Abstract
In this thesis we studied approximation methods for the radiative transfer equation, which has numerous important applications, see Chapter 1. For most of these applications the radiative transfer equation cannot be solved analytically and a wide variety of numerical methods has been developed.
As an important introductory step, we gave an overview of classical semidiscretizations in the angular component. The two most frequently used discretizations  the discrete ordinates and spherical harmonics methods are summarised in Chapter 1. While the spherical harmonics discretization allows to turn the radiative transfer equation into a system of linear equations with tridiagonal structure, approximating the boundary conditions effectively requires extra steps. Furthermore, since the spherical harmonics expansion is a global approximation method, it is not suited for approximating nonsmooth or discontinuous solutions, unlike the discrete ordinates method, which is a local approximation in the angle. The discrete ordinates method allows to obtain a consistent discretization of both the radiative transfer equation and the boundary conditions, though yielding dense scattering matrix.
A number of iterative techniques has been developed to tackle the difficulty that arises from the dense scattering matrix. A summary of some important methods was given in Chapter 2. We discussed two closely related methods  the first collision source method and the standard source iteration method, which is often accompanied by further preconditioning techniques, such as the diffusion synthetic acceleration technique.
In the first collision source method the radiative transfer boundary value problem is split into two equations for the uncollided and collided components, which can be separately approximated by different numerical methods. In general this can, however, introduce consistency errors, which are difficult to analyse. We turned, therefore, to the source iteration method, which can be discretized consistently and for which convergence results are available. On the basis of these methods we gave in Chapter 2 a description of a splitting technique, which is essentially an extension of the first collision source method.
These iterative methods, together with the aforementioned discretization techniques, provided an inspiration for the major part of our research.
In Chapter 3 we presented a discontinuous approximation in angle that allows for arbitrary partitions of the angular domain and arbitrary polynomial degrees on each element of that partition. As such, it can be understood as a generalization of the classical spherical harmonics approximation, where the angular domain is discretized by a single interval $(1,1)$ and polynomials of high degree, and the discrete ordinates method, where the angular domain is partitioned into several intervals with piecewise constant functions.
In particular, the approach described in Chapter 3 allows to account for the natural discontinuity of the solution at $\mu=0$. In Chapter 3 an $hp$discretization was applied to the evenparity formulation of the radiative transfer equation with isotropic scattering. Moreover, we developed and analysed an iterative solution technique that employs subspace correction as a preconditioner. Our approach was inspired by the DSA preconditioned source iteration method. It was shown that our iterative method exhibited convergence independent of the resolution of the computational mesh.
In Chapter 4 we then focused on an efficient iterative framework that is capable of accurately solving the system of linear equations that arises from the discretization of anisotropic radiative transfer problems. In case of forwardpeaked scattering the convergence of the standard DSApreconditioned source iteration method is slow, hence acceleration with the use of an appropriate preconditioning technique is necessary. In Chapter 4 we proposed a provably convergent iterative method, equipped with two preconditioners, one of which corresponds to the efficient approximate inversion of transport. The second preconditioner was used to improve the standard contraction rate $\\frac{\sigma_s}{\sigma_t}\_{\infty}$ in the source iteration method. The subspace correction is then constructed from low order spherical harmonics expansions  eigenfunctions corresponding to the largest eigenvalues of the anisotropic scattering operator. The method is shown to be efficient if the scattering operator is applied properly, for which we used $\mathcal{H}$ and $\mathcal{H}^2$matrix compression algorithms.
Finally, in Chapter 5 we considered a nontensor product discontinuous Galerkin discretization for the evenparity radiative transfer equations for the slab geometry. We proved stability and wellposedness for the symmetric interior penalty discontinuous Galerkin method. We also investigated the numerical convergence of the phasespace discontinuous Galerkin method. For piecewise smooth solutions the phasespace discontinuous Galerkin method with low order polynomials displays a linear rate of convergence. We show numerically that in case of nonsmooth solutions the use of adaptive mesh refinement allows for efficient approximation. The question of an appropriate choice of the error estimator remains an open question. Despite the similarity of the evenparity form of the RTE to standard elliptic problems, standard elliptic residualbased error estimators can not be generalized directly.
There are several problems, which are open for future research.
• Developing and analyzing proper aposteriori error estimators for our discontinuous Galerkin discretization of the radiative transfer equations in phasespace, allowing for $hp$adaptivity.
• Improving the error analysis of the preconditioned iterative schemes presented in Chapters $3$ and $4$, by proving precise quantitative rates of convergence.
• Developing multigrid methods as an alternative to the preconditioned source iteration method.
