A-to-Z Guide to Thermodynamics,
Heat & Mass Transfer, and Fluids Engineering
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## TRANSIENT PROBLEMS

Following from: Discrete ordinates and finite volume methods

Many practical problems involving combined heat transfer modes are transient, e.g., the boundary conditions may be time dependent. However, the timescales characteristic of convective and diffusive transport of mass, momentum, and energy are typically much larger than the timescale characteristic of radiation propagation. Photons travel at the speed of light, which implies that the characteristic timescale of radiative transport, defined as the ratio of a characteristic length to the speed of light, is very small. As a consequence, the transient term of the radiative transfer equation (RTE) may be neglected in most cases.

Recently, however, new technologies involving short-pulsed lasers have received significant development in a variety of applications such as optical tomography, biomedical diagnosis and treatment, optical remote sensing, and laser materials processing. In these applications, the order of magnitude of the duration of the laser pulse is equal to or smaller than the time required for the radiation to propagate through the entire domain. This means that the transient term of the RTE cannot be neglected, and so the transient RTE needs to be solved. This article describes the application of the discrete ordinates method (DOM) and finite volume method (FVM) to the solution of the transient RTE.

#### Application of the DOM to Transient Problems

The first application of the DOM to the solution of the transient RTE is reported by Mitra and Kumar (1999). They compared the accuracy of the P1 and P3 models, diffuse approximation, two-flux method, and DOM in the prediction of transient radiative transfer in a 1D slab, and found that the DOM predictions were more accurate than the others. Mitra and Churnside (1999) used the DOM to analyze 1D transient radiative transfer in oceanographic optical remote sensing. An accurate spatial discretization scheme is needed to resolve the steep radiation wave front with little numerical diffusion and oscillation errors. Sakami et al. (2000) used a piecewise parabolic method to achieve this goal, and report excellent agreement with the results obtained using a Monte Carlo or an integral formulation. Sakami et al. (2002a) extended the DOM to predict transient radiative transfer in a square medium subjected to a collimated beam. Sakami et al. (2002b) used the same method to investigate short-pulse laser propagation through tissues for the detection of tumors and inhomogeneities in tissues. Das et al. (2003) compared experimentally measured scattered optical signals originated 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 Sakami et al. (2002a). Guo and Kumar (2001) were the first to apply the DOM to 2D rectangular enclosures containing absorbing, emitting, and anisotropically scattering media subject to diffuse and/or collimated laser irradiation. Guo and Kumar (2002, 2003) extended the method to 3D problems.

The transient RTE for an emitting, absorbing, and scattering medium is written as follows: (1)

where c is the speed of light and t is the time. The spectral dependence of the radiation intensity and radiative properties has been omitted. The emission term is often neglected in the case of short-pulsed lasers, since the absorbed energy in the medium is too small to raise the temperature significantly, and any emission can be neglected in comparison to the intensity of the scattered incident pulse (Guo and Kumar, 2002). In other problems, the emission of the medium is considered, but the temperature of the medium is assumed to be constant during the short timescale of the problem (e.g., Guo and Kim, 2003). However, in some applications, such as short-pulsed laser heating on metals, the temperature of the medium increases due to the irradiated energy, and the spatial and temporal evolution of the temperature are quite different from those of nonemitting or constantly emitting media (Kim et al. 2010). Accordingly, the emission term will be retained here.

In many transient problems, the pulsed radiation penetrating into the medium is parallel (collimated radiation). In this case, it is convenient to split the total radiation intensity within the medium into two components. One is the remainder of the collimated radiation after partial extinction by absorption and scattering along its path Ic, and the other is a diffuse component Id, which includes the contribution of emission from the boundaries and from the medium, and the radiation scattered away from the collimated irradiation (Modest, 2003), (2)

The collimated component satisfies the following equation (Boulanger and Charette, 2005a): (3)

Inserting Eqs. (2) and (3) into Eq. (1) yields the governing equation for the diffuse component, (4)

The collimated component is obtained from the solution of Eq. (3). If a collimated square pulse radiation penetrates into the domain along sc direction, the last term of Eq. (4) is calculated as (5)

where s = |r-rw| is the distance from the point rw on the boundary to the point r along the direction of propagation, Io is the intensity leaving the boundary toward the medium, tp is the pulse duration, and H denotes the Heaviside step function.

