中子输运方程数值解与Burgers方程格子Boltzmann方法研究
摘要
本文分两部分。第一部分研究中子输运方程的数值方法，主要讨论
输运方程离散纵标方法，积分方程方法和球谐近似方法中当前大家
关注的研究课题；第二部分针对Burgers方程研究了格子Boltzmann方
法的收敛性和稳定性问题。
全文共七章。第一章为前言，简要介绍了中子输运方程和格子
Boltzmann理论的研究概况以及本文所讨论的基本内容。第二章研究
离散纵标方法。采用将角变量与空间变量的离散组合起来的方法，
给出了常见的输运方程数值格式的误差阶。结果表明，对标量通量
而言，这些数值方法的精度均达不到二阶。在第三章，对平板几何输
运方程，我们将解的奇异部分分离出来，应用算子逼近理论构造了
二阶精度的数值格式，并证明了解的存在唯一性。第四章研究球几
何输运方程的数值解问题，也分为独立的两部分。前一部分利用调
和分析方法精细地研究了解在球心附近的正则性和在边界附近的奇
异性，将解的奇异部分分离出来，并且构造了二阶精度数值格式。后
一部分考虑离散纵标方法中的内迭代问题，给出了步函数格式内迭
代的收敛性证明。第五章应用奇异摄动理论和Case的奇异本征值理
论，给出了P_2方程的边界条件，数值结果表明我们的条件较Marshak
的边界条件更精确。第六，七章分别研究了格子Boltzmann(LB)方法对
一维和二维Burgers方程的模拟，构造了带有修正项的BGK型的LB
方程。第六章，针对一维问题的特殊性，应用离散泛函分析方法，证
明了LB方程的解在变换后收敛到Burgers方程的解。第七章，我们证
明了二维LB方程满足极值原理和L~1压缩性，从而证明了解在L~1意
义下的稳定性。章末的数值例子表明LB方法的数值精度与经典的二
阶守恒方法符合得非常好。
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 Boltz
mann methods. LB equations fOr modeling 1D Burgers equations are constructed
using 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 the
solutions of 2D LB equations satisfy maxmum value principle and Ll constraction,
and so we get the stability of numerical solutions.
输运方程离散纵标方法，积分方程方法和球谐近似方法中当前大家
关注的研究课题；第二部分针对Burgers方程研究了格子Boltzmann方
法的收敛性和稳定性问题。
全文共七章。第一章为前言，简要介绍了中子输运方程和格子
Boltzmann理论的研究概况以及本文所讨论的基本内容。第二章研究
离散纵标方法。采用将角变量与空间变量的离散组合起来的方法，
给出了常见的输运方程数值格式的误差阶。结果表明，对标量通量
而言，这些数值方法的精度均达不到二阶。在第三章，对平板几何输
运方程，我们将解的奇异部分分离出来，应用算子逼近理论构造了
二阶精度的数值格式，并证明了解的存在唯一性。第四章研究球几
何输运方程的数值解问题，也分为独立的两部分。前一部分利用调
和分析方法精细地研究了解在球心附近的正则性和在边界附近的奇
异性，将解的奇异部分分离出来，并且构造了二阶精度数值格式。后
一部分考虑离散纵标方法中的内迭代问题，给出了步函数格式内迭
代的收敛性证明。第五章应用奇异摄动理论和Case的奇异本征值理
论，给出了P_2方程的边界条件，数值结果表明我们的条件较Marshak
的边界条件更精确。第六，七章分别研究了格子Boltzmann(LB)方法对
一维和二维Burgers方程的模拟，构造了带有修正项的BGK型的LB
方程。第六章，针对一维问题的特殊性，应用离散泛函分析方法，证
明了LB方程的解在变换后收敛到Burgers方程的解。第七章，我们证
明了二维LB方程满足极值原理和L~1压缩性，从而证明了解在L~1意
义下的稳定性。章末的数值例子表明LB方法的数值精度与经典的二
阶守恒方法符合得非常好。
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 Boltz
mann methods. LB equations fOr modeling 1D Burgers equations are constructed
using 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 the
solutions of 2D LB equations satisfy maxmum value principle and Ll constraction,
and so we get the stability of numerical solutions.
引文
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