As an important introductory step, we gave an overview of classical semidiscretizations in the angular component. The two most frequently used discretizations  the discrete ordinates and spherical harmonics methods are summarised in Chapter 1. While the spherical harmonics discretization allows to turn the radiative transfer equation into a system of linear equations with tridiagonal structure, approximating the boundary conditions effectively requires extra steps. Furthermore, since the spherical harmonics expansion is a global approximation method, it is not suited for approximating nonsmooth or discontinuous solutions, unlike the discrete ordinates method, which is a local approximation in the angle. The discrete ordinates method allows to obtain a consistent discretization of both the radiative transfer equation and the boundary conditions, though yielding dense scattering matrix.
A number of iterative techniques has been developed to tackle the difficulty that arises from the dense scattering matrix. A summary of some important methods was given in Chapter 2. We discussed two closely related methods  the first collision source method and the standard source iteration method, which is often accompanied by further preconditioning techniques, such as the diffusion synthetic acceleration technique.
In the first collision source method the radiative transfer boundary value problem is split into two equations for the uncollided and collided components, which can be separately approximated by different numerical methods. In general this can, however, introduce consistency errors, which are difficult to analyse. We turned, therefore, to the source iteration method, which can be discretized consistently and for which convergence results are available. On the basis of these methods we gave in Chapter 2 a description of a splitting technique, which is essentially an extension of the first collision source method.
These iterative methods, together with the aforementioned discretization techniques, provided an inspiration for the major part of our research.
In Chapter 3 we presented a discontinuous approximation in angle that allows for arbitrary partitions of the angular domain and arbitrary polynomial degrees on each element of that partition. As such, it can be understood as a generalization of the classical spherical harmonics approximation, where the angular domain is discretized by a single interval $(1,1)$ and polynomials of high degree, and the discrete ordinates method, where the angular domain is partitioned into several intervals with piecewise constant functions.
In particular, the approach described in Chapter 3 allows to account for the natural discontinuity of the solution at $\mu=0$. In Chapter 3 an $hp$discretization was applied to the evenparity formulation of the radiative transfer equation with isotropic scattering. Moreover, we developed and analysed an iterative solution technique that employs subspace correction as a preconditioner. Our approach was inspired by the DSA preconditioned source iteration method. It was shown that our iterative method exhibited convergence independent of the resolution of the computational mesh.
In Chapter 4 we then focused on an efficient iterative framework that is capable of accurately solving the system of linear equations that arises from the discretization of anisotropic radiative transfer problems. In case of forwardpeaked scattering the convergence of the standard DSApreconditioned source iteration method is slow, hence acceleration with the use of an appropriate preconditioning technique is necessary. In Chapter 4 we proposed a provably convergent iterative method, equipped with two preconditioners, one of which corresponds to the efficient approximate inversion of transport. The second preconditioner was used to improve the standard contraction rate $\\frac{\sigma_s}{\sigma_t}\_{\infty}$ in the source iteration method. The subspace correction is then constructed from low order spherical harmonics expansions  eigenfunctions corresponding to the largest eigenvalues of the anisotropic scattering operator. The method is shown to be efficient if the scattering operator is applied properly, for which we used $\mathcal{H}$ and $\mathcal{H}^2$matrix compression algorithms.
Finally, in Chapter 5 we considered a nontensor product discontinuous Galerkin discretization for the evenparity radiative transfer equations for the slab geometry. We proved stability and wellposedness for the symmetric interior penalty discontinuous Galerkin method. We also investigated the numerical convergence of the phasespace discontinuous Galerkin method. For piecewise smooth solutions the phasespace discontinuous Galerkin method with low order polynomials displays a linear rate of convergence. We show numerically that in case of nonsmooth solutions the use of adaptive mesh refinement allows for efficient approximation. The question of an appropriate choice of the error estimator remains an open question. Despite the similarity of the evenparity form of the RTE to standard elliptic problems, standard elliptic residualbased error estimators can not be generalized directly.
There are several problems, which are open for future research.
• Developing and analyzing proper aposteriori error estimators for our discontinuous Galerkin discretization of the radiative transfer equations in phasespace, allowing for $hp$adaptivity.
• Improving the error analysis of the preconditioned iterative schemes presented in Chapters $3$ and $4$, by proving precise quantitative rates of convergence.
• Developing multigrid methods as an alternative to the preconditioned source iteration method.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  2 Jun 2022 
Place of Publication  Enschede 
Publisher  
Print ISBNs  9789036553889 
DOIs  
Publication status  Published  2 Jun 2022 