If the transient term of the RTE is discretized using the fully implicit method and the step scheme is used for spatial discretization, the discrete form of Eq. (1) is written as follows: (6)

where IP m, o is the radiation intensity at grid node P and previous time step for the mth direction, and Δt is the time step. This equation reverts to Eq. (10) of the article “Mathematical formulation” for steady state problems.

The step scheme is highly diffusive, as discussed in the article “Spatial discretization schemes,” and does not satisfactorily simulate sharp gradients that are typical of short-pulse lasers. Guo and Kumar (2001, 2002) and Guo and Kim (2003) used the fully implicit method and the positive scheme (see article “Spatial discretization schemes”). Duhamel’s superposition theorem was proposed to solve transient problems with time-dependent boundary conditions. The dimensionless time step, defined as cΔt/L, where L is the characteristic length, was chosen to be smaller than the dimensionless length of the sides of the control volumes (Δx/L, Δy/L, and Δz/L for Cartesian coordinates), i.e., the traveling distance between two successive time steps should not exceed the size of the control volume. The method used by Guo and co-workers is only first-order accurate, and did not adequately capture the abrupt rising of the transmitted radiation pulse, which becomes wider.

An efficient and accurate spatial discretization scheme is needed to resolve the steep radiation wave front with little numerical diffusion and oscillation errors (Sakami et al., 2002a). In the case of a high-order resolution scheme, the discrete DOM equations may be written as (7)

This equation reverts to Eq. (16) of the article “Solution algorithm” for steady state problems. The subscript HO stands for any spatial discretization scheme other than the step scheme.

Sakami et al. (2000, 2002a,b) and Das et al. (2003) used a piecewise parabolic scheme along with a Strang-type splitting method. In the first half of a time step, the radiation intensities in the 2D rectangular domain are first updated along the x direction, row by row, setting to zero the y derivatives, and then updated along the y direction, column by column, setting to zero the x derivatives. In the second half of the time step, the order of the directional sweeps is reversed. The results obtained using this method are quite satisfactory. An incident pulse was found to travel and emerge on the other side of a 1D slab without any spurious diffusion.

Boulanger and Charette (2005a) adapted the formulation of Sakami and co-workers to multidimensional nonhomogeneous media of arbitrary optical distribution. The method was used in Boulanger and Charette (2005b,c) to solve inverse problems, namely, to determine the optical properties inside a medium from a given set of measurements at the boundaries.

Other spatial discretization schemes have been employed by Ayranci and Selçuk (2004). They compared several schemes in the method of lines (see article “Alternative formulations”) solution of the DOM equations for transient problems, namely, standard finite difference formulations derived from Taylor series expansion (in particular, a first-order two-point upwind scheme and a fourth-order five-point biased upwind scheme); TVD schemes with the CLAM and SUPERBEE flux limiters (see article “Spatial discretization schemes”); and third- and fifth-order weighted essentially nonoscillatory (WENO) schemes and monoticity-preserving weighted essentially nonoscillatory (MPWENO) schemes, which are variants of the essentially nonoscillatory (ENO) schemes. They concluded that the CLAM scheme performs the best from the viewpoints of accuracy and computational efficiency. The application of the DOM to the solution of the transient RTE in cylindrical coordinates may be found in Kim and Guo (2004).

#### Application of the FVM to Transient Problems

Chai (2003, 2004) and Chai et al. (2004) were the first to apply the FVM to the solution of the transient RTE for 1D, 2D, and 3D problems. If the transient term of the RTE is discretized using the fully implicit method, and the step scheme is used for spatial discretization, as in Ruan et al. (2010), the discrete form of Eq. (1) is written as follows: (8)

This equation reverts to Eq. (25) of the article “Mathematical formulation” for steady state problems. If a high-order resolution scheme is used, the discrete FVM equations may be written as (9)

This equation reverts to Eq. (17) of the article “Solution algorithm” for steady state problems. Chai and co-workers compared the performance of the step and CLAM schemes, and found that the latter is far more accurate. Kim et al. (2010) compared the step, diamond, second-order upwind, QUICK, and CLAM schemes applied to the solution of radiative transfer in a medium in radiative equilibrium exposed to diffuse or continuous collimated irradiation. They concluded that it is necessary to adopt a higher-order convection scheme to more accurately capture the discontinuity at the wave front, especially when the medium is exposed to collimated irradiation. Rousse (2007a) employed the CLAM and SUPERBEE schemes, and found that there is a low transmitted flux before the minimal physical time required by the radiation to leave the medium, even with high-order schemes. A description of the spatial discretization schemes mentioned here may be found in the article “Spatial discretization schemes.”

