Abstract
The neutron transport equation typically has seven independent variables, however to facilitate rigorous mathematical analysis we consider the monoenergetic, steadystate equation without fission, and with isotropic interactions and isotropic source. Due to its high dimension, the equation is usually solved iteratively and we begin by considering a fundamental iterative method known as source iteration. We prove that the method converges assuming piecewise smooth material data, a result that is not present in the literature. We also improve upon known bounds on the rate of convergence assuming constant material data. We conclude by numerically verifying this new theory.
We move on to consider the use of a specific, wellknown diffusion equation to approximate the solution to the neutron transport equation. We provide a thorough presentation of its derivation (along with suitable boundary conditions) using an asymptotic expansion and matching procedure, a method originally presented by Habetler and Matkowsky in 1975. Next we state the method of diffusion synthetic acceleration (DSA) for which the diffusion approximation is instrumental. From there we move on to explore a new method of seeing the link between the diffusion and transport equations through the use of a block operator argument.
Finally we consider domain decomposition algorithms for solving the neutron transport equation. Such methods have great potential for parallelisation and for the local application of different solution methods. A motivation for this work was to build an algorithm applying DSA only to regions of the domain where it is required. We give two very different domain decomposed source iteration algorithms, and we prove the convergence of both of these algorithms. This work provides a rigorous mathematical foundation for further development and exploration in this area. We conclude with numerical results to illustrate the new convergence theory, but also solve a physicallymotivated problem using hybrid source iteration/ DSA algorithms and see significant reductions in the required computation time.
Language  English 

Qualification  Ph.D. 
Awarding Institution 

Supervisors/Advisors 

Thesis sponsors  
Award date  29 Jul 2016 
Status  Published  Mar 2016 
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Keywords
 domain decomposition
 neutron transport
 Boltzmann equation
 diffusion acceleration
 iterative methods
 nuclear
 diffusion synthetic acceleration
 numerical analysis
 Convergence of numerical methods
Cite this
Domain Decomposition Methods for Nuclear Reactor Modelling with Diffusion Acceleration. / Blake, Jack.
2016. 191 p.Research output: Thesis › Doctoral Thesis
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TY  THES
T1  Domain Decomposition Methods for Nuclear Reactor Modelling with Diffusion Acceleration
AU  Blake,Jack
PY  2016/3
Y1  2016/3
N2  In this thesis we study methods for solving the neutron transport equation (or linear Boltzmann equation). This is an integrodifferential equation that describes the behaviour of neutrons during a nuclear fission reaction. Applications of this equation include modelling behaviour within nuclear reactors and the design of shielding around xray facilities in hospitals. Improvements in existing modelling techniques are an important way to address environmental and safety concerns of nuclear reactors, and also the safety of people working with or near radiation. The neutron transport equation typically has seven independent variables, however to facilitate rigorous mathematical analysis we consider the monoenergetic, steadystate equation without fission, and with isotropic interactions and isotropic source. Due to its high dimension, the equation is usually solved iteratively and we begin by considering a fundamental iterative method known as source iteration. We prove that the method converges assuming piecewise smooth material data, a result that is not present in the literature. We also improve upon known bounds on the rate of convergence assuming constant material data. We conclude by numerically verifying this new theory. We move on to consider the use of a specific, wellknown diffusion equation to approximate the solution to the neutron transport equation. We provide a thorough presentation of its derivation (along with suitable boundary conditions) using an asymptotic expansion and matching procedure, a method originally presented by Habetler and Matkowsky in 1975. Next we state the method of diffusion synthetic acceleration (DSA) for which the diffusion approximation is instrumental. From there we move on to explore a new method of seeing the link between the diffusion and transport equations through the use of a block operator argument. Finally we consider domain decomposition algorithms for solving the neutron transport equation. Such methods have great potential for parallelisation and for the local application of different solution methods. A motivation for this work was to build an algorithm applying DSA only to regions of the domain where it is required. We give two very different domain decomposed source iteration algorithms, and we prove the convergence of both of these algorithms. This work provides a rigorous mathematical foundation for further development and exploration in this area. We conclude with numerical results to illustrate the new convergence theory, but also solve a physicallymotivated problem using hybrid source iteration/ DSA algorithms and see significant reductions in the required computation time.
AB  In this thesis we study methods for solving the neutron transport equation (or linear Boltzmann equation). This is an integrodifferential equation that describes the behaviour of neutrons during a nuclear fission reaction. Applications of this equation include modelling behaviour within nuclear reactors and the design of shielding around xray facilities in hospitals. Improvements in existing modelling techniques are an important way to address environmental and safety concerns of nuclear reactors, and also the safety of people working with or near radiation. The neutron transport equation typically has seven independent variables, however to facilitate rigorous mathematical analysis we consider the monoenergetic, steadystate equation without fission, and with isotropic interactions and isotropic source. Due to its high dimension, the equation is usually solved iteratively and we begin by considering a fundamental iterative method known as source iteration. We prove that the method converges assuming piecewise smooth material data, a result that is not present in the literature. We also improve upon known bounds on the rate of convergence assuming constant material data. We conclude by numerically verifying this new theory. We move on to consider the use of a specific, wellknown diffusion equation to approximate the solution to the neutron transport equation. We provide a thorough presentation of its derivation (along with suitable boundary conditions) using an asymptotic expansion and matching procedure, a method originally presented by Habetler and Matkowsky in 1975. Next we state the method of diffusion synthetic acceleration (DSA) for which the diffusion approximation is instrumental. From there we move on to explore a new method of seeing the link between the diffusion and transport equations through the use of a block operator argument. Finally we consider domain decomposition algorithms for solving the neutron transport equation. Such methods have great potential for parallelisation and for the local application of different solution methods. A motivation for this work was to build an algorithm applying DSA only to regions of the domain where it is required. We give two very different domain decomposed source iteration algorithms, and we prove the convergence of both of these algorithms. This work provides a rigorous mathematical foundation for further development and exploration in this area. We conclude with numerical results to illustrate the new convergence theory, but also solve a physicallymotivated problem using hybrid source iteration/ DSA algorithms and see significant reductions in the required computation time.
KW  domain decomposition
KW  neutron transport
KW  Boltzmann equation
KW  diffusion acceleration
KW  iterative methods
KW  nuclear
KW  diffusion synthetic acceleration
KW  numerical analysis
KW  Convergence of numerical methods
M3  Doctoral Thesis
ER 