Abstract
This thesis introduces several innovative techniques to improve the numerical solutions for radiative transfer problems, focusing on adaptive methods, low-rank approximations, and accelerated iterative solvers.
The first contribution is the development of an adaptive discontinuous Galerkin method for solving the radiative transfer equation in slab geometry with isotropic scattering. Unlike traditional methods that use tensor-product discretizations, this approach employs a non-tensor strategy to allow local mesh refinement. Guided by error estimators, the method effectively handles phase-space singularities, offering optimal convergence for point singularities and consistent but sub-optimal convergence for line discontinuities. This results in significant computational efficiency while maintaining high accuracy.
Next, the thesis presents a low-rank tensor product framework aimed at reducing the computational costs and memory requirements of tensor-product discretizations. It combines preconditioned Richardson iteration with rank-compression, providing flexibility and error control in the energy norm, while supporting various discretizations.
Finally, the thesis introduces an acceleration technique for iterative solvers in multi-dimensional anisotropic radiative transfer problems. This method accelerates standard source iteration through residual minimization over appropriate subspaces, improving convergence, especially in scenarios with highly forward-peaked scattering. It offers a flexible and efficient solution, reducing the number of iterations needed for convergence.
Overall, these contributions represent a significant leap forward in the numerical treatment of radiative transfer, addressing challenges in adaptivity, computational efficiency, and solver performance.
The first contribution is the development of an adaptive discontinuous Galerkin method for solving the radiative transfer equation in slab geometry with isotropic scattering. Unlike traditional methods that use tensor-product discretizations, this approach employs a non-tensor strategy to allow local mesh refinement. Guided by error estimators, the method effectively handles phase-space singularities, offering optimal convergence for point singularities and consistent but sub-optimal convergence for line discontinuities. This results in significant computational efficiency while maintaining high accuracy.
Next, the thesis presents a low-rank tensor product framework aimed at reducing the computational costs and memory requirements of tensor-product discretizations. It combines preconditioned Richardson iteration with rank-compression, providing flexibility and error control in the energy norm, while supporting various discretizations.
Finally, the thesis introduces an acceleration technique for iterative solvers in multi-dimensional anisotropic radiative transfer problems. This method accelerates standard source iteration through residual minimization over appropriate subspaces, improving convergence, especially in scenarios with highly forward-peaked scattering. It offers a flexible and efficient solution, reducing the number of iterations needed for convergence.
Overall, these contributions represent a significant leap forward in the numerical treatment of radiative transfer, addressing challenges in adaptivity, computational efficiency, and solver performance.
| Original language | English |
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| Qualification | Doctor of Philosophy |
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| Supervisors/Advisors |
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| Award date | 22 Jan 2025 |
| Place of Publication | Enschede |
| Publisher | |
| Print ISBNs | 978-90-365-6413-7 |
| Electronic ISBNs | 978-90-365-6414-4 |
| DOIs | |
| Publication status | Published - Jan 2025 |