Time-dependent Linear Transport Research Articles

Dynamics and spectrum of the linear multiple scattering operator in the Banach lattice L1(R3× R3)

Abstract In the time dependent linear transport theory two kinds of problems have been studied in the Hilbert space L2(R3 × R3): The reactor problem (with a boundary condition) and the multiple scattering problem (without a boundary condition). The aim of this paper is to give a complete description of the dynamics and the spectrum of the multiple scattering operator in the Banach lattice L1(R3 × R3), because this space is more relevant to the physics of the problem. For a wide class of such operators the spectrum is a subset of the imaginary axis.

The behaviour of neutron bursts in matter

An exact method is developed for solving the time-dependent linear transport equation for neutrons. The problem of finding the behaviour of neutron bursts in matter have been considered. The method leads to a new kind of perturbation theory applicable to the transport theoretical reactor dynamics. Applications of the theory are given for discontinuously or continuously distributed initial values of the neutron population. The boundary and initial conditions are exactly fulfilled.

Scattering theory of the linear Boltzmann operator

In time dependent scattering theory we know three important examples: the wave equation around an obstacle, the Schrodinger and the Dirac equation with a scattering potential. In this paper another example from time dependent linear transport theory is added and considered in full detail. First the linear Boltzmann operator in certain Banach spaces is rigorously defined, and then the existence of the Moller operators is proved by use of the theorem of Cook-Jauch-Kuroda, that is generalized to the case of a Banach space.

An alternating direction implicit scheme for discrete ordinate transport equations

This paper concerns a finite difference approximation of the discrete ordinate equations for the time-dependent linear transport equation posed in a multi-dimensional rectangular parallelepiped with partially reflecting walls. We present an unconditionally stable alternating direction implicit finite difference scheme, show how to solve the difference equations, and establish the following properties of the scheme. If a sequence of difference approximations is considered in which the time and space increments approach zero, then the corresponding sequence of solutions has a subsequence which converges continuously to a strong solution of the discrete ordinate equations. Provided that the time increment is sufficiently small, independently of the space and velocity increment sizes: the solution of the difference equations is bounded by an exponential function of time; in the subcritical case the coefficient of t in this exponential bound is zero or negative; and if the constituent functions are all nonnegative, then the solution of the difference equations will also be nonnegative. This last result implies a monotonicity principle for solutions of related difference problems.

Time dependent linear transport I. Existence, uniqueness, and continuous dependence

This paper concerns the time dependent linear transport equation posed in a multidimensional rectangular parallelepiped with partially reflecting walls. We consider the continuous transport equation and the discrete ordinate equations simultaneously. Our boundary condition, partial specular reflection, includes both vacuum and reflecting boundaries as special cases. We define strong and weak solutions of the problem, strong solutions being solutions in the ordinary sense and weak solutions being distributions, and show that a weak solution is a strong solution if it has space and time derivatives almost everywhere. For weak solutions we establish existence, uniqueness, and continuous dependence upon the initial data and the other functions which define the problem.