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Discrete Form of the Governing Equations

The mathematical formulation of the discrete ordinates method (DOM) and finite volume method (FVM) has been presented in the article “Mathematical formulation.” In that article, it was shown that, in the case of Cartesian coordinates, and when the step scheme is employed, the discretized equations for the DOM may be written as follows for a control volume centered at grid node P:

(1)

for m = 1, 2, ... M. If the FVM is used, the corresponding discretized equations take the following form:

(2)

for m = 1, 2, ... M. In both cases, the equations may be rewritten as follows:

(3)

where the summation extends over all the neighbors of grid node P, which will be denoted as E (east), W ( ...

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References

  1. Axelsson, O., Iterative Solution Methods, New York: Cambridge University Press, 1996.
  2. Asllanaj, F., Feldheim, V., and Lybaert, P., Solution of Radiative Heat Transfer in 2-D Geometries by a Modified Finite-Volume Method Based on a Cell Vertex Scheme Using Unstructured Triangular Meshes, Numer. Heat Transfer, Part B, vol. 51, pp. 97−119, 2007.
  3. Balsara, D., Fast and Accurate Discrete Ordinates Methods for Multidimensional Radiative Transfer, Part I, Basic Methods, J. Quant. Spectrosc. Radiat. Transfer, vol. 69, pp. 671−707, 2001.
  4. Ben Salah, M., Askri, F., and Ben Nasrallah, S., Unstructured Control-Volume Finite-Element Method for Radiative Heat Transfer in a Complex 2-D Geometry, Numer. Heat Transfer, Part B, vol. 48, no. 5, pp. 477−497, 2005a.
  5. Ben Salah, M., Askri, F., Rousse, D., and Ben Nasrallah, S., Control Volume Finite Element Method for Radiation, J. Quant. Spectrosc. Radiat. Transfer, vol. 92, no. 1, pp. 9−30, 2005b.
  6. Brandt, A., Multi-Level Adaptive Solutions to Boundary Value Problems, Math. Comput., vol. 31, no. 138, pp. 333−390, 1977.
  7. Chai, J. C., Parthasarathy, G., Lee, H. S., and Patankar, S. V., Finite Volume Radiative Heat Transfer Procedure for Irregular Geometries, J. Thermophys. Heat Transfer, vol. 9, no. 3, pp. 410−415, 1995.
  8. Chui, E. H. and Raithby, G. D., Implicit solution Scheme to Improve Convergence Rate in Radiative Transfer Problems, Numer. Heat Transfer, Part B, vol. 22, pp. 251−272, 1992.
  9. Coelho, P. J., Bounded Skew High-Order Resolution Schemes for the Discrete Ordinates Method, J. Comput. Phys., vol. 175, pp. 412−437, 2002.
  10. Fiveland, W. A., Discrete-Ordinates Solutions of the Radiative Transport Equation for Rectangular Enclosures, J. Heat Transfer, vol. 106, pp. 699−706, 1984.
  11. Fiveland, W. A. and Jessee, J. P., Acceleration Schemes for the Discrete Ordinates Method, J. Thermophys. Heat Transfer, vol. 10, no. 3, pp. 445−451, 1996.
  12. Godoy, W. F. and Desjardin, P. E., On the use of Flux Limiters in the Discrete Ordinates Method for 3D Radiation Calculations in Absorbing and Scattering Media, J. Comput. Phys., vol. 229, pp. 3189−3213, 2010.
  13. Grissa, H., Askri, F., Ben Salah, M., and Ben Nasrallah, S., Three-Dimensional Radiative Transfer Modeling Using the Control Volume Finite Element Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 105, no. 3, pp. 388−404, 2007.
  14. Grissa, H., Askri, F., Ben Salah, M., and Ben Nasrallah, S., Prediction of Radiative Heat Transfer In 3D Complex Geometries Using the Unstructured Control Volume Finite Element Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 111, no. 1, pp. 144−154, 2010.
  15. Hutchinson, B. R. and Raithby, G. D., A Multigrid Method Based on the Additive Correction Strategy, Numer. Heat Transfer, vol. 9, no. 5, pp. 511−537, 1986.
  16. Joseph, D., El Hafi, M., Fournier, R., and Cuenot, B., Comparison of Three Spatial Differencing Schemes in Discrete Ordinates Method using Three-Dimensional Unstructured Meshes, Int. J. Thermal Sci., vol. 44, pp. 851−864, 2005.
  17. Koo, H.-M., Cha, H., and Song, T.-H., Convergence Characteristics of Temperature in Radiation Problems, Numer. Heat Transfer, Part B, vol. 40, pp. 303−324, 2001.
  18. Krishnaprakas, C. K., Narayana, K. B., and Dutta, P., Use of CGS Methods for the Efficient Solution of Matrix Problems Arising from the FVM Formulation of Radiative Transfer, Numer. Heat Transfer, Part B, vol. 40, pp. 515−533, 2001.
  19. Moder, J. P., Kumar, G. N., and Chai, J. C., An Unstructured-Grid Radiative Heat Transfer Module for the National Combustion code, AIAA Paper No, 2000−0453, 2000.
  20. Modest, M. F., Radiative Heat Transfer, New York: Academic Press, 2003.
  21. Murthy, J. Y. and Mathur, S. R., Finite Volume Method for Radiative Heat Transfer Using Unstructured Meshes, J. Thermophysics Heat Transfer, vol. 12, no. 3, pp. 313−321, 1998.
  22. Murthy, J. Y. and Mathur, S. R., Coupled Ordinates Method for Multigrid Acceleration of Radiation Calculations, J. Thermophysics Heat Transfer, vol. 13, no. 4, pp. 467−473, 1999.
  23. Raithby, G. D. and Chui, E. H., Accelerated Solution of the Radiation-Transfer Equation with Strong Scattering, J. Thermophysics Heat Transfer, vol. 18, no. 1, pp. 156−159, 2004.
  24. Saad, S. and Schultz, M. H., GMRES: A Generalized Minimal Residual Algorithm for solving Nonsymmetric Linear Systems, SIAM J. Sci. Stat. Comput., vol. 7, no. 3, pp. 856−869, 1986.
  25. Sonneveld, P., CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear Systems, SIAM J. Sci. Stat. Comput., vol. 10, pp. 36−52, 1989.
  26. Wesseling, P., An Introduction to Multigrid Methods, Chichester: John Wiley & Sons, 1992.
  27. Young, D. M. and Jea, K. C., Generalized Conjugate Gradient Acceleration of Non-symmetrizable Iterative Methods, Linear Algebra Appl., vol. 34, pp. 159−194, 1980.
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