Numerical models are paramount in describing the complex physical world around us. They are often based on non-linear functions, which have to be solved in order to run a simulation. The first goal of this thesis is to compare the Jacobian-free Newton-Krylov method against the regular Newton-Raphson method for solving non-linear equations. When doing this, preconditioning was only used for the Newton-Raphson method. As a second goal, the optimal parameters for solving a coupled system with the Schur complement method are determined. For computational purposes, these analyses are performed using PETSc. The comparison between JFNK and Newton-Raphson were performed by solving a heat equation. The physical system which was analyzed is a one-dimensional radiating rod, with a heterogeneous thermal conductivity and Dirichlet boundary conditions. Firstly this rod was modelled as one single system. For this case, it was found that Newton-Raphson outperformed the JFNK method by a factor of 5-1700, depending on the type of preconditioning used. Secondly, the rod was modelled as a coupled system by solving two parts of the rod separately. In this case, it was found that the JFNK method outperformed the preconditioned Newton-Raphson method by a factor of 9.1± 0.3. It was concluded that the JFNK is only favorable over regular Newton-Raphson for coupled systems. To achieve the second goal, an incompressible Navier-Stokes coupled system was solved using the Schur complement method. The coupled system originated from incompressible flow in a back-step pipe, using a finite element method. For the Schur complement parameters, it was shown that using an approximation of the Schur complement offered a significant increase in efficiency. The momentum and pressure subsystems were analyzed separately. The momentum subsystem performed best with a relative tolerance of τ_r =10^-4.5, while the pressure subsystem performed best with a tolerance of 10^-2. With these parameters, the Schur complement method had the same computational time as the pressure-correction method, which was used as a benchmark. For future research, there are three main recommendations. Firstly, to use preconditioning for the JFNK method. This was not done in this thesis because of technical constraints, but it is theoretically possible. Secondly, if the JFNK method is preconditioned, it could be used for the incompressible Navier-Stokes simulation. Finally, there are combinations of parameters which were not tested for the Schur complement method. Trying out more combinations might result in finding an even more optimized method.

The freeze plug is a key safety component of the molten salt fast reactor (MSFR), one of six next-generation nuclear reactor technologies being developed under the Generation IV International Forum (GIF). It should be designed to melt if an accident occurs, allowing the MSFR to drain before it incurs structural damage.
Two freeze plug concepts have been considered in recent years, in which the plug is melted either through the decay heat produced in the core, or through heat generated by special heating rings and stored in steel blocks adjacent to the freeze plug. Variations consisting of both a single freeze plug, and multiple smaller plugs contained in a metal plate, have been proposed. This work seeks to evaluate the feasibility of these designs and study how parameters such as the sub-cooling of the plug affect melting times. Additionally, an alternative, wedge-shaped freeze plug design is proposed for increased reliability. 
Simulations performed in COMSOL showed that the decay heat plug melts within 600 s only if placed within 0.01 m of the mixed core flow. Because such a placement makes the plug vulnerable to temperature and velocity fluctuations in the core during regular operation of the reactor, this design is considered unfeasible and is not recommended for further study. On the other hand, melting times under 600 s were possible with the heating ring design for a range of sub-cooling amounts and plug configurations, suggesting that this
design is promising. A thin frozen layer was shown to form on top of the metal grate in the multi-plug configurations, preventing heat transfer through the top of the plate. Although the melting behavior of this layer warrants further investigation, its insulating effect was found to generally cause the single-plug designs to melt faster than the multi-plug designs.

A simplified, isothermal model of the wedge-shaped plug was simulated using the enthalpy-porosity approach to account for convection. To model the sinking of the wedge, an extended Darcy term approach was developed based on an analytical solution which was validated experimentally, with good agreement. This model shows that melting of the wedge is unsteady, and that melting times depend linearly on the wedge angle and sub-cooling. Unfortunately, melting times of the wedge plug could not be estimated with realistic, non-isothermal, time-dependent boundary conditions. For future study, a customizable numerical solver such as OpenFOAM is recommended, which would allow the sinking of the solid phase to be modeled more robustly through an immersed boundary method.