A comparison of the DOM, FVM, and discrete transfer method in the calculation of the irradiation of a short-pulse laser is reported in Mishra et al. (2006). In all the studied cases, 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.

Muthukumaran and Mishra (2008a,b,c,d) studied the interaction of a short-pulse laser train of step or Gaussian temporal profiles in 1D and 2D media. The presence of inhomogeneities in the medium was investigated in Muthukumaran and Mishra (2008e,f,g,h). In all cases, the FVM was applied using the fully implicit method for time discretization, and the diamond scheme for spatial discretization.

#### Solution of Transient Problems in the Space-Frequency Domain

Measurements in biomedicine based on a collimated radiation beam, whose intensity is modulated in amplitude at a given frequency, have some advantages compared to time domain measurements. Moreover, the numerical approach described above, based on the space-time formulation, may induce transmitted fluxes that emerge earlier than the minimal time required by the radiation to leave the medium.

The transient RTE may be solved in the space-frequency domain, as demonstrated by Elaloufi et al. (2002). To accomplish this goal, all temporal variables are transformed into frequency-dependent variables by introducing the temporal Fourier transform of the radiation intensity, (10)

where ω is the angular frequency, and i is the imaginary unit. Applying the Fourier transform to the transient RTE [Eq. (1)] and neglecting the emission term yields (Balima et al., 2010) (11)

The radiation intensity may be split into a diffuse and a collimated component, according to Eq. (2). The collimated component satisfies the following equation: (12)

and the diffuse component is obtained from (13)

where the solution of Eq. (12) is used as a source term for the diffuse component. If a collimated square pulse radiation penetrates into the domain along the sc direction, the last term of this equation is calculated as (14)

where Îo(ω) is obtained from a temporal Fourier analysis of the incident pulse. Hence, the solution of the transient RTE is obtained by solving the steady state Eq. (13) for each frequency contained in the temporal Fourier decomposition of the pulse.

The standard DOM or the FVM may be used to solve the complex RTE [Eq. (13)], e.g., Francoeur and Rousse (2007). First, the temporal Fourier transform of the incident pulse is calculated, and the frequencies for which Eq. (13) is solved are determined. Then, Eq. (13) is solved for every discrete frequency using the standard DOM or FVM. Finally, the time-dependent radiation intensities are recovered by applying an inverse Fourier transform to the frequency-dependent intensities.

The space-time and the space-frequency formulations of the transient RTE were compared by Francoeur et al. (2005), Francoeur and Rousse (2007), and Rousse (2008) using the DOM. The space-frequency formulation of the transient RTE provides accurate solutions, without physically unrealistic transmitted radiation at early time periods (Francoeur et al., 2005; Rousse, 2007b). This precision cannot be achieved with a space-time formulation, even if high-order resolution schemes or flux limiters are used (Rousse, 2007b). However, the space-frequency formulation is time consuming, due to the large number of angular frequencies needed to correctly represent the incident pulse. Rousse (2007b) reported an increase of the computational time by a factor of approximately five when the frequency-based approach is used instead of the time-domain formulation.

#### Alternative Formulations

Transient problems have also been solved using alternative space-time formulations of the DOM for the spatial discretization (see article “Alternative formulations”), namely, the method of lines solution of the DOM equations (Ayranci and Selçuk, 2004), the least-squares finite element method (FEM) (An et al., 2006, 2007), the discontinuous FEM (Liu and Liu, 2007), the discontinuous spectral element method (Zhao and Liu, 2008), and different implementations of the meshless method (Tan et al., 2006; Kindelan et al., 2010). The space-frequency formulation of the RTE was solved by Balima et al. (2010, 2011a,b) using the least-squares finite element formulation for spatial discretization, and the DOM for angular discretization.

Transient problems in graded index planar media have been solved by Wu (2006) using the DOM, and Wang and Wu (2010) using a modified DOM. Liu and Hsu (2007) solved transient problems in 1D and 2D graded index media using the space-frequency formulation, the discontinuous FEM for spatial discretization, and the DOM for angular discretization.

#### REFERENCES

An, W., Ruan, L. M., Tan, H. P., Qi., H., and Lew, Y. M., Finite Element Simulation for Short Pulse Light Radiative Transfer in Homogeneous and Nonhomogeneous Media, J. Heat Transfer, vol. 129, pp. 353−362, 2007.

