Hybrid Methods Approach for Solving Problems in Transport Theory

Abstract

In this work the hybrid methods approach is introduced in order to solve some problems in Transport Theory for different geometries. The transport equation is written as: ∂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),inΩTψ(x,v,0)=φ0(x,v),in∂Ω×Vψ(x,v,t)=φ(x,v,t),in∂Ω×V×R(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. If ψ does not depend on the time, it is said that the problem (1) is a steady transport problem. Once the problem is defined, including the boundary conditions, it is disposed a set of chained methods in order to solve the problem. Between the different alternatives, an optimal scheme for the resolution is chosen. Two illustrations are given. For two-dimensional geometries it is employed a hybrid analytical and numerical method, for transport problems: conformal mapping first, then the solution in a proper geometry (rectangular for example). Each of the following two techniques is then applied, Krylov subspaces method or spectral-LTSN method. For three-dimensional problems also it is used a hybrid analytical and numerical method, for problems with more complex geometries: a homotopy between the original boundaries (piecewise surfaces) and another (a parallelepiped for example). Then each of two techniques are applied, Krylov subspaces method or nodal-LTSN method. In this case, the design of new geometries for reactors is a straightforward task. En each case, the domain consist of three regions, one of the source, other is the void region and the third one is a shield domain. The results are obtained both with an algebraic computer system and with a language of high level. An important extension is the study and treatment of transport problems for domains with irregular geometries, between them Lipschitzian domains. One remarkable fact of this work is the combination of different modeling and resolution techniques to solve some transport problems.