高维中子输运方程的离散格式与并行算法研究
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摘要
针对高维中子输运方程的数值模拟，基于在应用中提出的问题和未来发展的需求，本文研究了二维离散格式的“对称性”问题，并对三维差分方程作了离散解的先验估计以及并行和加速收敛算法的设计、应用，得到了若干具有理论和实际意义的成果。
全文共分八章。
第一章为前言，介绍了粒子输运计算方法的研究背景和发展动态。国际上介绍了美、俄等国的一些研究状况，其中以美国Los Alamos国家实验室半个多世纪以来在相关算法研究和程序研制方面的代表性成果为重点；特别是结合并行计算机的发展简述了输运问题并行算法的研究发展动态；关于国内的动态，以我所为主作了简要叙述；同时还概述了本文的研究工作及其创新。
第二章介绍了中子输运方程及其离散格式。从粒子输运的基本概念和输运方程的一般形式入手，重点讨论了三维直角坐标中子输运方程和隐式差分格式，以及二维柱坐标中子输运方程和间断有限元格式。
在第三章中，运用离散泛函分析的手段，对三维中子输运方程的差分解及其时间、空间差商进行了先验估计，从而得到了差分解的稳定性和收敛性，也为三维问题的应用研究打下了初步的理论基础。
第四章深入讨论了二维柱几何非定态中子输运方程离散格式的“一维球对称性”问题，这是多年来在实际应用中存在的疑难问题。通过理论上的探索，得到了对实际应用具有指导意义的成果。
第五章研究了基于几何空间区域分解的三维非定态中子输运差分方程的并行迭代算法，该算法达到了多处理机并行的高并行度和可扩展性。同时，还给出了串、并行迭代的算法实现方式和误差估计。
为加速迭代收敛，第六章研究了三维非定态中子输运差分方程的多重网格算法，给出了细致的算法描述和算法实现步骤。在此基础上，将几何区域分解与多重网格算法相结合，提出了多重网格的区域分解并行算法，这也是本文的重要成果之一。
高维中子输运方程的离散格式与并行算法研究
第七章介绍了实现以上算法的串、并行程序编制和计算的情况，给出了多个
模型的串、并行数值计算结果以及对相关数值结果的分析和比较。
第八章为本文研究工作的主要结论。
Motivated by issues from applications and requirements in the future, this paper is focused on the numerical simulations for the multi-dimensional neutron transport equation. Firstly, the "symmetry" property of the two-dimensional discrete schemes and the prior estimates of the three-dimensional discrete solutions have been studied in theory. Then, the parallel algorithms and the convergence acceleration methods have been designed and applied. Some significant results have been achieved.
Eight chapters are included in this paper.
The background and the development of the numerical methods for particle transport are introduced in the first chapter. The research status of USA and Russia etc. are investigated, with the emphasis on the representative achievements on the methods and codes in LANL in recent 50 years. Especially, the development of the parallel algorithms for transport problems combined with that of the parallel computers is described briefly. The domestic status, mainly being in IAPCM, is introduced. Furthermore, the work and innovation of this paper are summarized.
The neutron transport equations and their discrete schemes are introduced in Chapter 2. Starting with the basic concept about particle transport and the universal form of transport equation, the discuss is concentrated upon the transport equation and the implicit difference scheme under 3-D Cartesian coordinate, as well as the transport equation and the discontinuous finite element scheme under 2-D cylindrical geometry.
In Chapter 3, the prior estimates about the 3-D difference solution and its difference quotients on time space and geometry space are made by means of discrete functional analysis, so that the stability and the convergence of the 3-D difference solution are obtained.
The "1-D sphere symmetry" property of the discrete schemes for the time-dependent neutron transport equation under 2-D cylindrical geometry, which is a
difficult issue existing in practical applications for many years, is discussed thoroughly in Chapter 4. By the theoretical analysis, some results which can guide the applications are acquired.
The domain decomposition parallel iterative algorithm for the 3-D time-dependent neutron transport difference equation is presented in Chapter 5. This algorithm is highly parallelizable and scalable. Furthermore, the implementation and the error estimates for both the serial and the parallel iteration are provided.
In order to accelerate iterative convergence, the multigrid algorithm for the 3-D time-dependent neutron transport difference equation is researched in Chapter 6. The detail description for this algorithm and the steps o f the implementation are given. Combining the domain decomposition with the multigrid algorithm, the domain decomposition multigrid parallel algorithm is posed.
