Sea breezes are low-level atmospheric flows that move from the sea to land. They typically form during warm days when the temperature difference between the land and sea surfaces becomes large enough. This temperature difference occurs because seawater, having a higher heat capacity, heats up more slowly than land. In this thesis, we study a non dimensional mathematical model that describes the horizontal velocity of sea breezes in two regions: the Gulf of Carpentaria (Australia) and Calgary (Canada). The model was derived in [4] by writing the Navier-Stokes equations in rotating spherical coordinates. After non dimensionalisation and assuming a specific form of the solution, they obtained the non dimensional model. Furthermore, the model was complemented by two homogeneous Dirichlet boundary conditions, representing the no-slip condition at Earth’s surface and the presence of a thermal inversion layer. In [11], their work was extended by studying the corresponding Sturm-Liouville problem for various mass density functions to determine the existence and uniqueness of the solutions to the model. Building on the work in [11], we analysed the solvability of the model and determined the numerical solutions for two physically relevant mass density functions. These functions were chosen to ensure that zero is not an eigenvalue. The Fredholm alternative then implies that the solution is unique. On the other hand, if zero were an eigenvalue, then the Fredholm alternative implies that we could have an infinite number of solutions, which is physically unrealistic. For the first mass density function, the eigenvalue problem was written as the hyper geometric equation. For the Gulf of Carpentaria (where β > 0), we proved analytically that zero is not an eigenvalue. However, since the parameter β is negative for the Calgary region, we were unable to prove this analytically. Instead, we demonstrated numerically that the eigenvalues closest to zero were λ807 = −828 and λ808 = 790, confirming that zero is not an eigenvalue. Using the finite difference method, we then computed the velocity profiles for three different forcing functions. We first considered the breezes in the Gulf of Carpentaria. In the first case we assumed zero forcing, the result showed that in the absence of any forcing, sea breezes do not occur. For the second case we assumed a non-zero constant forcing, resulting in a uniform velocity profile. Lastly, with an increasing quadratic forcing function, the velocity increased quadratically with height, reaching its maximum just below the thermal inversion layer. Similarly, we computed the velocity profiles for the Calgary region. From these computations, we found that the breezes were flowing in the opposite direction and with higher velocities. These differences arise due to the Coriolis force, which forms due to Earth’s rotation. This force deflects winds to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. Furthermore, the Coriolis force is zero at the equator and grows with latitude, which explains the higher velocities. Finally, we considered the model for the second mass density function, which includes the weak effect the boundaries have on the flow. For the Gulf of Carpentaria it was again proven analytically that zero was not an eigenvalue, while for the Calgary region, the eigenvalues closest to zero were λ501 ≈ −351 and λ502 ≈ 3030. Similar to the first mass density function, we used the finite difference method to compute the velocity profiles. The results showed that the profileswere identical to those obtained with the first mass density function. This similarity occurs because the two mass density function exhibit similar parabolic behaviour. Moreover, the same forcing functions were used in both cases.

The field of Computational Fluid Dynamics (CFD) is constantly finding new ways to improve simulation results. One of the large challenges in CFD, is advection. Advection dominated flows are governed by hyperbolic equations, which pose severe stability issues. Stabilized methods usually add significant numerical diffusion, which alters the solution. Another issue are discontinuities in solutions, such as found in shock waves and acoustics. Besides solving these issues, conservative properties have to be satisfied at all costs. In order to further improve results, new methods are still developed today. The mathematical tools to do such developments are also further expanded. For instance, in the last decades the branch of differential geometry has steadily grown. This branch of mathematics can be extremely useful, since a clear distinction is made between geometric depend and geometric independent operators. These operators become more in use for numerical methods. Of all the available operators, the Lie derivative is important for advection. In junction with newly developed edge polynomials, a scheme for advection is created. The new edge polynomials are special, as these partly scale with a variable integrated over the cell and partly scale with the values on the cell boundaries. In order to do this, one polynomials integrates to 1 over the cell, while the other polynomials do not contribute, hence integrate to 0 over a cell. This gives several benefits. Conservation is clearly defined in the cells and not in points, hence conservation is locally defined. Also an extinction is made between conservative values and fluxes. One can also use the property of not contributing to conservation by correcting point values for stabilization or improvement of the solution. These polynomials are used within a Galerkin Finite Elements (FE) framework. Besides correction methods, upwind methods can be used as well. The approach is tested on several equation sets. On a simple linear advection problem, excellent results are found. Four test cases are tried, from a sine wave, two discontinuous functions and a hat function. With the proposed polynomials, the shapes are conserved extremely well, also in comparison to widely used schemes. On the inviscid Burgers’ equations, the stabilization methods become of importance. In the inviscid Burgers’ equation, initially smooth problems can become discontinuous. When a discontinuity forms, several issues can occur. For one, spurious oscillations can become apparent, or the velocity of the discontinuity can have errors. With several approaches, competitive results can be obtained. The discontinuity velocity is computed well, but spurious oscillation usually do occur. With several methods, the oscillations are suppressed. Not all approaches succeed in doing so. The third and final equation set to be tested, are the Euler equations. Although solutions can be found, all solutions contain errors. Several error sources are indicated, which should be addressed in further research. In comparison to a Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) scheme, results need to be further improved to be on par. In conclusion, the suggested polynomials show potential. Several methods to improve solutions showed their usefulness. In linear advection, the new polynomials performed better compared to existing widely used methods. In non-linear advection, the results are on par with reference methods. For the Euler equations, further improvements are necessary. With further