<P> In this thesis, we mainly study geodesics on various two dimensional surfaces. All the background material needed throughout the thesis is provided, including an explanation of the theory of geodesics. We will calculate geodesics using two numerical methods: Euler's method and Runge-Kutta method of fourth order. Using Maple, we will test the accuracy of the numerical methods on a test case surface, the Poincare half plane. Later, we proceed to investigate several interesting surfaces by numerically calculating geodesics. From the investigated surfaces, we will draw similarities between the human cerebral cortex and certain surfaces. The human cerebral cortex is the most intensely studied part of the brain and it is believe that their exists a relation between the function and structure of the cortex. Geodesic analysis can possibly be an essential tool in better understanding the cortical surface as it is in many disciplines of science to understand the nature of physical based problems. </P> / Thesis / Master of Science (MSc)

The main objective of this research work is to apply the discontinuous Galerkin method to a classical partial differential equation to investigate the properties of the numerical solution and compare the numerical solution to the analytical solution by using discontinuous Galerkin method. This scheme is applied to 1-D non-linear conservation equation (Burgers equation) in which the governing differential equation is simplified model of the inviscid Navier-stokes equations. In this work three cases are studied. They are sinusoidal wave profile, initial shock discontinuity and initial linear distribution. A grid and time step refinement is performed. Riemann fluxes at each element interfaces are calculated. This scheme is applied to forward differentiation method (Euler's method) and to second order Runge-kutta method of this work.