SPATIAL DISCRETIZATION SCHEMES
Following from: Discrete ordinates and finite volume methods
In the article “Mathematical formulation,” we have seen that both the discrete ordinates method (DOM) and the finite volume method (FVM) require the evaluation of the radiation intensity at the cell faces of the control volume of the mesh. In that article, the radiation intensity at the cell faces was obtained using the step scheme, which is the counterpart of the upwind differencing scheme in computational fluid dynamics (CFD). The main advantages of the step scheme are its simplicity and stability, never yielding physically unrealistic negative radiation intensities. However, excessive numerical smearing is introduced by this scheme (see article “Ray effects and false scattering”), and it should therefore not be used if accurate solutions are sought (Leonard and Drummond, 1995). In the present article, other spatial discretization schemes are described.
Basic Spatial Discretization Schemes
The diamond scheme (Carlson and Lathrop, 1968; Fiveland, 1984) is also widely used for spatial discretization in the DOM and FVM. It is similar to the central differencing scheme in CFD. In the diamond scheme, the radiation intensities at the west and east cell faces of the control volume depicted in Fig. 1, for the direction shown (positive x component), are related to the radiation intensity at grid node P as
Figure 1. Control volume and grid nodes.
(1) |
where the coefficient γ_{x} is equal to 0.5. The lowercase subscripts identify the cell faces, while the uppercase subscripts stand for grid nodes. Similar equations may be written for other cell faces. Notice that the step scheme is recovered if γ_{x} = 1.
The diamond scheme reduces the numerical smearing, but it may yield overshoots and undershoots of the boundary intensities, and negative intensities may appear (Fiveland, 1984; Chai et al., 1994a). These negative intensities may be eliminated using the negative intensity fix-up procedure proposed by Carlson and Lathrop (1968), which sets them equal to zero. However, spatially oscillating, physically unrealistic intensities may still occur. Fiveland (1984) proposed guidelines to choose the size of the control volumes in order to maintain positive intensities and avoid unphysical oscillations. However, Chai et al. (1994a) showed that such guidelines do not necessarily ensure positive intensities, no matter the grid size.
The positive scheme (Lathrop, 1969) ensures positive radiation intensities, but not necessarily bounded ones. The positive scheme may also be expressed by Eq. (1) with γ_{x} given by
(2) |
where ξ^{m} and η^{m} are the direction cosines of s^{m}, β is the extinction coefficient and Δx and Δy the dimensions of the control volume, as shown in Fig. 1.
Negative radiation intensities may also be prevented using a variable weight scheme (Jamaluddin and Smith, 1988) that combines the step and the diamond schemes. The diamond scheme is used if negative intensities are not found. Otherwise, a weighted average of the two schemes is used according to Eq. (1), and the weight γ_{x} is selected by trial and error to enforce positive values.
Exponential Schemes
An exponential scheme was proposed by Carlson and Lathrop (1968). In this scheme, Eq. (1) still holds, and γ_{x} is given by
(3) |
A slightly modified version of this scheme is proposed in Chai et al. (1994b). This exponential scheme is potentially more accurate in 1D computations, but not in multidimensional ones, where unbounded solutions may occur, as discussed in Chai et al. (1994a).
All these spatial discretization schemes treat the radiation across a control volume face as locally 1D, i.e., the radiation intensity at a cell face is calculated from the radiation intensity at points that lie along the normal to the cell face. Other exponential schemes have been proposed to account for the multidimensional nature of radiation, i.e., schemes that calculate the radiation intensity at a cell face based on the radiation intensity at points that lie along the direction of propagation of radiation. The simplest of these schemes determines downstream cell face intensities by integrating the radiative transfer equation (RTE) from appropriate upstream locations, where the radiation intensity is evaluated from interpolation between nodes, up to those downstream locations. In this way, if ξ^{m} > 0 then the east cell face intensity will be evaluated as follows:
(4) |
where Δs is the distance from the upstream to the downstream location and ω = σ_{s} / β is the scattering albedo. The upstream location must be chosen carefully to avoid negative radiation intensities and wiggles. In this scheme, it is assumed that the source term, which includes the radiative emission and the in-scattering term of the RTE, remains constant between the upstream and the downstream locations. A similar scheme, which differs only in the evaluation of the upstream intensity, is presented in El Wakil and Sacadura (1992). A more accurate high-order formulation, which allows for the variation of the source term, is proposed in Raithby and Chui (1990). These skewed schemes are potentially more accurate than the locally 1D ones mentioned before.
