E. Sauter

Solution of the one-dimensional transport equation by the vector Green function method: Error bounds and simulation

In this work we solve the general anisotropic transport equation for an arbitrary source with semi-reflexive boundary conditions. First we present a complete existence theory for this problem in the space of continuous functions and in the space of @a-Holder continuous functions. As a result of our analysis we construct integral operators which we discretize in a finite dimensional functional space, yielding a new robust numerical method for the transport equation, which we call Greens function decomposition method (GFD). As well, we demonstrate a convergence theorem providing error bounds for the reported method. Finally we provide numerical results and applications.

Eigenvalues of the anisotropic transport equation in a slab

The critical eigenvalues of the transport equation play an important role in the description of the dynamics of transport problems both in nuclear physics as well as in radiative transport theory. This article treats the problem of calculating numerically the critical spectrum of the transport equation with semireflecting boundary conditions. The eigenvalue problem is solved using spectral methods and numerical results are presented. The scattering kernel is considered to be one of three types, namely, isotropic, linearly anisotropic, or Rayleigh scattering, even although more general kernels could be considered.

Green’s Function Decomposition Method for Transport Equation

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.

Existence theory for one-dimensional quasilinear coupled conductive–radiative flows

Abstract The paper deals with the coupled conductive–radiative problem with a diffusion coefficient depending on the temperature. The technique of upper and lower solutions is used to generate a solution for this nonlinear problem in the space of Holder continuous function by Pao (1992,2007) [1,2] together with certain integral representations given in Azevedo et al. (2011) [3] . We also produce numerical results using GFD N method, the Green Function Decomposition of the order N , coupled with the Crank–Nicolson method and the Newton–Raphson method. The GFD N methodology arises from the integral representation involved and does not involve any a priori discretization on the angular variable μ .

Green Function Formulation and Finite Element Discretization for Solving the Heat Radiative Transfer in a Slab

ABSTRACT The radiative transport problem involving the nonlinear coupling between temperature and radiative transfer poses several difficulties to be solved numerically. In this paper, we sought numerical solutions for the transient one-dimensional model by applying the Green Function Decomposition Method to the radiative transport equation and a finite element scheme to the energy balance equation. We apply this new numerical implementation to calculate the evolution of temperature in a slab of glass. The proposed scheme has been found computationally efficient and we have achieved better accuracy than the previous results from the literature.