An, W., Ruan, L. M., Tan, H. P., and Qi., H., Least-Squares Finite Element Analysis for Transient Radiative Transfer in Absorbing and Scattering Media, J. Heat Transfer, vol. 128, pp. 499−503, 2006.

Ayranci, I. and Selçuk, N., MOL Solution of DOM for Transient Radiative Transfer in 3-D Scattering Media, J. Quant. Spectrosc. Radiat. Transfer, vol. 84, pp. 409−422, 2004.

Balima, O., Boulanger, J., Charette, A., and Marceau, D., New Developments in Frequency Domain Optical Tomography. Part I: Forward Model and Gradient Computation, J. Quant. Spectrosc. Radiat. Transfer, vol. 112, pp. 1229−1234, 2011a.

Balima, O., Boulanger, J., Charette, A., and Marceau, D., New Developments in Frequency Domain Optical Tomography. Application with a L-BFGS Associated to an Inexact Line Search, J. Quant. Spectrosc. Radiat. Transfer, vol. 112, pp. 1235−1240, 2011b.

Balima, O., Pierre, T., Charette, A., and Marceau, D., A Least Square Finite Element Formulation of the Collimated Irradiation in Frequency Domain for Optical Tomography Applications, J. Quant. Spectrosc. Radiat. Transfer, vol. 111, pp. 280−286, 2010.

Boulanger, J. and Charette, A., Numerical Developments for Short-Pulsed Near Infra-Red Laser Spectroscopy. Part I: Direct Treatment, J. Quant. Spectrosc. Radiat. Transfer, vol. 91, pp. 189−209, 2005a.

Boulanger, J. and Charette, A., Numerical Developments for Short-Pulsed Near Infra-Red Laser Spectroscopy. Part I: Inverse Treatment, J. Quant. Spectrosc. Radiat. Transfer, vol. 91, pp. 297−318, 2005b.

Boulanger, J. and Charette, A., Reconstruction Optical Spectroscopy Using Transient Radiative Transfer Equation and Pulsed Laser: a Numerical Study, J. Quant. Spectrosc. Radiat. Transfer, vol. 93, pp. 325−336, 2005c.

Chai, J. C., One-Dimensional Transient Radiation Heat Transfer Modeling using a Finite-Volume Method, Numer. Heat Transfer Part B, vol. 44, pp. 187−208, 2003.

Chai, J. C., Transient Radiative Transfer in Irregular Two-dimensional Geometries, J. Quant. Spectrosc. Radiat. Transfer, vol. 84, pp. 281−294, 2004.

Chai, J. C., Hsu, P-F. and Lam, Y. C., Three-Dimensional Transient Radiative Transfer Modeling using the Finite-Volume Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 86, pp. 299−313, 2004.

Das, C., Trivedi, A., Mitra, K., and Vo-Dinh, T., Short Pulse Laser Propagation Through Tissues for Biomedical Imaging, J. Phys. D, vol. 36, pp. 1−8, 2003.

Elaloufi, R., Carminati, R., and Greffet, J. J., Time-Dependent Transport Through Scattering Media: from Radiative Transfer to Diffusion, J. Opt. A, vol. 4, pp. S103−S108, 2002.

Francoeur, M. and Rousse, D. R., Short-Pulsed Laser Transport in Absorbing and Scattering Media: Time-Based Versus Frequency-Based Approaches, J. Phys. D, vol. 40, no. 18, pp. 5733−5742, 2007.

Francoeur, M., Vaillon, R., and Rousse, D. R., Theoretical Analysis of Frequency and Time-Domain Methods for Optical Characterization of Absorbing and Scattering Media, J. Quant. Spectrosc. Radiat. Transfer, vol. 93, pp. 139−150, 2005.

Guo, Z. and Kim, K., Ultrafast-Laser-Radiation Transfer in Heterogeneous Tissues with the Discrete-Ordinates Method, Appl. Opt., vol. 42, no. 16, pp. 2897−2905, 2003.

Guo, Z. and Kumar, S., Discrete-Ordinates Solution of Short-Pulsed Laser Transport in Two-Dimensional Turbid Media, Appl. Opt., vol. 40, no. 19, pp. 3156−3163, 2001.