Schemes Based on the Normalized Variable Diagram Formulation
The transport of radiative energy in the RTE is similar to the transport of momentum, as far as the numerical discretization is concerned. Accordingly, discretization schemes formerly developed for the discretization of the convective term of the momentum equations may be readily applied to the RTE. This was first demonstrated by Liu et al. (1996) and Jessee and Fiveland (1997), who applied to the RTE several high-order resolution schemes formulated according to the normalized variable diagram (NVD), proposed by Leonard (1988), namely, the MINMOD (Harten, 1983), CLAM (Van Leer, 1974), MUSCL (Van Leer, 1979), and SMART (Gaskell and Lau, 1988) schemes. In these schemes, the radiation intensity at a cell face f of a control volume, whose grid node is denoted by C, is related to the radiation intensities at grid node C and at the grid nodes upstream and downstream of that control volume, which are denoted by U and D, respectively, as shown in Fig. 2. If a normalized radiation intensity is defined as
Figure 2. Grid nodes notation for NVD-based schemes.
(5) |
then the function _{f}(_{C}) defines the particular scheme (the superscript m denoting the direction is dropped from now on for conciseness). This functional relationship may be plotted in a NVD, illustrated in Fig. 3. Gaskell and Lau (1988) formulated a convection boundedness criterion that prevents the appearance of overshoots or undershoots. A scheme is bounded if the following conditions are simultaneously satisfied: function _{f}(_{C}) is continuous; _{f} = _{C} if _{C} ≤ 0 or _{C} ≥ 1; and _{C} ≤ _{f} ≤ 1 if 0 < _{C} < 1. The functional relationship of bounded schemes must lie in the shaded region of Fig. 3 for 0 < _{C} < 1, and coincide with _{C} elsewhere. Moreover, all schemes that are defined by a function _{f}(_{C}) that satisfies these conditions and, in addition, satisfies _{f}(0.5) = 0.75, are at least second-order accurate (Leonard, 1988). If, in addition, the slope of the functional relationship at _{C} = 0.5 is equal to 0.75, then the scheme is third-order accurate (Leonard, 1988). The step scheme is the only linear scheme that satisfies these boundedness criteria.
Figure 3. Normalized variable diagram (NVD) and convection boundedness criteria (shaded area).
The functional relationships of several schemes that satisfy the boundedness criteria are given below for the case of uniform grids (all of them satisfy _{f} = _{C} if _{C} ≤ 0 or _{C} ≥ 1).
