The Rigorous SP3 Theory and Study on its Numerical Verification

On the premise of no mathematical approximation, the rigorous SP3 theory which is deduced from the theoretical basis of the 3rd-order spherical harmonic method and the simplified harmonic method has more definite physical meaning. Compared with the conventional SP3 theory, the interface and boundary conditions of the rigorous SP3 theory are obtained according to the continuity of angular flux, and the angular flux is also established a relationship with each order scalar flux, which provides the theoretical solution to the discontinuous factors applied to the SP3 equation. However, the interface and boundary conditions of the rigorous SP3 theory bring in the high-order partial derivative related to the transverse leakage, which makes the current nodal method difficult to solve the SP3 equation and easily leads to numerical instability. The rigorous SP3 theory is summarized and compared with the conventional SP3 theory. Combining the special modality of the rigorous SP3 equation and the current nodal method, three methods to verify the rigorous SP3 theory are proposed, each of which has advantages and disadvantages. Developing codes and analyzing the calculation process and the results, comprehension and assumptions are concluded for the implementation of the rigorous SP3 theory.

Finding the Minimun of the Quadratic Functional in Variational Approach in Transport Theory Problems

In this work it is reviewed the variational approach for some Transport Problems. Let X be a convex domain in Rn, and V a compact set. For that, it is considered the following equation: ∂ψ∂t(x,v,t)+v·∇ψ(x,v,t)+h(x,μ)ψ(x,v,t)==∫Vk(x,v,v′)ψ(x,v′,t)dv′+q(x,v,t)(1) where x represents the spatial variable in a domain D, v an element of a compact set V, Ψ is the angular flux, h(x,v) the collision frequency, k(x,v,v’) the scattering kernel function and q(x,v) the source function. It is put the attention in the construction of the quadratic functional J which appears in variational approaches for transport theory (for example, the Vladimirov functional). Some properties of this functional in a proper functional framework, in order to determine the minimum for J are considered. First, the general formulation is studied. Then an algorithm is given for minimizing the functional J for two remarkable problems: spherical harmonic method and spectral collocation method. A program associated to this algorithm is worked in a computer algebraic system, and also was depeloped a version in a high level language.