Guo, Z. and Kumar, S., Three-Dimensional Discrete Ordinates Method in Transient Radiative Transfer, J. Thermophys. Heat Transfer, vol. 16, no. 3, pp. 289−296, 2002.

Kim, K. and Guo, Z., Ultrafast Radiation Heat Transfer in Laser Tissue Welding and Soldering, Numer. Heat Transfer, Part A, vol. 46, pp. 23−40, 2004.

Kim, M. Y., Menon, S., and Baek, S. W., On the Transient Radiative Transfer in a One-Dimensional Planar Medium Subjected To Radiative Equilibrium, Int. J. Heat Mass Transfer, vol. 53, pp. 5682−5691, 2010.

Kindelan, M., Bernal, F., González-Rodríguez, P., and Moscoso, M., Application of the RBF Meshless Method to the Solution of the Radiative Transport Equation, J. Comput. Phys., vol. 229, no. 5, pp. 1897−1908, 2010.

Liu, L. H. and Liu, L. J., Discontinuous Finite Element Approach for Transient Radiative Transfer Equation, J. Heat Transfer, vol. 129, pp. 1069−1074, 2007.

Liu, L. H. and Hsu, P.-f., Analysis of Transient Radiative Transfer in Semitransparent Graded Index Medium, J. Quant. Spectrosc. Radiat. Transfer, vol. 105, no. 3, pp. 357−376, 2007.

Mishra, S. C., Chugh, P., Kumar, P., and Mitra, K., Development and Comparison of the DTM, the DOM and the FVM Formulations for the Short-Pulse Laser Transport Through a Participating Medium, Int. J. Heat Mass Transfer, vol. 49, pp. 1820−1832, 2006.

Mitra, K. and Kumar, S., Development and Comparison of Models for Light-Pulse Transport Through Scattering-Absorbing Media, Appl. Opt., vol. 38, no. 1, pp. 188−196, 1999.

Mitra, K. and Churnside, J. H., Transient Radiative Transfer Equation Applied to Oceanographic Lidar, Appl. Opt., vol. 38, no. 6, pp. 889−895, 1999.

Muthukumaran, R. and Mishra, S. C., Effect of a Step Short-Pulse Laser Train on an Inhomogeneous Planar Participating Medium, Int. Commun. Heat Mass Transfer, vol. 35, pp. 1073−1078, 2008a.