CLAM (Van Leer, 1974):
(6a) |
CUBISTA (Alves et al., 2003):
(6b) |
GAMMA (Jasak et al., 1999):
(6c) |
MINMOD (Harten, 1983):
(6d) |
MUSCL(Van Leer, 1979):
(6e) |
NOTABLE (Pascau et al., 1995):
(6f) |
SMART (Gaskell and Lau, 1988):
(6g) |
VONOS (Varonos and Bergeles, 1998):
(6h) |
WACEB (Song et al., 2000):
(6i) |
The NVD methodology has been extended to nonuniform grids by Darwish and Moukalled (1994), where it is referred to as normalized variable and space formulation. According to this formulation, a normalized radiation intensity is defined by Eq. (5) and a normalized coordinate is defined as
(7) |
The functional relationships given by Eq. (6) may be generalized for the case of nonuniform grids. These schemes have been applied to a few radiative heat transfer benchmark problems in Coelho (2008). None of the schemes can be considered as consistently superior to all the others, regarding the accuracy and computational requirements. The CLAM scheme, which was recommended by Jessee and Fiveland (1997), is not among the most accurate ones, but it is rather stable and relatively economical, while other good NVD schemes (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 1D. Bounded skew high-order resolution schemes were developed for CFD (Moukalled and Darwish, 1997) and applied to thermal radiation problems by Coelho (2002). The skewed schemes are more computationally demanding, particularly for fine grids. However, these are not generally required due to the accuracy of the schemes. Whether the superior accuracy of skewed schemes compensates their higher computational requirements or not is likely to be problem dependent.
Total Variation Diminishing Schemes
Another important group of high-order schemes are the total variation diminishing (TVD) schemes, e.g., Hirsch, (1990), which combine accuracy with monotonicity and entropy preservation. These 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. Several TVD schemes exist, which may be defined using a TVD diagram, somewhat similar to the NVD diagram (see Fig. 4). In TVD schemes, originally developed for hyperbolic equations, the radiation intensity at a cell face f (see Fig. 2) may be calculated from (Lien and Leschziner, 1998)
Figure 4. Total variation diminishing (TVD) diagram and second-order TVD criteria (shaded region).
(8) |
where parameter k controls the order of the scheme, ψ(r_{f}) is a flux limiter, and r_{f} is defined as
(9) |
In the case of symmetric flux limiters, which satisfy the condition ψ(r_{f}) = r_{f} ψ(1/r_{f}), Eq. (8) takes the following form:
(10) |
TVD schemes satisfy the following conditions: 0 ≤ ψ(r) ≤ 2r and 0 ≤ ψ(r) ≤ 2. Second-order accurate TVD schemes satisfy the following more stringent requirements: r ≤ ψ(r) ≤ 2r if 0 ≤ r ≤ 1, 1 ≤ ψ(r) ≤ r if 1 ≤ r ≤ 2, and 1 ≤ ψ(r) ≤ 2 if r > 2 (shaded region in Fig. 4). These conditions may be written in terms of normalized variables defined by Eq. (5) yielding more restrictive conditions than the NVD boundedness criteria. In fact, TVD schemes imply that the following conditions are satisfied in the domain [0,1]: I_{C} ≤ I_{f}, I_{f} ≤ 2I_{C}, and I_{f} ≤ 1. Hence, some schemes, such as MINMOD, CLAM, and MUSCL, satisfy both the NVD and the TVD constraints, and may be developed using both formulations. Other schemes, such as SMART, do not satisfy the TVD criteria. Flux limiters for several TVD schemes are given below.
CLAM (Van Leer, 1974):
(11a) |
KOREN (Koren, 1993):
(11b) |
MINMOD (Harten, 1983):
(11c) |
where
MUSCL (Van Leer, 1979):
(11d) |
OSHER (Chakravarthy and Osher, 1983):
(11e) |
SWEBY (Sweby, 1984):
(11f) |
SUPERBEE (Roe, 1985):
(11g) |
UMIST (Lien and Leschziner, 1988):
(11h) |
VAN ALBADA (Van Albada et al., 1982):
(11i) |
These schemes have been applied to several radiative heat transfer benchmark problems by Balsara (2001), Coelho (2008), and Godoy and Desjardin (2010). SUPERBEE yielded the most accurate results among the TVD schemes for the tests carried out in Coelho (2008), but it is computationally demanding. Godoy and Desjardin (2010) found that most limiters have approximately the same performance, and are superior to the step scheme. They further concluded that the SUPERBEE limiter is slightly more accurate for moderate pure absorbing-emitting and isotropic media, but not in the case of anisotropic media.
