The Boltzmann transport equation describes neutral and charged particle transport phenomena. Numerical solution of the Boltzmann transport equation finds application in different fields such as nuclear reactor design, radiation shielding calculations, radiative transfer in stellar atmospheres, semiconductor device design, radiation oncology, and high energy physics, to name a few. There are two classes of computational techniques that are used to solve the transport equation. In the first class, deterministic methods, the transport equation is discretized using a variety of methods and then solved directly or iteratively. Different types of discretization give rise to different deterministic methods [Lew93, Dud79], such as discrete ordinates (SN), spherical harmonics (PN), collision probabilities, nodal methods, and others. The second class of techniques, Monte Carlo methods, constructs a stochastic model in which the expected value of a certain random variable is equivalent to the value of a physical quantity to be determined [Car75, Ham64, Lux91, Rub81, Spa69]. The expected value is estimated by the average of many independent samples representing the random variable. Random numbers, following the distributions of the variable to be estimated, are used to construct these independent samples. There are two different ways to construct a stochastic model for Monte Carlo calculations. In the first case the physical process is stochastic and the Monte Carlo calculation involves a computational simulation of the real physical process. In the other case, a stochastic model is constructed artificially, such as the solution of deterministic equations by Monte Carlo.
Both deterministic and Monte Carlo methods have errors, but the source of errors is different for each method. In the treatment of deterministic computational methods the computing errors are systematic. They arise from the discretization of the time-space-angle-energy phase space and the approximate geometry. The present state of the art does not allow full representation of complicated three-dimensional geometries for deterministic transport methods. Monte Carlo methods, on the other hand, can treat continuous energy, space, and angle, and hence avoid discretization errors. The errors in Monte Carlo methods take the form of stochastic uncertainties. Estimation of the statistical uncertainty of Monte Carlo results requires understanding of properties of random variables such as expectation values, variance, and the central limit theorem. Deterministic methods are computationally fast but may sacrifice accuracy; whereas Monte Carlo methods are computationally slow yet arbitrarily accurate.
We conclude these introductory remarks on the numerical solution of the Boltzmann transport equation, by noting that the rest of this chapter is divided into following sections: in section 1.2 we explain the motivation and objective for this research work. In section 1.3 we introduce the ideas of Monte Carlo eigenvalue calculation, which are vital to Monte Carlo eigenvalue perturbation calculations. In section 1.4 different Monte Carlo perturbation methods for the calculation of reaction rates and eigenvalues are introduced. Section 1.5 introduces parallel Monte Carlo algorithms. Section 1.6, the last section of this chapter, gives an outline of the remainder of this dissertation.