The Numerical Solution Of Neutron Transport Equation And The Burgers Cheng Gezai Boltzmann Method Research

This paper consists of two parts. The first part is devoted to the study of nu- merical solutions of neutral transport equations, and the other to the investigation of Lattice Boltzmann (LB) methods for modeling Burgers equation. The developments of the transport equation and their numerical methods, and Lattice Boltzmann methods have been introduced in Chapter 1, in which main work of this paper is also described. In Chapter 2 the error estimates for some common used schemes of the transport equations with combined spatial and angular approximations are considered. The conclusions show that error order of scalar flux in all of these schemes can not get second order accurate. In Chapter 3, we consider the properties of solution of transport equation and decompose the singularities of solution near boundary and interface. Furthermore, we con- struct the second order accurate scheme and prove the existence and uniqueness of numerical solutions. Numerical solution problems of transport equation in sphere geometry are discussed in Chapter 4 which is divided into two parts too. In the first part, we research the regularities of solution near the spherical center and the singu- larities near the spherical surface. We get the coefficients of weak singular term and construct a second order accurate numerical scheme. In the second part we discuss the inner iteration schemes of discrete ordinate equations and prove the convergence of iterative sequence. In Chapter 5 we study the boundary conditions of P2 equations using singu- lar perturbation method and Case singular eigenvalue technologies. The numer- ical examples show that asymptotic boundary condition is more accurate than Marshak boundary condition. In Chapter 6 and 7, we research the mathematics theories of Lattice Boltzmann methods. LB equations fOr modeling 1D Burgers equations are constructedusing modified BGK model in Chapter 6. By discrete functionaI analysis method.we prove the convergence of solutions for LB equation. In Chapter 7, we prove thesolutions of 2D LB equations satisfy maxmum value principle and Ll constraction,and so we get the stability of numerical solutions.