Essentially Non-Oscillatory Schemes
The accuracy of TVD schemes decreases in the vicinity of smooth extrema. The essentially nonoscillatory (ENO) schemes (Harten and Osher, 1987a,b) 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, such as the UNO2 (Harten and Osher, 1987a) and SONIC (Huynh, 1989) schemes, involve three upstream nodes and two downstream nodes. The UNO2 scheme was obtained by applying the ENO ideas to the MINMOD scheme. The schemes of the SONIC family are the ENO extensions of formerly developed TVD schemes, being the SONIC-A and the SONIC-B extensions of the MUSCL and SUPERBEE schemes, respectively. Using these schemes, the radiation intensity at a cell face is calculated as follows: SONIC-A (Huynh, 1989):
(12a) |
where ψ^{MM}(r) is the function ψ(r) computed using the MINMOD scheme and
Subscripts UU and DD identify the grid nodes immediately upstream of grid node U and downstream of grid node D, respectively.
SONIC-B (Huynh, 1989):
(12b) |
UNO2 (Harten and Osher, 1987a):
(12c) |
These ENO schemes were tested in Coelho (2008), and no improvement over the TVD schemes was found for the test cases reported there.
Genuinely Multidimensional Schemes
Schemes based on the NVD and TVD concepts involve stencils with grid nodes aligned along a line normal to the cell face under consideration [P, N, S, E, W, NN, SS, EE, and WW in Fig. 5(a)]. In contrast to those schemes, the genuinely multidimensional schemes involve stencils with grid nodes that do not lie along that line, but which are placed in a quadrangular arrangement around the central grid node [P, N, S, E, W, NE, NW, SE, and SW in Fig. 5(b)]. In the case of a homogeneous governing equation, i.e., if the medium neither emits nor scatters, and assuming that the direction cosines ξ and η of the direction of propagation of radiation intensity are positive (if they are not, the corresponding equations may be obtained by symmetry arguments), then the genuinely multidimensional first-order N scheme (Roe and Sidilkover, 1992) and the second-order S scheme (Sidilkover and Brandt, 1993) are given by
(a) | (b) |
Figure 5. (a) Stencils for NVD and TVD discretization schemes in 2D domains; (b) stencils for genuinely multidimensional schemes in 2D domains.
N scheme:
(13a) |
(13b) |
S scheme:
(14a) |
(14b) |
(15) |
In these equations, ψ(R) is a flux limiter, which may be any of those presented above for the TVD schemes, and the symbols (I_{e})_{NSC} and (I_{n})_{NSC} denote the radiation intensity at the east and north cell faces, respectively, calculated using the N scheme. If the medium emits and/or scatters, the RTE is not homogeneous, and the previous expressions should be replaced by the following ones in order to maintain the same order of accuracy:
N scheme:
(16a) |
(16b) |
S scheme:
(17a) |
(17b) |
(18) |
where the symbols (I_{e})_{NSC – NH} and (I_{n})_{NSC– NH} denote the radiation intensity at the east and north cell faces, respectively, calculated using the N scheme for nonhomogeneous equations, i.e., calculated using Eqs. (16a) and (16b), respectively; s_{e} and s_{n} are the source terms of the RTE (emission and in-scattering terms) integrated over a control volume, calculated at east and north cell faces, respectively, which are obtained by linear interpolation from the neighboring grid nodes; and A_{e} and A_{n} are the area of the east and north cell faces, respectively.
The S-Van Albada scheme was originally employed to solve the RTE by Balsara (2001), who found that it has several advantages over the TVD schemes, namely, minimal cross-stream dissipation and dispersion errors. Ismail and Salinas (2004) 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 (2008), who concluded that they perform particularly well for problems with discontinuities. They are rather stable, but computationally demanding. As a consequence, the N scheme is not competitive in comparison with NVD or TVD schemes, in contrast to the S schemes, especially the S-SUPERBEE scheme.
