The differential-discrete-ordinate method for solutions of the equation of radiative transfer

S. Kumar, A. Majumdar, C. L. Tien

Research output: Contribution to journalArticlepeer-review


This paper introduces a powerful but simple methodology for solving the general equation of radiative transfer for scattering and/or. absorbing one-dimensional systems. Existing methods, usually designed to handle specific boundary and energy equilibrium conditions, either provide crude estimates or involve intricate mathematical analysis coupled with numerical techniques. In contrast, the present scheme, which uses a discrete-ordinate technique to reduce the integro-differential equation to a system of ordinary differential equations, utilizes readily available software routines to solve the resulting set of coupled first-order ordinary differential equations as a two-point boundary value problem. The advantage of this approach is that the user is freed from having to understand complicated mathematical analysis and perform extensive computer programming. Additionally, the software used is state of the art, which is less prone to numerical instabilities and inaccuracies. Any degree of scattering anisotropy and albedo can be incorporated along with different conditions of energy equilibrium or specified temperature distributions and boundary conditions. Examples are presented where the radiative transfer is computed by using different quadratures such as Gaussian, Lobatto, Fiveland, Chebyshev, and Newton-Cotes. Comparison with benchmark cases shows that in a highly forward scattering medium Gaussian quadrature provides the most accurate and stable solutions.

Original languageEnglish (US)
Pages (from-to)424-429
Number of pages6
JournalJournal of Heat Transfer
Issue number2
StatePublished - May 1990


  • Numerical methods
  • Radiation
  • Radiation interactions

ASJC Scopus subject areas

  • General Materials Science
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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