Muir, Dean2024-12-052024-12-05https://hdl.handle.net/1885/733728434In this thesis, we derive a provably stable numerical method for solving the anisotropic diffusion equation in magnetic fields. Specifically, we are interested in an approach suitable for problems in magnetic confinement fusion scenarios, where the equation is often used to model the temperature; in these settings, fast scale diffusion along the magnetic field lines is orders of magnitude higher than the diffusion across them. Confinement fusion devices require temperatures of millions of degrees centigrade to operate efficiently. Modelling temperature is therefore crucial, in order for devices to be designed such that they are not damaged by direct interaction with the plasma. Chaotic magnetic fields make it difficult to build a grid which conforms to them; therefore numerical techniques are required that allow for non-conforming grids to still accurately solve the equation. The equation can be written in a form where the fast scale parallel and slow scale perpendicular components are isolated, so efficient solvers can be applied to the two terms. For the slow scale diffusion, the \ac{sbp} finite difference operators, which mimic integration by parts discretely, are used to discretise the perpendicular diffusion, and boundary conditions are enforced weakly using so-called \ac{sat}. We introduce a unique penalty approach for enforcing the fast scale diffusion along magnetic field lines, which is constructed by field line tracing. This penalises the solution to the equation based on the misalignment of the grid with the magnetic field. Our approach to constructing the method is systematic. First, we explore the method using a model one dimensional problem, and then we introduce the two dimensional problem in Cartesian geometry. Finally, we extend the two dimensional method to curvilinear geometry, which will allow us to extend the methods to domains more relevant to confinement fusion. At each stage we prove the stability of our semi-discrete scheme by matching the energy estimates for the continuous problem. We also show unconditional stability for the fully discrete scheme. Numerical tests are presented which validate the convergence of the scheme in each case; experiments show the method has promise for the more complicated geometries involved in confinement fusion devices. We conclude by outlining the extensions of the method derived in this thesis, including the extension to geometries relevant to confinement fusion and other potential future work.en-AU202410.25911/92HX-JN74