Discretization Schemes for Unstructured Grids
Most of the discretization schemes mentioned above are readily applicable to curvilinear structured grids, either orthogonal or nonorthogonal. The genuinely multidimensional schemes are an exception, since they have been designed to rectangular geometries, but there are similar schemes applicable to unstructured grids (Sidilkover, 1994; Hubbard, 2008). However, the generality of the works on radiative heat transfer in curvilinear grids has relied on the basic spatial discretization schemes or on the exponential schemes mentioned above.
There are relatively few works on the solution of radiative heat transfer problems using the DOM or FVM on unstructured grids. Most of these rely on the step scheme (Murthy and Mathur, 1998a,b; Liu et al., 2000; Kim et al., 2001, 2008) and on slightly different versions of the exponential scheme (Raithby and Chui, 1990; Sakami and Charrete, 1998; Sakami et al, 1998; Asllanaj et al., 2007). A skew upwinding scheme was reported in Rousse and Lassue (2008) and Rousse and Asllanaj (2011). The diamond scheme is not directly applicable to unstructured grids, but a modified version referred to as diamond mean flux was proposed by Ströhle et al. (2001). In this method, the radiation intensity at grid node P is given by
(19) |
where γ = 0.5 for the diamond mean flux scheme and
(20a) |
(20b) |
Joseph et al. (2005) reported a comparison of the step, diamond mean flux and exponential scheme of Sakami and Charrete (1998).
Recently, Capdevilla et al (2010) tested four different high order spatial schemes for the discretization of the RTE using the FVM. These schemes are defined as follows (see Fig. 6):
Figure 6. Notation for the spatial discretization schemes proposed by Capdevila et al. (2010) for unstructured grids.
(21a) |
(21b) |
(21c) |
(21d) |
The radiation intensity I_{f ′′} in Eq. (21c) is calculated in a way similar to I_{f ′} in Eq. (21a), while Δs in Eq. (21d) stands for the distance between f ′ and f. The gradient of the radiation intensity at the cell center was evaluated using methods described in Pérez-Segarra et al. (2006), and it was concluded that the calculation of the gradient based on the Gauss divergence theorem along with the midpoint integration rule is very stable. A disadvantage of these schemes is that they are not bounded. Unphysical negative radiation intensities were set equal to zero, following Carlson and Lathrop (1968). It was found that the scheme given by Eq. (21c) presents convergence problems.
The schemes based on the NVD, TVD, and ENO concepts have not been applied to radiative transfer problems in unstructured meshes, but they have been applied to such meshes in CFD problems (see, e.g., Jasak et al., 1999; Darwish and Moukalled, 2003; Wolf and Azevedo, 2007; Tsui and Wu, 2009). Therefore, it should be possible to apply them to radiative heat transfer problems.
Implementation of High-Order Resolution Schemes in the DOM and FVM
Any discretization scheme may be easily taken into account in the discretization procedure of the DOM or FVM described in the article “Mathematical Formulation” using the deferred correction procedure, originally suggested by Khosla and Rubin (1974), to evaluate the radiation intensity at a cell face. According to this procedure, the radiation intensity at a cell face is evaluated by adding to the radiation intensity evaluated from the step scheme a correction term that is equal to the difference between the radiation intensities calculated from the selected scheme and from the step scheme,
(22) |
Here, the superscripts U and HO denote the upwind scheme and the selected discretization scheme, respectively. Although HO actually stands for a high-order resolution scheme, such as the schemes based on the NVD, TVD, or ENO concepts, this procedure is quite general, and so it is also applicable to the basic and to the exponential schemes mentioned above. The first term on the right side of Eq. (22) is treated implicitly, i.e., the upstream radiation intensity appearing in the evaluation of (I_{f})^{U} is an unknown. In contrast, the term in square brackets on the right side of that equation is handled explicitly. This means that the radiation intensities at grid nodes that appear in the evaluation of that term are calculated using the values available from the previous iteration of the solution algorithm. The radiation intensity at the cell face will be that evaluated by the HO scheme when Eq. (22) is used and a converged solution is obtained. This procedure guarantees the diagonal dominance of the system of discretized governing equations, and therefore ensures convergence of stationary and Krylov subspace iterative methods for the solution of that system (see article “Solution algorithm” for further details).