In the 7th chapter, the numerical experiments concerned in the above serial and parallel algorithms are introduced, and the experimental results about various models as well as the analysis and comparison are given.
The conclusions for the research work of this paper are surveyed in the last chapter.
全文共分八章。
第一章为前言，介绍了粒子输运计算方法的研究背景和发展动态。国际上介绍了美、俄等国的一些研究状况，其中以美国Los Alamos国家实验室半个多世纪以来在相关算法研究和程序研制方面的代表性成果为重点；特别是结合并行计算机的发展简述了输运问题并行算法的研究发展动态；关于国内的动态，以我所为主作了简要叙述；同时还概述了本文的研究工作及其创新。
第二章介绍了中子输运方程及其离散格式。从粒子输运的基本概念和输运方程的一般形式入手，重点讨论了三维直角坐标中子输运方程和隐式差分格式，以及二维柱坐标中子输运方程和间断有限元格式。
在第三章中，运用离散泛函分析的手段，对三维中子输运方程的差分解及其时间、空间差商进行了先验估计，从而得到了差分解的稳定性和收敛性，也为三维问题的应用研究打下了初步的理论基础。
第四章深入讨论了二维柱几何非定态中子输运方程离散格式的“一维球对称性”问题，这是多年来在实际应用中存在的疑难问题。通过理论上的探索，得到了对实际应用具有指导意义的成果。
第五章研究了基于几何空间区域分解的三维非定态中子输运差分方程的并行迭代算法，该算法达到了多处理机并行的高并行度和可扩展性。同时，还给出了串、并行迭代的算法实现方式和误差估计。
为加速迭代收敛，第六章研究了三维非定态中子输运差分方程的多重网格算法，给出了细致的算法描述和算法实现步骤。在此基础上，将几何区域分解与多重网格算法相结合，提出了多重网格的区域分解并行算法，这也是本文的重要成果之一。
高维中子输运方程的离散格式与并行算法研究
第七章介绍了实现以上算法的串、并行程序编制和计算的情况，给出了多个
模型的串、并行数值计算结果以及对相关数值结果的分析和比较。
第八章为本文研究工作的主要结论。
Motivated by issues from applications and requirements in the future, this paper is focused on the numerical simulations for the multi-dimensional neutron transport equation. Firstly, the "symmetry" property of the two-dimensional discrete schemes and the prior estimates of the three-dimensional discrete solutions have been studied in theory. Then, the parallel algorithms and the convergence acceleration methods have been designed and applied. Some significant results have been achieved.
Eight chapters are included in this paper.
The background and the development of the numerical methods for particle transport are introduced in the first chapter. The research status of USA and Russia etc. are investigated, with the emphasis on the representative achievements on the methods and codes in LANL in recent 50 years. Especially, the development of the parallel algorithms for transport problems combined with that of the parallel computers is described briefly. The domestic status, mainly being in IAPCM, is introduced. Furthermore, the work and innovation of this paper are summarized.
The neutron transport equations and their discrete schemes are introduced in Chapter 2. Starting with the basic concept about particle transport and the universal form of transport equation, the discuss is concentrated upon the transport equation and the implicit difference scheme under 3-D Cartesian coordinate, as well as the transport equation and the discontinuous finite element scheme under 2-D cylindrical geometry.
In Chapter 3, the prior estimates about the 3-D difference solution and its difference quotients on time space and geometry space are made by means of discrete functional analysis, so that the stability and the convergence of the 3-D difference solution are obtained.
The "1-D sphere symmetry" property of the discrete schemes for the time-dependent neutron transport equation under 2-D cylindrical geometry, which is a
difficult issue existing in practical applications for many years, is discussed thoroughly in Chapter 4. By the theoretical analysis, some results which can guide the applications are acquired.
The domain decomposition parallel iterative algorithm for the 3-D time-dependent neutron transport difference equation is presented in Chapter 5. This algorithm is highly parallelizable and scalable. Furthermore, the implementation and the error estimates for both the serial and the parallel iteration are provided.
In order to accelerate iterative convergence, the multigrid algorithm for the 3-D time-dependent neutron transport difference equation is researched in Chapter 6. The detail description for this algorithm and the steps o f the implementation are given. Combining the domain decomposition with the multigrid algorithm, the domain decomposition multigrid parallel algorithm is posed.
In the 7th chapter, the numerical experiments concerned in the above serial and parallel algorithms are introduced, and the experimental results about various models as well as the analysis and comparison are given.
The conclusions for the research work of this paper are surveyed in the last chapter.
引文
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