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#### References

1. An, W., Ruan, L. M., Tan, H. P., Qi., H., and Lew, Y. M., Finite Element Simulation for Short Pulse Light Radiative Transfer in Homogeneous and Nonhomogeneous Media, J. Heat Transfer, vol. 129, pp. 353−362, 2007.
2. An, W., Ruan, L. M., Tan, H. P., and Qi., H., Least-Squares Finite Element Analysis for Transient Radiative Transfer in Absorbing and Scattering Media, J. Heat Transfer, vol. 128, pp. 499−503, 2006.
3. Ayranci, I. and Selçuk, N., MOL Solution of DOM for Transient Radiative Transfer in 3-D Scattering Media, J. Quant. Spectrosc. Radiat. Transfer, vol. 84, pp. 409−422, 2004.
4. Balima, O., Boulanger, J., Charette, A., and Marceau, D., New Developments in Frequency Domain Optical Tomography. Part I: Forward Model and Gradient Computation, J. Quant. Spectrosc. Radiat. Transfer, vol. 112, pp. 1229−1234, 2011a.
5. Balima, O., Boulanger, J., Charette, A., and Marceau, D., New Developments in Frequency Domain Optical Tomography. Application with a L-BFGS Associated to an Inexact Line Search, J. Quant. Spectrosc. Radiat. Transfer, vol. 112, pp. 1235−1240, 2011b.
6. Balima, O., Pierre, T., Charette, A., and Marceau, D., A Least Square Finite Element Formulation of the Collimated Irradiation in Frequency Domain for Optical Tomography Applications, J. Quant. Spectrosc. Radiat. Transfer, vol. 111, pp. 280−286, 2010.
7. Boulanger, J. and Charette, A., Numerical Developments for Short-Pulsed Near Infra-Red Laser Spectroscopy. Part I: Direct Treatment, J. Quant. Spectrosc. Radiat. Transfer, vol. 91, pp. 189−209, 2005a.
8. Boulanger, J. and Charette, A., Numerical Developments for Short-Pulsed Near Infra-Red Laser Spectroscopy. Part I: Inverse Treatment, J. Quant. Spectrosc. Radiat. Transfer, vol. 91, pp. 297−318, 2005b.
9. Boulanger, J. and Charette, A., Reconstruction Optical Spectroscopy Using Transient Radiative Transfer Equation and Pulsed Laser: a Numerical Study, J. Quant. Spectrosc. Radiat. Transfer, vol. 93, pp. 325−336, 2005c.
10. Chai, J. C., One-Dimensional Transient Radiation Heat Transfer Modeling using a Finite-Volume Method, Numer. Heat Transfer Part B, vol. 44, pp. 187−208, 2003.
11. Chai, J. C., Transient Radiative Transfer in Irregular Two-dimensional Geometries, J. Quant. Spectrosc. Radiat. Transfer, vol. 84, pp. 281−294, 2004.
12. Chai, J. C., Hsu, P-F. and Lam, Y. C., Three-Dimensional Transient Radiative Transfer Modeling using the Finite-Volume Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 86, pp. 299−313, 2004.
13. Das, C., Trivedi, A., Mitra, K., and Vo-Dinh, T., Short Pulse Laser Propagation Through Tissues for Biomedical Imaging, J. Phys. D, vol. 36, pp. 1−8, 2003.
14. Elaloufi, R., Carminati, R., and Greffet, J. J., Time-Dependent Transport Through Scattering Media: from Radiative Transfer to Diffusion, J. Opt. A, vol. 4, pp. S103−S108, 2002.
15. Francoeur, M. and Rousse, D. R., Short-Pulsed Laser Transport in Absorbing and Scattering Media: Time-Based Versus Frequency-Based Approaches, J. Phys. D, vol. 40, no. 18, pp. 5733−5742, 2007.
16. Francoeur, M., Vaillon, R., and Rousse, D. R., Theoretical Analysis of Frequency and Time-Domain Methods for Optical Characterization of Absorbing and Scattering Media, J. Quant. Spectrosc. Radiat. Transfer, vol. 93, pp. 139−150, 2005.
17. Guo, Z. and Kim, K., Ultrafast-Laser-Radiation Transfer in Heterogeneous Tissues with the Discrete-Ordinates Method, Appl. Opt., vol. 42, no. 16, pp. 2897−2905, 2003.
18. Guo, Z. and Kumar, S., Discrete-Ordinates Solution of Short-Pulsed Laser Transport in Two-Dimensional Turbid Media, Appl. Opt., vol. 40, no. 19, pp. 3156−3163, 2001.
19. Guo, Z. and Kumar, S., Three-Dimensional Discrete Ordinates Method in Transient Radiative Transfer, J. Thermophys. Heat Transfer, vol. 16, no. 3, pp. 289−296, 2002.
20. Kim, K. and Guo, Z., Ultrafast Radiation Heat Transfer in Laser Tissue Welding and Soldering, Numer. Heat Transfer, Part A, vol. 46, pp. 23−40, 2004.
21. Kim, M. Y., Menon, S., and Baek, S. W., On the Transient Radiative Transfer in a One-Dimensional Planar Medium Subjected To Radiative Equilibrium, Int. J. Heat Mass Transfer, vol. 53, pp. 5682−5691, 2010.
22. Kindelan, M., Bernal, F., GonzÃ¡lez-RodrÃ­guez, P., and Moscoso, M., Application of the RBF Meshless Method to the Solution of the Radiative Transport Equation, J. Comput. Phys., vol. 229, no. 5, pp. 1897−1908, 2010.
23. Liu, L. H. and Liu, L. J., Discontinuous Finite Element Approach for Transient Radiative Transfer Equation, J. Heat Transfer, vol. 129, pp. 1069−1074, 2007.
24. Liu, L. H. and Hsu, P.-f., Analysis of Transient Radiative Transfer in Semitransparent Graded Index Medium, J. Quant. Spectrosc. Radiat. Transfer, vol. 105, no. 3, pp. 357−376, 2007.
25. Mishra, S. C., Chugh, P., Kumar, P., and Mitra, K., Development and Comparison of the DTM, the DOM and the FVM Formulations for the Short-Pulse Laser Transport Through a Participating Medium, Int. J. Heat Mass Transfer, vol. 49, pp. 1820−1832, 2006.
26. Mitra, K. and Kumar, S., Development and Comparison of Models for Light-Pulse Transport Through Scattering-Absorbing Media, Appl. Opt., vol. 38, no. 1, pp. 188−196, 1999.
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