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Roe, P. L. and Sidilkover, D., Optimum Positive Linear Schemes for Advection in Two and Three Dimensions, SIAM J. Numer. Anal., vol. 29, pp. 1542−1568, 1992.
Rousse, D. R. and Asllanaj, F., A Consistent Interpolation Function for the Solution of Radiative Transfer on Triangular Meshes. I — Comprehensive Formulation, Numer. Heat Transfer, Part B, vol. 59, no. 2, pp. 97−115, 2011.
Rousse, D. R. and Lassue, S., A Skew Upwinding Scheme for Numerical Radiative Transfer, Proc. of Int. Symp. on Advances in Computational Heat Transfer, Marrakech, Morocco, 2008.
Sakami, M. and Charette, A., A New Differencing Scheme for the Discrete Ordinates Method in Complex Geometries, Rev. Gén. Thermique, vol. 37, pp. 440−449, 1998.
Sakami, M., Charette, A., and Le Dez, V., Radiative Heat Transfer in Three-Dimensional Enclosures of Complex Geometry by Using the Discrete-Ordinates Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 59, no. 1−2, pp. 117−136, 1998.
Sidilkover, D., A Genuinely Multidimensional Upwind Scheme and an Efficient Multigrid for the Compressible Euler Equations, ICASE Report No. 94−84, 1994.
Sidilkover, D. and Brandt, A., Multigrid Solution to Steady-State Two-Dimensional Conservation Laws, SIAM J. Numer. Anal., vol. 30, pp. 249−274, 1993.
Song, B., Liu, G. R., Lam, K. Y., and Amano, R. S., On a Higher-Order Bounded Discretization Scheme, Int. J. Numer. Methods Fluids, vol. 32, no. 7, pp. 881−897, 2000.
Ströhle, J., Schnell, U., and Hein, K. R. G., A Mean Flux Discrete Ordinates Interpolation Scheme for General Coordinates, 3rd Int. Symp. on Radiative Heat Transfer, Antalya, 2001.
Sweby, P., High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws, SIAM J. Numer. Anal., vol. 21, pp. 995−1011, 1984.
Tsui, Y.-Y. and Wu, T.-C., Use of Characteristic-Based Flux Limiters in a Pressure-Based Unstructured-Grid Algorithm Incorporating High-Resolution Schemes, Numer. Heat Transfer, Part B, vol. 55, pp. 14−34, 2009.
Van Albada, G. D., Van Leer, B., and Roberts, Jr., W. W. A Comparative Study of Computational Methods in Cosmic Gas Dynamics, Astron. Astrophys., vol. 108, pp. 76−84, 1982.
Van Leer, B., Towards the Ultimate Conservation Difference Scheme. II. Monotonicity and Conservation Combined in a Second-Order Scheme, J. Comput. Phys., vol. 14, pp. 361−370, 1974.
Van Leer, B., Towards the Ultimate Conservation Difference Scheme. V. A Second-Order Sequel to Godunov´s Method, J. Comput. Phys., vol. 32, pp. 101−136, 1979.
Varonos, A. and Bergeles, G., Development and Assessment of a Variable-Order Non-Oscillatory Scheme for Convection Term Discretization, Int. J. Numer. Methods Fluids, vol. 26, no. 1, pp. 1−16, 1998.
Wolf, W. R. and Azevedo, J. L. F., High-Order ENO and WENO Schemes for Unstructured Grids, Int. J. Numer. Methods Fluids, vol. 55, no. 10, pp. 917−943, 2007.
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