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Green’s Function Decomposition Method for Transport Equation

In: Integral Methods in Science and Engineering

Author

Listed:
  • F. S. Azevedo

    (Federal University of Rio Grande do Sul, Institute of Mathematics)

  • E. Sauter

    (Federal University of Rio Grande do Sul, Institute of Mathematics)

  • M. Thompson

    (Federal University of Rio Grande do Sul, Institute of Mathematics)

  • M. T. Vilhena

    (Federal University of Rio Grande do Sul, Institute of Mathematics)

Abstract

The Green’s Function Decomposition Method is a methodology to solve the trans-port equation in a slab with specular reflexion at the boundaries. Nomerical solutions face in general at least three difficulties: (1) the domain is not finite; (2) the scattering kernel is not a nonnegative function and may assume large values; (3) the reflection coefficient may not vary smoothly with the angular variable. The first difficult is overcome by truncating the domain into a finite interval taking into account some analitical estimates. The second difficulty means that well-known iterative methods will not converge easily outside the spectral radius. The third difficulty implies a large number of ordinates in case of angular discretization. The present method makes use of the Green’s Function Decomposition Method (GFD) with the following features: (1) It is not iterative. (2) It does not involve any discretization of the angular variable. In this work we present the GFD method to solve numerically the transport equation in a slab with anisotropic scattering kernel and specular reflection at the boundary. We present the original problem and solve it by reformulation as an integral operator equation. Finally, the integral operators involved are discretized yielding a finite approximation of the problem which can be solved numerically. We present numerical results for a broad range of applications.

Suggested Citation

  • F. S. Azevedo & E. Sauter & M. Thompson & M. T. Vilhena, 2013. "Green’s Function Decomposition Method for Transport Equation," Springer Books, in: Christian Constanda & Bardo E.J. Bodmann & Haroldo F. de Campos Velho (ed.), Integral Methods in Science and Engineering, edition 127, chapter 0, pages 15-39, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-7828-7_2
    DOI: 10.1007/978-1-4614-7828-7_2
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