium with transparent
boundaries and optical thickness τL. The medium is subject
to a collimated short pulse at normal incidence. The
predicted results using the space–frequency formulation
were in close agreement with a Monte Carlo solution [198]
for anisotropy factors of 0, 0.9 and 0.9. In the case of an
optical thickness of unity, the minimum dimensionless
time (defined as βct, where β is the extinction coefficient
100
Transmittance
10-1
10-3
10-2
10-3
10-4
-4
10
10-5
0
10
20
30
40
50
t*
60
70
80
90 100
10-6
0
1
2
3
4
5
t*
Fig. 9. Comparison between the frequency-domain method and (a) a time-domain Monte Carlo formulation [198]; (b) a time-domain DOM using the Van
Leer flux limiter [194].
140
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
and c the speed of light) required by the radiation to leave
the medium is t nP ¼ 1. The results obtained using the
space–frequency formulation exactly predict that time,
while calculations performed using the space–time formulation underpredict that value, despite a TVD scheme
with the Van Leer flux limiter (CLAM scheme) being used.
The divergence of the radiative transfer equation is
needed whenever the energy equation is also solved. Most
authors have calculated this divergence in the same way
for both transient and steady-state problems. However,
Rath and Mahapatra [199] pointed out that the divergence
of the radiative heat flux as commonly used in steady-state
conditions needs to be modified for transient problems,
and derived a new formulation that is valid for both cases.
They have shown that the new formulation predicts
accurately the temperature of the medium, while the
steady-state formulation underpredicts that temperature
in the time scale where the transient radiation effect is
predominant.
11. Application to non-gray media
The literature on radiative transfer in non-gray media is
very extensive. The purpose of the present section is to
briefly survey a few works that describe how the DOM and
FVM are applied to these media, rather than attempt to
review such literature. Applications of the DOM and FVM
methods to non-gray gaseous media have generally been
carried out using global models or the correlated
k-distribution (CK) method [5,6]. A few works have used
the line-by-line method [5,6], but these are generally
restricted to one-dimensional problems, e.g., [200,201]
due to very high computational requirements of this
method. Exceptions are the works carried out by Menart
[202] and Chu et al. [203] who applied the DOM along
with the line-by-line method to two-dimensional problems. The classical narrow-band and wide-band models
[5,6] cannot be used along with the DOM and FVM for
multidimensional geometries, and their application to
one-dimensional geometries is cumbersome, as discussed
by Kim et al. [204], unless the non-correlated formulation
is employed [205]. However, the accuracy of this formulation is not good for general applications [206], and therefore it is seldom used.
The CK and the statistical narrow-band correlated-k
(SNBcK) methods have been used along with the DOM or
FVM by several authors, e.g., [207–210]. Radiation from
both participating gases and soot has been considered in
[211–213]. The CK method was also applied to wide bands
in the late nineties using several different approaches. One
of them, reported in [214], relies on a correlation in closed
form for the reordered wave number that closely approximates the four-region expression for the wide-band
absorption. This approach has been compared in [215]
with various computational implementations of the wideband model using the DOM, and further assessed in
[216,217].
Global models are much more economical than band
models, and they can easily be coupled with the DOM
and FVM. The classical weighted-sum-of gray gases model,
and the more recent and accurate spectral line-based
weighted-sum-of gray gases, absorption distribution function and full spectrum correlated-k models [5,6] have been
widely employed along with the DOM and FVM for both
academic and industrial configurations (see, e.g., [208–
210,218,219]). Some of these applications include radiation
from non-gray gases with soot and other gray particles
[24,213,220–222].
12. Application to media with variable refractive index
Some problems involve two or more semi-transparent
media with uniform but different refractive indices, e.g.,
the atmosphere and the ocean. These problems may be
handled using Snell's law and Fresnel equations at the
interface [223–225]. In other problems, the refractive
index of the medium may vary continuously along the
medium, as a result of changes in the chemical and
physical properties of the medium, e.g., the concentration
of salt in the ocean, the variation of density in planetary
atmospheres and in biological tissues. In these media,
referred to as graded index media, the photons propagate
along curved trajectories that depend on the local refractive index of the medium, and which minimize the
travel time.
Lemonnier and Le Dez [226] were the first to apply the
DOM to a graded index medium. They split the streaming
operator (derivative of the ratio of the radiation intensity
to the square of the refractive index) along the direction of
propagation of the photons into two parts, one that
accounts for spatial variations at a constant angle, and
the other that considers the angular variation at a fixed
position. An angular redistribution term, somewhat similar to that appearing in cylindrical geometries and uniform
refractive index media, is present in the RTE for these
media due to the propagation of radiation along curved
paths. The method was applied to radiative transfer in a
one-dimensional semi-transparent slab with a transverse
continuous and monotonic variation of the refractive
index. Chang and Wu [227] studied azimuthally dependent radiative transfer in an anisotropically scattering slab
with variable refractive index and oblique irradiation using
a similar formulation. An extension of this formulation to
multidimensional problems in graded index media is
described in Liu [228], who transformed the original RTE
to allow the use of the divergence theorem and the FVM.
Asllanaj and Fumeron [229] applied the FVM to twodimensional complex geometries using a slightly different
procedure, relying on finite differences, to discretize the
angular redistribution terms.
Transient problems in graded index media have also
been solved using the DOM. Wu [230] solved the transient
RTE for a planar medium subjected to pulse irradiation.
Wang et al. [231] compared the performance of the DOM,
along with a first-order discretization scheme, with the
modified DOM and the Monte Carlo method for a similar
problem. They found that the modified DOM almost
eliminates numerical diffusion, in contrast with the standard DOM, which yields some early transmitted radiation
from the slab.
P.J. Coelho / Journal of Quantitative Spectroscopy & Radiative Transfer 145 (2014) 121–146
13. Concluding remarks
Progress in the DOM and FVM for the solution of
radiative transfer problems in participating media has
been reviewed. These methods, although not as flexible
and general as the Monte Carlo method, which is considered the method of reference for radiative transfer in
participating media, have achieved a degree of maturity
that allows accurate solutions to be obtained at a moderate
computational cost for a wide range of problems. The
following remarks summarize the current status of development of the methods, point out still unsolved problems
and suggest research directions:
1. Complex geometries may be handled in the framework
of the DOM and FVM using either blocked-off regions
or body-fitted structured or unstructured grids. Powerful techniques formerly developed in CFD, such as
embedded boundaries, multi-block grids, local grid
refinement and grid adaptation further enhance the
flexibility of the methods in dealing with complex
geometries and/or problems with strong gradients.
However, the multi-block strategy for angular discretization has received little attention, while adaptivity has
not been addressed.
2. The spatial discretization may be carried out using
advanced schemes that provide accuracy comparable
to that achieved in CFD, and largely overcome the
numerical smearing. However, there is still room for
improvement in the case of unstructured grids, and in
discretization schemes with an order of accuracy
greater than two, as often used in large eddy simulation
of fluid flow problems.
3. The angular discretization, despite the availability of
many quadratures, remains a weakness of the DOM and
FVM, since the modifications proposed to mitigate ray
effects are not entirely satisfactory, due to lack of
generality or to the significant increase of complexity
and computational requirements. There seems to be at
present no general remedy for the ray effects, which
adversely influence the accuracy of the methods whenever discontinuities or sharp gradients are present in
the boundary conditions or in the temperature or
radiative properties of the medium. Other difficulties
that may appear in radiative transfer problems, such as
collimated radiation or strongly anisotropic scattering
phase functions, may be satisfactorily treated using
strategies developed to account for them.
4. The standard space-marching solution algorithm may
become too slow if the coupling between different
discrete directions is strong. Other iterative solution
algorithms of the Krylov subspace family may be
employed to solve the system of discrete equations
resulting from the DOM or FVM discretization of the
RTE, which may be preferable in that case. However,
there is limited experience in the application of these
algorithms to solve the RTE, and there is no rule to
decide whether they are faster or not than the spacemarching algorithm for a particular problem.
5. Alternative formulations of the DOM and FVM have
been proposed. Although some formulations present
6.
7.
8.
9.
141
advantages that have been identified for particular
problems, e.g., in the mitigation of ray effects, they
are not widely employed, and are often more complicated than the standard formulations.
Significant progress has been achieved in the development of parallelization algorithms. Earlier algorithms
for the domain decomposition strategy, which is commonly used in CFD, yielded a sharp decrease of the
parallel efficiency with the increase of the number of
processors. However, recent strategies are able to overcome this drawback, and provide very good speedup,
for both structured and unstructured grids. Nevertheless, they have not been widely employed, and their
performance for participating media needs to be
further investigated. The parallel efficiency may be
further improved by combining the domain decomposition with the angular decomposition parallelization
methods. Good parallel efficiencies have also been
reported for other solution algorithms. The parallelization using GPU needs to be investigated.
One of the areas where more progress has been
achieved in the past few years is the solution of
transient radiative transfer problems. It has been
shown that these problems may be effectively solved
using the DOM and FVM, relying on either the space–
time formulation or on the space–frequency formulation. However, accurate spatial discretization schemes
are needed to provide reliable results. Applications to
biological media, including layers with different radiative properties and refractive indices, complex boundary and interface conditions, and the inclusion of
inhomogeneities, has been receiving increased attention, due to their practical relevance in biomedicine.
The DOM and the FVM may be easily applied to nongray media using the line-by-line method, the correlated k-distribution method or global models. However,
the classical narrow-band and wide-band models are
not compatible with the DOM and FVM, except in
one-dimensional case and at the expense of a significant increase of complexity. The application of both
methods to graded index media has been demonstrated, and requires the inclusion of an angular redistribution term in the RTE, which may be discretized
using a procedure formerly developed to solve axisymmetrical problems.
The accuracy of the DOM and FVM has generally been
assessed by means of comparison of the results with
analytical solutions for simple benchmark problems or
with numerical solutions obtained by reference methods, namely, the Monte Carlo method. There is a need
to develop or apply tools recently developed, e.g.,
polynomial chaos, to quantify the uncertainty of the
solutions predicted by these methods when reference
solutions are not available.
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